I recognize this is off-topic for my blog, but I love math. And I also discovered that Greek grammar has some features of math properties or laws, so it’s only mostly off-topic.
I know this is a Greek language blog, but math uses Greek characters sometimes, so there’s at least a tenuous connection. And all truth is God’s truth, even in Mathematics, for which God created the principles, properties, and laws. Being a former high school math teacher myself, I was disappointed at the lack of knowledge of fundamental laws and properties in math that led to many people thinking Expression 1 did not equal 1 in certain viral social media threads. In an effort to restore some truth to people’s mathematical knowledge, I present the following proof that the answer to Expression 1 is ALWAYS AND FOREVER 1.
Expression 1
8 ÷ 2(2 + 2)
Conventions vs. Laws
Many people in the social media chains trying to tackle this problem were claiming that following the order of operations they learned in and impressively remembered from grade school was the correct way to approach the problem:
Parentheses
Exponents
Multiplication/Division LTR
Addition/Subtraction LTR
This order is commonly known in American math as PEMDAS and recalled by the sentence “Please Excuse My Dear Aunt Sally.” I do not deny the importance of PEMDAS, but the reality of the problem is, any basic math problem like this can only have one correct answer. It’s not and never a matter of personal interpretation. Otherwise, the foundations of mathematics would crumble into oblivion, and not even Common Core could save us (not that it ever did anyone any good). This is math; it doesn’t care about and is never affected by your feelings about it.
PEMDAS is only a tool for organizing the operations in the problem, but by itself, it is not sufficient to solve the problem correctly. In fact, PEMDAS isn’t a mathematical law at all. It is merely an agreed-upon convention to work “linear” math problems. Math does have many laws or properties that come into play and must be considered in the PEMDAS process, and PEMDAS is subservient to these laws. Nothing about the correct solution I’m about to show you violates PEMDAS, provided you correctly interpret the forms of the individual expressions within the larger expression and how the various laws and principles apply.
If you think back to your primary school math lessons, you may have a vague memory of a set of laws[1] about the relationships of numbers in certain types of expressions. For example, the Associative Properties of addition and multiplication say that no matter how you group the numbers in their respective equations, the sum (addition) or product (multiplication) will always be the same. The Commutative Properties for these two operations are similar; the order of the order or arrangement of the elements in an expression does not affect the value of either expression. These Associative Properties are represented in Expressions 2 and 3, while the Commutative Properties are in 4 and 5:
Expression 2: Associative Property of Addition:
(a + b) + c = a + (b + c)
Expression 3: Associative Property of Multiplication:
(a * b) * c = a * (b * c)
Expression 4: Commutative Property of Addition
a + b = b + a
Expression 5: Commutative Property of Multiplication.
ab = ba
The expressions on either side of the equal sign in the respective equations above reveal another principle of math, that of identical expressions. They look different, but regardless of the values assigned to each variable, they will always be equal. This is also called an identity.[2]
The other important thing to know is that PEMDAS, unlike the Associative and Commutative Properties, is not a law! It is merely a convention for solving a problem that is subject to these laws. PEMDAS does NOT usurp these laws. This is where people are getting tripped up on solving Expression 1 or similar expressions for that matter. I will demonstrate how the correct application of these laws within the framework of PEMDAS will ALWAYS yield the answer of 1, NOT 16 or some other number.
Solving the Expression
One other law must be brought to the fore to solve this expression: the Distributive Property. This is slightly different from the other four laws, in that it involves both addition and multiplication, and it establishes a common equation form that must be worked the same way every time it is found within an equation. Wolfram Research is considered one of the premier math knowledge platforms in the world, so I will draw on their examples of the Distributive Property to make my point. If anyone wants to challenge me on my conclusions drawn from this source, you’ll have to do better than a cheesy homework help Web site. The Wolfram Web sites have two different ways of writing the Distributive formula. BOTH equations are identical expressions and should be solved the same way every time regardless of where they fall in an equation.
Expression 6: Distributive with intervening multiplication operator
a * (b + c) = ab + ac (and of course, if you’re using all real numbers, combine like terms).[3],[4]
Expression 7: Distributive without intervening multiplication operator
a(b + c) =ab + ac (and of course, if you’re using all real numbers, combine like terms).[5]
Whether the expression has the multiplication operator or not, you would treat both as an expression to be solved BEFORE leaving the P step in PEMDAS. The actions UPON the parenthetical result must be completed BEFORE leaving the P step.
For purposes of demonstration later on, we can also apply the Commutative Property of Multiplication to the Distributive property form. We have two “factors” (the a and the (b + c)), so we can rearrange them and still have the same result. In the case of the current form, if we put the a term to the left as written in Expression 7 above, this form of the expression is said to be left distributive (i.e., the a multiplies through from left to right). If the a term is to the right of the parentheses, then the form is called right distributive.[6] See Expression 8 below. The right distributive form of the expression is an identical expression to the left distributive form. I will use this to demonstrate that PEMDAS is not consistent if you don’t first solve the expression in distributive property form.
Expression 8
(b + c)a = ba + ca (and combine like terms if using all real numbers).
Are you with me so far? Maybe you see where I’m going with this? The expression to the right of the division sign must be processed as and simplified to an individual, inseparable term, because it is in the form of a Distributive Property expression. It has parentheses after all, so it must be dealt with before being divided into 8. So here’s the explanation of solving the equation as written:
This then leaves you with the final expression (Expression 10) to be solved:
Expression 10
8 ÷ (8) = 1
QED
Why the Answer Is NEVER 16 or Any Other Number
I am going to offer several proofs or citations that demonstrate why PEMDAS is not sufficient by itself to solve this problem. The first citation comes from a 1935 textbook for advanced algebra.7 Here is what the authors say:
“If the multiplication of two or more numbers is indicated, as in 4m or 5a2, without any symbol of multiplication, it is customary to think of the multiplication as already performed.
Thus 4m2 ÷ 2m = 4m2/2m, not (4m2/2)m.”
This equation (original to the authors’ text) has the same basic form of Expression 1, with the only difference being all real numbers are used in Expression 1. I’m guessing that all of you agree that the expression to the immediate right of the equal sign in the example above is the correct way to interpret the expression on the Left. And of course, the expression on the right simplifies down to simply 2m. The other form, which you get if you do strict PEMDAS without any other consideration, simplifies to 2m3. You all know the 2m is correct, right? That’s the way we all learned how to process variables with coefficients. So if m = 2, we should expect an answer of 4, not 16. 4(4) ÷ 2(2) = 16 ÷ 4 = 4. If you do it the strict PEMDAS only way, then you get 4(4)/2 * 2 = 16/2 * 2 = 16. Wrong answer, therefore, the wrong method to solve.
[Additional notes and evidence added 08/05/2023 (italicized).]
From Wolfram MathWorld, the following article on Precedence, which you get to from the WolframAlpha description of “Order of Operations,” discusses the concept of “advanced operations” that “bind more tightly.” It then contrasts that with “simple operations.” Although Wolfram does not detail what those two concepts embrace, multiplication by juxtaposition (or “implied multiplication”) would be one of those advanced operations that “bind more tightly” and thus have precedence (what is an exponent or factorial if not implied multiplication?), while a “simple operation” would include any expression with all operational signs extant and leave nothing to be implied.
In Wolfram MathWorld’s discussion of the “solidus,” they confirm that most textbooks teach that an equation in the form of a ÷ bc should be interpreted as a ÷ (bc) and NOT ac ÷ b, and then acknowledge that most computational languages, including Wolfram, treat the expression without parentheses in a strictly linear manner without acknowledging the implied multiplication. They say that in order to solve the problem the way the textbooks teach it, or the way most people who actually use real-world equations where it makes a difference, you MUST enter the parentheses in most computational languages (that is the context of MathWorld), including in most calculators. Here’s the definition:
This is solid evidence that the answer to Expression 1 should be 1 and not 16. I’m not blowing smoke, and I’m not trolling. Those who get the answer 16 should rethink their logic.
[Here ends the addition of new material on 08/05/23]
Let’s make the expression in question look a little more like the example I just gave, and remember, that is from an advanced algebra book written by a couple math professors from Columbia U and the U of Southern California. For the expression in question, let m = (2 + 2) and substitute it into the expression, giving us Expression 11.
Expression 11
8 ÷ 2m
You Sixteeners should see right away the error of your PEMDAS-only ways. We don’t break the coefficient away from the variable, so we wouldn’t break it away from what we substitute into the variable. It works both ways. Just like the problem from the textbook, we can clearly see that the answer to “simplified” expression is not 4m, but 4/m. Since we let m = (2 + 2), 4/(2 + 2) = 4/4 = 1. QED.
If that historical example isn’t enough to convince you that PEMDAS alone isn’t correct, consider the following based on my discussion of right- and left-distributive above. As the original equation is written, I’ve already thoroughly demonstrated that the part of the main expression right of the division sign must be treated as an inseparable expression. But for the sake of argument, let’s consider the contention that PEMDAS alone applies without calling on the Distributive Property. As many Sixteeners have demonstrated, this works out to (8/2) * (2 + 2), or 16. However, if we substitute the right-distributive form of the expression in question for the left distributive form, we get Expression 12. Remember, whether right or left, the two expressions are considered equal, or identical.
Expression 12
8 ÷ (2 + 2)2
Expression 12 is, by definition, identical to Expression 1, so we should expect the same answer, right? However, if you apply the PEMDAS-only method on this form of the equation, you get (8/4) * 2, or 4. This PROVES that PEMDAS alone is not sufficient to solve the whole expression, because you get different answers for identical expressions! That is logically impossible in a first-order math equation with real numbers. NOTE: Because I demonstrated that the expressions are themselves equal or identical before solving them, you can’t turn around and say they’re not identical because they get different answers with PEMDAS-only. Distributive property is a law; PEMDAS is a convention. Law trumps convention.
QED
A Quick Note About Your Calculators
Most basic calculators don’t typically recognize the Distributive Property from what I’ve seen. In fact, if you read the manuals of most scientific calculators, you’ll find them admitting that you may need to use parentheses to force it to act according to the laws of mathematics in some instances. (See definition of Solidus above.) So don’t trust your calculators. In fact, I’m willing to bet whoever submitted that problem in the first place most likely knew that about calculators and is rolling on the floor laughing their butts off that many were fooled by the calculator, thus resulting in the social media melee over the problem.
A Comparison to Greek
Since this is primarily a blog about interpreting the Greek New Testament (and occasionally the Hebrew/Aramaic Old Testament), I couldn’t help but notice that the Commutative and Distributive Properties apply to Greek adjectives and nouns. In Greek, if the definite article is with the adjective and the adjective modifies a noun, then it doesn’t matter which comes first. The phrases are still translated the same way (article+adjective+noun = noun+article+adjective). The same goes for the noun. If the article is with the noun, then the noun is the subject of the phrase and the adjective is the predicate: (article+noun+adjective = adjective+article+noun). The grammatical case, number, and gender of the noun (subject, object, possessive, etc.) distribute through the article and any adjectives associated with it. Who knew solving a math problem would lead me to discover that Greek grammar has some mathematical logic to it!
My opinions are my own, but my well-reasoned conclusions are indisputable!
Scott
Addendum (added 5/14/23)
PEMDAS is shortsighted. It ignores mathematical properties (that is, laws), which take precedence over order of operations like PEMDAS (an accepted convention, not a law or property). I would suggest the “P” in PEMDAS should not only stand for Properties first, but secondarily for Parentheses. Properties are to PEMDAS what the U.S. Code is to subregulatory guidance in legislative speak. Subregulatory guidance has no authority without the force of law behind it. PEMDAS has no power without the force of mathematical properties behind it.
If you use strictly PEMDAS without recognizing the Distributive Property, you wind up with the following, as the sixteeners are interpreting this:
8 ÷ 2(2+2)
Becomes
8 ÷ 2(4)
Becomes
8 ÷ 2 * 4
But let’s not stop there. This becomes, using the multiplicative inverse to change from division to multiplication:
(8 * 1/2) * 4
And applying the Associative property of multiplication, this becomes
8 * (1/2 * 4)
Substituting back in the original parenthetical expression
8 * (1/2 * (2+2))
Look at what happens to the last half of the equation: You’re now dividing (2+2) by 2 when the syntax of the original expression clearly indicates they should be multiplied together using the Distributive Property. The sixteeners have fundamentally changed the syntax of the original expression, which is a violation of the whole process of solving the equation. That’s why 16 is incorrect!
You can’t simply dismiss the Distributive Property here. A property is a law of math that essentially requires no proof. It is superior to and has precedence over PEMDAS, which is not a property at all and, to my knowledge, has never been proven to account for all things could be going on in an expression. That’s why Property should come before parentheses in PEMDAS.
Why You Can’t Trust Graphing Calculators
Look at how Wolfram Alpha handles the basic form of the expression in question as we gradually add information. All images grabbed from Wolfram Alpha on 5/14/23 around 10:30 pm CST). NOTE: I have an acknowledged request to Wolfram Alpha to investigate why the following is happening.
Now provide the values for x, a, b.
Substitute in 8 for x.
Add parentheses with the substitution; same result; solution is 1.
Then put in the 2+2 directly. Solution again is 1. Note that it’s NOT (8/a) * (2+2)
But when I directly enter a value for a, the logic changes.
And the expression as written behaves similarly.
So clearly PEMDAS by itself is not sufficient to process the problem, because you can see that even some of the best graphing calculators don’t process the basic form of the equation consistently. Arguments from your graphing calculator don’t cut it for me. Those are just AI. This is a real demonstration of faulty logic in certain formats. You can’t have two different answers to the same expression. The answer 1 is right; 16 is wrong.
Still Not Convinced? (Added weekend of 5/20/23)
What happens to the way you solve the problem if you change the 2 to a negative sign?
8 ÷ -(2 + 2)
Because the negative sign implies that what is inside the parentheses is multiplied by-1, would you PEMDAS-only proponents then have (8 ÷ -1) x (2 + 2) = -32? ABSOLUTELY NOT! Even the Sixteeners have to admit that the problem should be read as 8 ÷ -4 = -2, thus proving my point that what is outside a parenthetical expression (in this case, -1) and multiplied by implication must be solved first to fully deal with the parentheses.
Or how about if the problem is 6 ÷ 3!? The expression 3! represents an implied multiplication relationship, just as 2(2+2) is. So if it’s implied multiplication, is it then deconstructed to 3*2*1 or 1*2*3? Do you see the problem if you deconstruct it like that? Which order?!? According to the logic of the Sixteeners, it should be. But of course that’s silly. You wouldn’t break up the factorial, just as you shouldn’t break up the two factors of 2(2+2) and make one a divisor.
Making It Real
Here are a few examples of how applying PEMDAS-only to real-world formulas could potentially be disastrous. I posted the following examples on a Facebook post dedicated to one of the viral equations and got no end of criticism for proving PEMDAS wasn’t relevant to solving the problems, because the values now had units of measurement applied, which automatically groups the expressions without the need to resort to extra parentheses or brackets. (I cleaned this up a bit because of the limitations of responding on an iPhone that doesn’t have ready access to the obelus symbol that I’ve found.)
What is the context of the equation? Is it a velocity formula, where v= d/t with d = 36 miles and t = 6(2 + 2 + 2) hours = 36 hours? Then v = 1 mile/hour. Going 36 miles in 36 hours does NOT yield a velocity of 36 miles/hour!
Is it a density formula, where D = m/V with m = 36 kg and V = 6(2 + 2 + 2) cubic cm = 36 cubic cm? Then the answer is 1 kg/cubic cm. Who says the obelus doesn’t have grouping powers!
Johnny, Freddy, Rita, Ginger, Gary, and Nancy each have two apples, two oranges, and two bananas to share with their classmates. The class has 36 people including themselves and the teacher. How many pieces of fruit may each person have? 36 classmates ÷ 6(2 + 2 + 2) classmates*pieces of fruit/classmate) = 36 classmates / 36 pieces of fruit = 1 classmate for every 1 piece of fruit. It’s not 36 pieces of fruit for each classmate!
The resulting equations are PEMDAS-naive, proving that PEMDAS is not always necessary to solve these types of expressions.
[1] Reitz, H. L. & Crathorne, A. R. College Algebra, Third Ed. New York: Henry Holt and Co., 1929, pp. 4–5.
[3] If the parenthetical part of the expression has an exponent, you would follow PEMDAS process the exponent before distributing the a through the result (e.g., a(b + c)2 = a(b2 + 2bc + c2) à ab2 + 2abc + ac2.
[4] See the definition at Distributive — from the MathWorld Classroom (wolfram.com) (accessed 04/27/23), where Wolfram also indicates the concept is part of 5th grade math standards in California. The fact that it is a 5th grade standard may explain why the multiplication sign is used.
[5] See a more detailed description at distributive – Wolfram|Alpha (wolframalpha.com) (accessed 04/28/23). This more detailed description includes both expressions, with and without the multiplication sign.
Oh, yeah. I can get pretty nerdy sometimes. I wrote a 25-page paper about the variant readings in Acts 2:42 involve whether there should be a comma, or the word “and,” or nothing.
flour for 3 ppl when 2 ppl needed 6 cups of flour… So i need 5 less cups for more ppl?
The confusion with the equation comes in at the 6÷2(3) part… Universally you have to perform the operations in the parenthesis first…
But after that whether or not you perform the distributive property first depends on what is being asked by the creator of the question…
Since 6÷2(3) is literally the same as 6÷2*3. If I’m asking how many cups of flour do i need to add for 3 ppl instead of 2… Then I’m looking for you to use PEMDAS and go left to right when you see any numbers outside parenthesis
But if i want to know how many more servings 3 cups of flour will give me… Then i would use the distributive property… Which would tell me that 3 cups of flour will add 1 more serving…
In that case 6÷2(3) gives me 1
The problem with the equation is it’s poorly written. It’s unambiguous. It assumes you know what the author is asking. But we don’t. The correct answer all depends on what is being asked by the equation.
I’m not following how you’re assigning the units to each form of the expression. Would you mind mapping that out for me each way? It seems you’re confusing units.
I think I see what you’re saying here, but you don’t need to make the equation so complicated in this case. If 6 cups of flour are necessary for every 2 persons, then you have 6 cups/2 persons, or 3 cups/person. Then if you want to know how many cups of flour for three people, just multiply 3 cups/person by 3 persons and get 9 cups. By defining the units that way, you are correct in treating the 6/2 separately, and as such, that should be in parentheses since that is the ratio you need to determine how much flour for however many people, which is my point. If you use units to define the values in the expression, then it makes sense to organize the expression the way it fits the units.
I agree the question is ambiguous IF you just focus on operations and NOT on the linguistic clues in the syntax of the expression. I’m keen to the linguistics of the expression, partly because of my historical experience with similarly formatted expressions and partly because I can see the connection between the construction of implicit multiplication and the implicit multiplication of exponents. What I and others are saying (Stephen Worlfram is one who I think would agree with me) is that there’s more to PEMDAS than just the operations. There’s another level that must be discerned, and that we must teach people to discern, if there’s ever going to be any agreement. Still, the best way to avoid ambiguity is to group accordingly using parentheses.
Then there’s the whole issue of established formulas, as in the following example:
Jerry’s Custom Jars has an intelligent robotic machine that can apply a label or ribbon of any given height around the circumference of a round mason jar and cut it exactly so it fits on the jar with no gaps or overlaps. They have an order for 35 mason jars with a simple yellow ribbon completely around the middle of each jar with no gaps or overlaps for a local VFW dinner honoring veterans. The jars will be centerpieces at the tables. How many jars can Jerry completely wrap with 500″ of ribbon for jars with a 2″ radius in the middle?
I would write the equation as
500″ ÷ 2π(2)”/jar,
because I have an instinctive, historical understanding that 2πr is the circumference of a circle, so why would I need to bother putting parentheses around that, given that I know it represents a single unit or distinct formula? But because some people focus solely on the operation and not the syntax or linguistics of the formula or aren’t smart enough to recognize the circumference formula, I’m expected to apply parentheses. I would to make it unambiguous to those with a limited understanding of PEMDAS, but I wouldn’t need it myself.
So the true answer to the question is 39 jars with some ribbon left over. If someone works that from a perspective of operations ONLY and doesn’t discern the circumference formula, they’d get an answer of 1570 jars, which is clearly absurd given the parameters of the expression. Those of us who see it differently are unwittingly biased against the long history we have of seeing the deeper principles behind PEMDAS. Thank you for reading.
You are truly an idiot. PEMDAS Is and has always been sufficient. The answer is and always has been 16. You probably failed a test in college and it pissed you off so bad you thought you would get on the Internet and lie to people and convince them that you’re not an idiot….. but you are. DUMBASS
Comment by Jason — June 27, 2023 @ 2:11 pm
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Jason, thank you for your feedback. I decided to approve your unprofessional and vile comment to showcase that many people like you are more emotional about PEMDAS than intellectual. So thank you for modeling that and making your position look extremely weak juxtaposed to my intellectual analysis of the issue. If you or anyone else can’t behave yourselves more professionally and politely in future comments, I’ll think twice about approving them. FYI, I was a straight-A student in math throughout junior high and high school, and I passed my two semesters of college calc and one semester of differential equations before switching majors. So I know math, and I know how I solved problems. If you want to offer a more constructive analysis of my position like Frank did below, I’d be happy to respond to that. Otherwise, I’ll let your comment serve as a warning of how not to behave in my blog. Peace to you.
Expressions 1-8 are all fine. You start losing it on expression 9 though. While 2(2+2) is in fact 8, that is not the equation. In expression 7 you have a(b+c) = ab + bc which is true, but the a in this equation is 8/2, not 2. So you must multiply 8/2 by (2+2), giving an equivalent of 8/2*2 + 8/2*2, which is 8 + 8, which is indeed 16. There is nothing implied or anything about the denominator 2 and the sum (2+2). 2(2+2) is just a normal multiplication 2 times (2+2), so the term to multiply using the distributive property is 8/2. The only way to group the 2 to the 2+2 would be to use parenthesis, 8/(2(2+2)), this would equal 1.
Expression 11. 8/2m can be interpreted as either (8/2)m or 8/(2m). How can we know which it is? There has to be a way to distinguish ambiguities. Enter PEMDAS! We have an order of operations that we use to distinguish in which order the operations occur. In this case it is division and multiplication, left to right. So, 8/2m is 8 divided by 2, multiplied by m, or 4m not 4/m.
Expression 12. Your equation, 8/(2+2)2, is not equivalent to 8/2(2+2). You cannot just move a denominator. Dividing by 2 is equivalent to multiplying by 1/2 or 0.5. Rewriting this equation would be 8 * 0.5(2+2), then you can move it, 8*(2+2)0.5. This reduces to (8*2 + 8*2)0.5, or (16 + 16)0.5, or (32)0.5, or 16.
Addendum (added 5/14/23). I’m not sure what you’re trying to say here. 8*(1/2*(2+2)) is indeed 16, you can either add first, 8(1/2(4)), or 8(2), or 16, or distribute first, 8((1/2*2)+(1/2*2)), or 8(1+1), or 8(2), or 16. Those both check out to 16. I’m not sure what you’re doing to divide (2+2) by 2?
Still Not Convinced? (Added weekend of 5/20/23). I still don’t know what you’re trying to say here. To get -1 you need to multiply the 1/2 by -2, 1/2*-2 = -1. So you must then multiply the other side also by -2. So 16 * -2 is indeed equal to -32. To go further with this point, you got an answer of -2. -2 * -2 is 4, not the 1 you claim.
Making It Real. Using your first example, v= d/t with d = 36 miles and t = 6(2 + 2 + 2) hours. Here, t is the quantity of 6(2+2+2), so that has to be in parenthesis. So, the equation is not 36/6(2+2+2), the equation is, 36/(6(2+2+2)), which is equal to 1, not 36. The same applies to your other equations here as well.
So in the end, 8/2(2+2) is indeed equal to 16, not 1.
Comment by Frank — June 27, 2023 @ 2:08 pm
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Thank you for your response, Frank. Your tone represents the kind of dialogue I would expect on this topic, unlike Jason’s vitriol above. I will answer your response starting from the top.
You claim the obelus only refers to the number immediately after it, and you use the obelus to turn 8 ÷ 2 into an improper fractional coefficient of (2 + 2). Yet the obelus, and the solidus (/) for that matter, in printed math both typically represent a linear replacement for the vinculum (fraction bar), and thus both would take on the “grouping” function of the vinculum, that is, everything that comes after the obelus, especially since it is the sole extant symbol in the expression, goes “after” (i.e., under) the vinculum. As such, there is NO logical or technical explanation of why the obelus would be treated differently by only treating the first number after it as the divisor. The very form of the obelus itself demonstrates that its horizontal line represents the vinculum, while the dot above represents the dividend before it, and the dot below represents the divisor that comes after it.
As such, by kidnapping the 2 from its partner (2 + 2), you’re changing the fundamental nature of the relationship between the implicitly multiplied elements. You’re forcing a divorce and pitting one against the other in an improper fraction (8/2). Even the term “improper” fraction implies that something untoward is going on with your methodology. What you’re doing to the expression is immoral and deceptive, if unwittingly so.
Note that here, I am using the terms “dividend” and “divisor” intentionally for the obelus. The expression is not yet in fractional form at this point, so it is technically wrong to refer to the elements as “numerator” and “denominator.” And on a side note, the entire expression is one “term,” because the addition sign is within parentheses, and what’s in the parentheses is subject to multiplication. You can’t break out just what comes after the addition sign as a separate term. So the argument that some make that there’s more than one term here is disproven.
Now the other issue at play here is the “sacred” connection of implied multiplication, especially in print form. WolframAlpha acknowledges this in its definition of the “solidus”:
“Whereas in many textbooks, “a/bc” is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages (sic) such as the Wolfram Language, it means (a/b) × c.”
So Wolfram Alpha supports the fact that many of us learned from textbooks (especially before calculators were the norm) that the answer to the expression is 1, that the product after the obelus is the divisor as Lennes wrote in 1917. However, when the calculator and computer languages came along, they were not programmed with the intuition and linguistic instinct that many of us learned in print (there are exceptions among and within the brands). When people started plugging these expressions into such devices, the devices had no regard for the sacred connection represented by implicit multiplication. The devices treated the ENTIRE expression as a linear equation without regard to parentheses, unless they were intentionally added. In contrast, our (those of us who get the answer 1) intuition tells us the parentheses are implied around the WHOLE product after the obelus. As a result, some of you chose to allow the calculator or computer to do your thinking for you, and you lost or never developed the instinctive connection represented by implicit multiplication. As such, the literal PEMDAS you preach is inferior to the methods we learned by developing our intuition because it derives from a corrupted form of artificial intelligence (AI). That answers your concern about Expression 11.
Now, back to the expression at hand. Given that the 2(2 + 2) represents a sacred bond that cannot and should not be broken, I’m not doing anything wrong with the “denominator” as you call it. So when you say “You cannot just move a denominator” with respect to Expression 12, you’re failing to recognize the sacred connection of implicit multiplication. I’m not moving a denominator because, as I demonstrated above and supported with older and newer academically sound sources, the whole product after the obelus is the divisor, or in your words, the denominator. I’ve merely commuted the elements of the denominator.
As for the addendum on 5/14, my point there was to show how the sacred connection of implied multiplication is changed by the way strict PEMDAS disciples work the problem. I’ve already demonstrated that above, so I don’t need to revisit it here. The only thing I would add is that, to make explicit what our instincts tell us, we would rewrite it:
8
———
2(2 + 2)
If no parentheses are needed for that, then why would we need them for the obelus? The vinculum is a vertical form of implied division, just as juxtaposition to parentheses is a form of implied multiplication. Of course, the expression with the vinculum would be reduced to a proper fraction, or in this case, a whole number, 1.
The point about the negative sign was only to illustrate that when a negative sign appears before parentheses without an express coefficient, the signs of the terms within the parentheses are supposed to change. That is, the coefficient is an implied -1 multiplied through the parentheses. Your strict PEMDAS would treat that the same as the 2 after the obelus and get a completely different answer than the expression implies.
Finally, as for the “Making It Real” section, my point there is that, since the expression has context and I know what the terms before and after stand for, I don’t need to put parentheses around the divisor/denominator because I know what it stands for. If I were plugging it into a “dumb” calculator, then yes, I would need to put parentheses around the divisor/denominator for the calculator to work according to my instincts (and isn’t that the way we should use a calculator?). But some of us can still do math without a calculator, and some of us can even do it with a slide rule, if you know what that is, so we don’t always need the calculator. The same would go for any such established formulas. Revolutions of a wheel = d/2πr is one such example.
The bottom line here is that calculators need the parentheses for clarity. As humans who have a certain instinctual and linguistic understanding of the expressions we see in a math problem, we don’t always need those. We recognize a certain form and solve accordingly. Our way of thinking is the highway, because it depends on human intelligence and insight. The strict view of PEMDAS promoted by some is a capitulation to the flawed AI of the computer age and seems to lack the kind of reflective analysis and critical thinking I’ve demonstrated here.
Peace to you, and again, thank you for your polite response.
Thanks for your reply. I don’t agree with the name-calling above. If you don’t agree with someone doesn’t mean you need to start calling names. Sometimes people don’t agree, and that’s ok.
With that said, I don’t think we are going to see eye to eye, but I’ll try anyway.
The majority of what we are saying is, is 8/2(2+2) equal to 8/(2(2+2)) or (8/2)(2+2). You’re saying the former, I’m saying the latter. I say the latter as that follows the order of operations. 8 should divided 2 before multiplying because dividing comes before multiplying in the order of operations of this equation, left to right. Since there is no grouping on the 2(2+2) it should just go left to right.
From what I understand of your justification, what you’re saying is that because 2(2+2) is an implied multiplication that the 2(2+2) is part of the denominator.
Now to your reply.
First, the obelus and the solidus are the same thing. There is no difference, they both divide in the same way. 2÷2*2 and 2/2*2 are the same statement. However, as you have shown yourself, the vinculum is different as it does imply grouping. For example,
2
—-
2*2
This is not the same as 2/2*2. An equal statement to the above would be 2/(2*2). The vinculum implies that since the 2*2 is under the bar that it is part of the denominator. Why does this imply grouping, because it is actually under the bar.
Also, you said, “As such, there is NO logical or technical explanation of why the obelus would be treated differently by only treating the first number after it as the divisor.” So let me ask you, what is 2/4/2?
Is it, (2/4)/2, or is it 2/(4/2), is it 1/4 or is it 1? According to your definition above (“everything that comes after the obelus”) this would be 1, but according to the order of operations this would be 1/4. If you say that 2/4/2 is 1 then I think we’re done here and you’re probably just trolling at this point. However, if you say that it’s 1/4, then that goes against what you said in your last reply.
Second, implied multiplication is exactly what it says and nothing else, two values are implied that they multiply. 2(2) is the same as 2*2. Just because 2(2) is after a division 2/2(2) does not mean that 2(2) is in the denominator. It literally just means that it multiplies, 2/2*2. 2/2(2) and 2/2*2 are the same statements, equal to 2 not 1/2.
Third, changing the value to be a negative doesn’t change how the statement evaluates. -(2+2) is the same as -1(2+2), so 8/-1(2+2) still is 8 divided by -1, times the quantity 2+2. Again, to change the equation to get a -1, you must multiply by -2. 1/2 * -2 is equal to -1. Then to get the equation to solve you also need to multiply the other side by -2, so -2 * 16 is equal to -32. Where is this a problem? You also still failed to mention how your equation doesn’t equate when you multiply by the -2. -2 * -2 is 4, not the original 1 that you said this equation was.
Fourth, your “Making it real” section you were trying to imply that your v=d/t would look like 36/6(2+2+2) and then try to prove that it is incorrect, but your assumption was wrong because you have to input the value of t into the equation of v=d/t but an equivalent of v=d/t is v=(d)/(t) which would make the substituted equation v=(36)/(6(2+2+2)) which is 1 not 36 like you implied. Substituting without the parenthesis is incorrect and gives you the wrong answer.
Fifth, you just admitted that we need parenthesis for clarity, but since you are smart (“certain instinctual and linguistic understanding of the expressions”) you know what everyone always intended?
8/2(2+2) is intentionally ambiguous just so people like you and I would debate it, but, there is still one clear answer when you follow the order of operations and that is 16.
Yeah, we probably won’t see eye-to-eye, but I appreciate the discussion. My sticking point is the implicit multiplication. I view that as inseparable, like a prefix on a word or an inseparable prefix on a German verb. I was taught to recognize that as an inseparable term. So the parentheses aren’t really an issue. In my mind, I’m trained to put implied parentheses around it, if you will. Many of us who get the answer 1 do that instinctively. And with one extant operational sign, that seals it for me. It’s just a simple A/B problem for me at that point.
As for your 2/4/2 example, I would actually agree with you that it’s 1/4. There are two extant operational signs, so I’d follow OOO at that point, in the absence of parentheses. I would rewrite it as multiplicative inverses and multiply to prove its 1/4 either way: (2/1) * (1/4) * (1/2). Apply the associative property either way, and you get the same answer.
Peace to you, and thank you for the comment.
If 2/4/2 is 1/4 then “everything that comes after the obelus” is an incorrect way to look at an equation. Let me try another question, what is -4x^2? Is it (-4*x)^2, or is it -4*(x^2)? Lets say x is 2, is it -8^2 or 64, or is it -4*4 or -16? If you say it’s -16 then you’d have a hard sell on why you think your previous claim is true. If you claim the answer is 64, I’ll just say you’re wrong.
I am a person who always kept my old textbooks in case anything ever needed to be looked up again. This is the first time I think I’ve had to look up something in my old college math textbook. There is no mention whatsoever of Implied Multiplication (Or Juxtaposition). I wonder why that may be, do you have any reason why the author of a college textbook would leave out such an important rule?
No, not really. Although I admit I’m still working some kinks out. When I said everything after the obelus, I was referring specifically to that kind of expression, with one extant sign.
Someone else pointed out to me the -4x^2 issue, and I have to admit that the exponent takes precedence in that, so it’s (-1)(4)(x^2).
That doesn’t really change my position on grouping, but a I need to qualify it a bit. Grouping takes precedence over LTR MDAS with extant operational signs. But I would provide an order of precedence for grouped expressions as well. I define grouped expressions as anything without an operational sign that suggests some sort of function upon or between the numbers. It’s a linguistic structure. So as in PEMDAS, parentheses first, then exponents and factorials, then implied or proximate multiplication.
I’m also tempted to define a fraction with a vinculum as implied division as well, different from signed division with an obelus or a solidus. Although it’s a convenient way to express division, especially when you have polynomial expressions, I don’t think it’s intended to be a universal substitute for another form of a division sign. It’s kind of complicated to explain. I’ve already said I don’t think it’s right to assume that in 8 ÷ 2(2 + 2), the 8 ÷ 2 is a fractional coefficient of the expression. First of all, it would be an improper fraction, so that’s bad form for an expression. Second, the obelus is a simple operational sign, so I don’t think it’s rational to think it only applies to part of an implicitly multiplied element. That seems to imply some sophistication for the sign.
Again, thank you for the discussion. This has been an exercise in “thinking out loud,” which is how the scientific method should work. Peace to you.
Thanks for your reply. I have (I think) one last question for you involving implied multiplication. I don’t think I’ve even really heard about it growing up all the way through college. I’ve seen it, but it’s always been “just multiply” the numbers. So my question is, how is it not just multiply, why do we have this extra rule where it doesn’t seem like one should exist?
I’ll even go out on somewhat of a simple proof.
If there is a number x, such that a(b+c) = x and a*(b+c) = x, for any value a, b and c, then the x must be the same value for both equations. Thus meaning that a(b+c) = a*(b+c) for all numbers a, b and c.
Now, if you can show me any number that disproves the above statement, i.e. find any numbers for a, b and c where those equations are not equal and I will bow down and call you a mathematical god. If you cannot, how can you possibly say that they are different and have different rules?
I don’t have any of my high school textbooks, and my college calculus textbook doesn’t even have an obelus in it as far as I can tell. I think what’s going on in my mind is that I’m used to seeing a standardly written division problem (especially one with variables) written with the vinculum as such:
9a^2
——
3a
which of course when written that way everyone agrees the answer is 3a. But when someone rewrites that with an obelus and doesn’t use parentheses to “group” the denominator, my mind still sees the same problem with the same answer: 3a. The same goes with the problem at hand. I’m accustomed to two standardized forms that aren’t used in the expression. First, if someone wants to say 8÷2 is the coefficient of the expression, that’s definitely NOT a standard way to write a fractional coefficient, especially if you don’t put parentheses around it. If someone wants me to think that, they’d have to write the expression as follows:
8
— (2 + 2)
2
But that creates a related problem, in that a coefficient would also be in its simplest form, so you have an improper fraction used as a coefficient where a whole number would do. The improper fraction would be okay if there’s no way to simplify it to a whole number, as in the equation for the volume of a sphere:
4
— πr^3
3
The second standard I would expect is that if you want two of the three values in the expression to be multiplied together, you wouldn’t make them the first and last values in the expression. You’d write the expression in such a way to avoid any confusion as to what you expected to be multiplied together:
8(2 + 2)
———
2
So whether intentionally or not, the expression is an extremely poor example of a mathematics problem, and people shouldn’t be using such a poorly written expression to promote something that they expect to be a standard way of working equations. That being said, I’m not inclined to give props to the people who try to push a poorly written expression as the basis for a standard. I’ll fight it tooth and nail if I have to, because it’s bad pedagogy.
One final thing: PEMDAS is for the most part applied to hypothetical, that is, non-real-world equations. A good example is the formula for a pitcher’s earned run average (ERA), taken from the MLB Web site: “The formula for finding ERA is: 9 x earned runs / innings pitched.” Let i = innings pitched and R = earned runs. We wouldn’t write that equation 9 ÷ i(R), right? In your way of thinking, you’d get the right answer, of course, but it doesn’t fit the logic of how the stat is formally calculated and what its significance is. The best way to write it is 9R/i or
9R
—
i
My point here is that most standardized formulas for things we encounter daily (whether we see them or not) are written in such a way that you don’t have to worry about PEMDAS. They’re simplified enough that the “order” is inherent in the formula. And if that formula happens to come after a division sign (as in calculating the number of revolutions of a wheel over a certain distance) instead of under a vinculum, most people will know to keep the formula intact and not start playing PEMDAS games with it.
Again, thank you for the dialogue and feedback. It helps me clarify and solidify my position even more. Peace to you.
Scott
Is 3a. I’ve mentioned that in my previous post that the vinculum does imply grouping because you can clearly see that the 3a is under the bar. However, the above equation is not equal to 9a^2/3a. This equation would be,
9a^2
—- a
3
Which becomes 3a^2*a, or 3a^3. Although, I also do agree that the equation itself is silly and should just be written 9a^2(a)/3, or 9a^3/3, or 3a^3, but I disagree that it’s incorrect. In mathamatics it is part of the journey to get to the reduced state, rarely do you start with a reduced function.
For example, this statement, a-b where b is a-1 is equal to 1. You could have this statement, or you could just call it 1, but 1=1 doesn’t really mean anything whereas a-b where b is a-1 is equal to 1 implies something about a and b. This could be extrapilated out to a-b where b is a-c is equal to c.
My point is, just because something seems silly doesn’t mean it’s incorrect. We have the order of operations only to tell us in which order the operations are calculated in.
It sounds like maybe you’re starting to agree that 8/2(2+2) is 16 which is,
8
—(2+2)
2
whereas
8
——
2(2+2)
is 1.
I agree that 8/2(2+2) could be written differently, 8(2+2)/2, or 4(2+2) but that’s not what was given to us, but all three of these statements are the same and the only way to get there is to use the order of operations.
So hopefully you can see where the people who get 8/2(2+2) = 16 come from, and I think that is where I’m going to leave it.
This is all silly. The only thing and debate here is the order of operations. You mention an order of operations that I was unaware of, two numbers next to each other, without an operation symbol gets precedence over left to right. I am not aware of sexual. I don’t think it’s correct, but if it is, you’re right (1)and if it’s not, you’re wrong (16).
The math is the math, and that does not change. Order of operations is a language syntax issue. It’s scarcely different than how the placement of a comma can change the meaning of sentence.
Finally, your velocity example is completely wrong. I am extremely confident that anytime a variable is represented by a letter, that variable must be calculated in full and not broken up into pieces as you have done, which would be completely insane and make algebra, completely unworkable in any sense.
Comment by Rich, fellow math nerd — June 29, 2023 @ 5:49 pm
| Reply
Thank you for your response. The implicit multiplication represented by the right side of the expression is akin to a prefixed word in a regular sentence, or in German, a verb with an inseparable prefix. If you strip the prefix from the word and put it in a different place in the sentence, you change the meaning of the sentence. Same goes for the coefficient here. The coefficient is not intended to be separated by OOO. OOO was never intended to alter those fundamental relationships.
If you think my velocity equation is wrong, then please suggest a different way to construct it. Put your keystrokes where your statements suggest.
Thank you for your response. Peace.
Comment by Erin Kastenschmidt — September 14, 2023 @ 8:37 pm
| Reply
Glad you enjoyed it. I need to update it, because I’ve found several discipline-specific style manuals that further support a juxtaposed or implied multiplication expression after a sign of division should be taken as the whole divisor.
Thank you.
Very clear explanation.
My only suggestions are about terms and factors, that haven’t been mentioned much. Factors multiplied are single terms.
8 is a single term.
8/2 is 2 terms.
Euler, Elements of Algebra
Page 34, para 89.
“when we multiply both its terms, or its numerator and denominator,”
8/2(3) is 2 terms, as is 8/2(2+2).
storyofmathematics.com
Definition
A term is any single number or variable. An expression contains two or more terms. If two numbers or variables or both
are multiplied by each other, we consider them as a single term.
“0:51 So in this example, you have three terms.
0:56 The first term is 2 times 3.
0:59 The second term is just the number 4.
1:01 And the third term is 7 times y.”
Khan Academy Terms, Factors and Coefficients Transcript.
So the expression is the term 8, divided by the term 2(4) or 2(2+2), to give the
answer of 1.
Comment by Doug Hendrie — October 30, 2023 @ 1:29 pm
| Reply
Thank you, Doug. I appreciate the references you cited. Glad you enjoyed the read.
Scott
You’re wrong and willfully ignorant. You have misapplied the Properties and Axioms of math and you’re too mathematically incompetent to realize it…
Let me explain the Distributive Property… Using a similar expression 6÷2(1+2)= 9 not 1
The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division.
The Distributive Property is congruent with the Order of Operations it doesn’t supercede the Order of Operations… The Order of Operations work because of the Properties and Axioms of math not in spite of them…
The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in… If you can’t draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven’t applied the Distributive Property correctly…
The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication…
The axiom a(b+c)= ab+ac however the variable “a” represents the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that “a” is a single numeral…
This can be further demonstrated using the vinculum….
6
——(1+2)= 6÷2(1+2)= 9
2
6
———— = 6÷(2(1+2))= 1
2(1+2)
A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator….
________
2(1+2) = (2(1+2)) two grouping symbols each
That over bar (vinculum) is a grouping symbol
_______ _________
2(1+2) = 2×1+2×2
(2(1+2))= (2×1+2×2)
Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation…
If you choose to Distribute the 2 into the parentheses by itself you have to do one of two things. Either take the division symbol with it, as division is right side Distributive or change the division to multiplication by the reciprocal…
÷2= ×0.5
So… 6÷2(1+2)= 6(1÷2+2÷2) still equals 9
Or… 6÷2(1+2)= 6(0.5×1+0.5×2) still equals 9
Variables can represent more than just a numeral and it’s important to understand that when you replace a variable with a constant value or a set of operations that represent a constant value that you apply grouping symbols where called for by the Order of Operations and the basic rules and principles of math… example 6÷a does not have parentheses BUT a= 2+4 so 6÷a = 6÷(2+4) not 6÷2+4. BUT if a=2×3 and we have a÷2 we can write 2×3÷2 because we evaluate Multiplication and Division equally from left to right…
ab+ac =
12÷3×2×3+12÷3×2^2=
4×2×3+4×4=
8×3+16=
24+16=
40. <<< same answer
What most people don't understand is that you can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression… 6÷2×1+6÷2×2+6÷2×3^2-6÷2×4=
6÷2(1+2+3^2-4) as the LIKE TERM 6÷2 was factored out of the expanded expression.
I hope this helps you understand the issue a little better….
You're confusing and conflating an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity by this convention to Parenthetical Implicit Multiplication. They are not the same thing…
a/bc = a/(bc) by Algebraic Convention
a/b(c)= ac/b by the Commutative and Distributive Properties…
a/b(c) = a×b⁻¹(c) NOW the expression is Associative…
Wolfram Alpha supports the fact that 8÷2(2+2)= 8÷2×2+8÷2×2 …. Wolfram supports the fact that 8÷2(2+2)= (2+2)8÷2. Wolfram supports the fact that 8÷2(2+2)= 8×2⁻¹(2+2) AND Wolframalpha supports the fact that…
8
——-(2+2) = 8÷2(2+2)
2
Comment by Richard — November 1, 2023 @ 9:21 pm
| Reply
Mr. Smith: I will approve your post because I believe in the First Amendment, but I suspect at least part of your post was plagiarized from Facebook. For now, I’ll give you the benefit of the doubt that you’re the Richard Smith from the Math Challenge page on Facebook where I documented that at least a portion of your response was copied from and pasted here. (Yes, it’s plagiarism if you quote yourself from another previously published source and don’t acknowledge the original source.)
Before I get into responding to your short-sighted view of the Distributive Property, I want to cite some historical evidence that the ambiguity of such expressions as are promulgated on social media was called out nearly 100 years ago in Florian Cajori’s seminal work, A History of Mathematical Notation (1928), section 242, on the “Order of Operations.”
“242 Order of Operations in terms containing both ÷ and ×. –If an arithmetical or algebraical term contains ÷ and ×, there is at present no agreement as to which sign shall be used first. ‘It is best to avoid such expressions.’ [M. A. Bailey, American Mental Arithmetic (New York, 1892), p. 41)].” After citing three textbook examples of the day that recommend different ways of addressing the issue (i.e., in order as they occur or multiplication first; one textbook says to put the divided numbers in parentheses), he notes that “an English committee [“The Report of the Committee on the Teaching of Arithmetic in Public Schools” Mathematical Gazette, Vol. VII (1917), p. 238. See also p. 296] recommends the use of brackets to avoid ambiguity in such cases.” Because such ambiguity still exists today, as evidenced by the rightful detractors from your position, your failure to avoid this historically documented ambiguity of the expression leaves the answer at best open to interpretation, and at worst, demonstrates you’re just trolling people with your convoluted explanation. I’m pretty sure there are very few fifth graders today who could speak in terms of “monomials” and your nuanced definition of “term.”
Note here that this applies to expressions where the multiplication symbol is extant. In section 238, in discussing Leibniz’ Notation, he equates
a:(bc) with a:bc
(the colon being the symbol of division for Leibniz), so it’s clear that, in a division problem where the denominator consists of two implicitly multiplied factors, such implicitly multiplied factors are considered the ENTIRE denominator, with or without parentheses around the factors, and has been that way for over 100 years. I have already demonstrated in Wolfram’s description of Solidus that such occurrences in standardized formulas that do not have the parentheses in written form need to typically be enclosed in parentheses when input into a calculator or online problem-solving format.
The less convoluted way to look at this, and one much easier for a fifth grader to understand, is that given there is only one operational sign, the obelus, what comes before the obelus is the dividend, and the entire expression after the obelus is the divisor. Without clarifying parentheses, the problem is open to such interpretation, because the ENTIRE expression is itself one “term” as you define it. You can’t on the one hand say “parentheses first” (which would bypass the distributive property) and then claim the distributive property somehow gives you two terms to work with. Additionally, you can’t, on the one hand, claim the vinculum is a grouping symbol, and then not group underneath it everything that comes after the obelus! That’s a glaring contradiction in what you attempted to demonstrate above. If the obelus and vinculum have different functions as you claim (one doesn’t group; the other does; please cite a credible primary source for such an assertion, because I don’t believe that assertion to be true), then replacing one with the other is NOT making an equivalent substitution, and you’re changing the nature of the expression in doing so. With obelus, it’s clear what is the dividend and what is the divisor. When you insert the vinculum, you split the divisor of the original expression!
Finally (for the time being, anyway), implicit multiplication should be considered in the same category with other expressions of implicit multiplication, namely the “exponents” step. What is an exponent if not a representation of implicit multiplication (3^3 = (3)(3)(3)); a factorial is also an expression of implicit multiplication (3!=(3)(2)(1)). So why shouldn’t 2(2+2) or, in your plagiarized example, 2(1+2), be considered at the exponent level as well? No one has ever given a decent explanation of why that shouldn’t be the case.
I’ll deal with your misguided conception of the distributive property at another time. Suffice it for now to say I’ve never ever seen the distributive property defined in such a way as to include an obelus. It’s only about multiplication and addition. Again, cite a credible, primary source if you can find one.
Let me address your misconception of the Distributive Property, Mr. Smith. If the distributive property is defined in terms of a(b + c) = ab + ac, then if you’re substituting something into the variable that isn’t a single integer or a different single-letter variable, it would be customary to place the substituted expression in parentheses to indicate without a doubt the entire expression replaces the variable. In the case of your example, then, it’s not clear, and perhaps intentionally and maliciously ambiguous, to not put 6 ÷ 2 in parentheses to stand for the single variable “a” of the generic description of the Distributive property. It leaves the question open, as I stated above, about the role of the divisor. If your side wants to insist that the 2(1 + 2) must have parentheses around it to be understood as the whole divisor, then our side is equally justified to insist you must put parentheses around the 6 ÷ 2 to indicate it stands for a single entity before the parentheses. All your pedantic babbling about monomials and terms is irrelevant at that point. If you want to be clear and be clearly understood, use parentheses. If you want to be an Internet troll, continue on as is.
If we look at it as a simple dividend ÷ divisor problem, then we would say the divisor is in the form of the Distributive Property, which is then divided into the dividend (6), thus obtaining the answer of 1. But since you fail to follow standard procedure of placing parentheses around the 6 ÷ 2, you cannot claim unequivocally that that expression is the coefficient of what is in the parentheses. It’s a simple as that really, and it shows why your convoluted explanation lacks any theoretical validity. Your conception of the problem is misguided and just plain wrong. The answer to the example you use and the expression I discuss is 1 in both instances, indisputably and indubitably.
Let me also say something about the linguistics of the expression. Someone may look at your example and say “Six divided by twice the sum of (1 + 2)” and come up with the correct answer of 1. This is, in fact, how Wolfram solves the text version of the problem, 6/(2 (1 + 2)). It places the entire divisor (without parentheses) under the vinculum in graphic form. The form you see here with the solidus and extra parentheses is how it’s copied into WordPress (I copied the answer with the vinculum and directly pasted it here without any editorial intervention), because the Comments section here doesn’t handle graphics. So again, this is one more nail in the coffin of your argument.
Scott writes: “I do not deny the importance of PEMDAS, but the reality of the problem is, any basic math problem like this can only have one correct answer. It’s not and never a matter of personal interpretation. Otherwise, the foundations of mathematics would crumble into oblivion, and not even Common Core could save us (not that it ever did anyone any good). This is math; it doesn’t care about and is never affected by your feelings about it.”
PEMDAS alone can’t save us and never could. Think of PEMDAS as a basic set of training wheels and nothing more. It is a ruleset put into place at an early time in our lives when math simplicity was key, not completeness. PEMDAS is nothing more than a basic math helper tool, but one that has an incomplete understanding of mathematics.
Mathematics, like any computer language, has syntax rules. If that syntax isn’t absolutely clear, the math problem can’t be solved. No, not even one answer. Like failing the syntax check when writing C language, failing a syntax check when writing a math problem results in the same exact dilemma, there can be no answer because of a failed syntax check.
However, with 8 ÷ 2(2 + 2), people seem to wish to try their hand at attempting to solve this syntactically problematic equation anyway. There is only ONE logical way to solve this problem IF you choose to ignore the glaring RED syntax error and forge ahead anyway. PEMDAS was devised for use by students BEFORE they have an understanding of implied multiplication. Once multiplication by juxtaposition (implied multiplication) becomes understood during Algebra, a different more advanced ruleset (aka. style guide) comes into play. What ruleset is that?
That ruleset is that 8 ÷ 2(2 + 2) is the same as 8 / 2(2 + 2). Most algebra texts have style guides that state that any equation delimited by a slash (/) automatically moves what’s left of the slash into the numerator and what’s right of the slash into the denominator… all of it (unless there’s an operator that stops this). Thus, 8/2(2+2) is the same as the fraction…
8
——-
2(2+2)
where 8/2*(2+2) (using the explicit * operator) is understood as
8
– * (2+2)
2
The above proves why using the correct syntax is critical. The * halts the Algebra rule which applies to /.
PEMDAS doesn’t agree with Algebra text rules when an equation includes juxtaposition. That disagreement is mostly because PEMDAS isn’t even aware of multiplication by juxtaposition.
Thus, the assumed solving logic for the equation 8 ÷ 2(2 + 2) is to apply advanced math rules ONLY. Why? Simple, because it contains multiplication by juxtaposition, an advanced math Algebra concept. Thus, logic dictates that the basic grade school learned PEMDAS rules can’t and don’t apply. Thus, any answer conceived using PEMDAS is, indeed, incorrect because PEMDAS has no knowledge of multiplication by juxtaposition.
With all of this said, grade school math instructors WILL teach their students using PEMDAS and WILL interpret and thus solve this problem as though it were written (8 ÷ 2) * (2 + 2) using PEMDAS rules, even though the original version of this equation 8 ÷ 2(2 + 2) contains the more advanced Algebra concept, math by juxtaposition. Regardless, the syntax error still persists.
Thank you. I’ve seen similar responses to this problem that raise the issue of syntax from a computational language perspective as you have, and they always follow your line of thinking. Your point about PEMDAS being a beginning math paradigm is something I had just concluded in responding to Richard (see his comment below) on one of his Facebook posts the other day. It’s almost like PEMDAS is used as a justification for a poorly constructed expression or an excuse to give kids “busy work” in math class instead of fostering a little more critical and theoretical thinking about math that is sorely lacking from our modern education system these days. I’ve made the point several times on Facebook that most standard, real-world equations are written in such a way that PEMDAS or Order of Operations is really a moot point. The most anyone needs to know for those is the first two steps (parentheses and exponents); after that, the rest of the standardized formula/equation falls into place quite naturally, reading from left to right. As an editor and erstwhile linguist, I’ve come to appreciate the value of a concisely written “plain language” instruction or explanation. It occurs to me that perhaps math texts for grade schoolers should adopt that same principle as well instead of expecting the unsuspecting to know how to “juggle” the elements of an expression according to their strict (and misguided) application of PEMDAS/OOO.
PEMDAS, BODMAS, BEDMAS and BIDMAS all have their place when learning early mathematics. The point in using these helpers is in simplifying the rationale of how to solve simple left-to-right formatted mathematical equations. These tools are used at a time when students are between 6 and 11 years of age. At these ages, teaching a more advanced concept like multiplication by juxtaposition would probably prove impossible. To begin throwing Algebra concepts at an 7 year old who can barely grasp what 6 ÷ 2 means would make for a challenging grade school situation. A few students may be able to grasp these more advanced math concepts this early, but many more would not. At that level of math understanding, these advanced math concepts also aren’t really needed yet. Trying to teach 7 year old students the rules of Algebra is tantamount to throwing a student into the deep end of a pool before they know how to properly swim.
PEMDAS is intended to cater to all students of all maturity levels. There’s nothing wrong with using PEMDAS (et al) as long as they’re used to teach students with the type of mathematical problems that are best designed for use with PEMDAS. When advanced mathematical concepts, such as juxtaposition become involved, PEMDAS more or less makes way for more advanced mathematical understanding and rules. Attempting to teach these more advanced rulesets to 6-11 year olds is more than likely to confuse them at time when they need clarity and simplicity to grasp the basics.
Instead, mathematics courses need to become more clear as we move through them. For example, Algebra class should explain that while PEMDAS is useful for left-to-right formatted equations, many non-linear written Alegbra problems cannot be solved using PEMDAS alone. This needs to become a point of clarity and discussion when beginning advanced math classes like Algebra, Trigonometry and Calculus. However, I guess by the time students reach to these advanced math classes, it is assumed that they will “learn the ropes” on their own and learn as they “go along”.
That’s a valid solution. I did just read your blog article on the expression and really appreciate your perspective. I recently looked up the various journal style guides to see how they handle it, and found it exactly as you said. The professional journals give precedence to implied/juxtaposed multiplication.
Yep, which is why so many calculators also give juxtaposition this precedence. It also seems that the multiplication by juxtaposition rule before division dates back to at least the 1920s or possibly earlier… it seems before the PEMDAS teaching term was even coined. Just some history there.
I cited some of that history in my response to Richard on this blog a couple days ago. I recently bought Florian Cajori’s book “History of Mathematical Notation” for a little “light” reading.
Good find. I just found that Cajori’s book is available in a Kindle edition, but it’s ~$20, same as a used copy. I’ve just added it to my Amazon holiday list in case someone wants to gift me a copy. Otherwise, I’ll check out Half-Price Books soon and see if I can find a copy there. I’d like to read that book.
Totally wrong.. its 16. always 16. wolfram alpha gets you 16 too.
Comment by Scott A pike — November 29, 2023 @ 7:11 am
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Wolfram Alpha is way more than just a homework help site. If you enter “eight divided by twice the sum of two plus two” in the input field, it returns the answer of “1”. Also look at its description of Solidus. It explains that juxtaposed multiplication after a sign of division requires parentheses to be properly understood, and gives some examples of real-world formulas to prove it.
I have occasionally used
F=ma. Newtons 2nd law of motion.
Divide both sides by ma. Now
F/ma = 1. Assign F = 60 Newtons,
m=15 kilograms and a = 4 meters
per second per second.
Then
60/15(4). If the strict PEMDAS folks
don’t get an answer of 1 then I have no idea what the value represents or the units to assign to it.
Comment by Ray Morse — December 30, 2023 @ 1:04 pm
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Mathematicians are trained not to use implied multiplication. They write the text books used in their classes. They know how to write expressions and equations without using it. Engineers and Scientists however use the notation for simplifying their math. Math professors know that, but they don’t teach Engineering and science classes.
Comment by Ray Morse — December 30, 2023 @ 1:25 pm
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FAQs: “How do you identify a monomial? Monomials are the product of a coefficient, and a variable or variables.”
A monomial has a single value so it’s as if it already has parentheses around it, in much the same way an exponent with a base number never needs to be encased inside parentheses. In both cases, you’re told how many times to multiply a quantity.
That means they’re teaching students that even though a division sign (obelus) is used to indicate “divided by,” the statement should be treated as a top-and-bottom vertical fraction, with 15a^2b^3 as the numerator & 5b as the denominator — with NO PARENTHESES anywhere in the statement.
The “2” in “2x” is not a stand-alone number — it’s the coefficient in a monomial which tells you how many times to multiply a quantity (which is actually adding the quantity to itself), in much the same way that an exponent tells you how many times to multiply the base number by itself. Just as the exponent is “attached” to the base number, the coefficient of a monomial is “attached” to the variable (factor). Thus, the monomial division statement “2x divided by 2x” is: [ x + x ] ÷ [ x + x ]. Notice that the coefficient “disappears” when the statement is written out in its most basic form (as the indicated additions of the quantity). That proves, once and for all, that “peeling off” the coefficient of the monomial (the “2” in “2x”) & using it in some other operation is not valid.
In the monomial division statement “2x ÷ 2x,” if x equals 4 [expressed as (2+2) ], it is:
2(2+2) ÷ 2(2+2)
which is also written as
8 ÷ 2(2+2) or 8 / 2(2+2)
which is the same as…
8 ______ 2(2+2)
…which has a quotient of 1.
A monomial has one single value (not two separate values that can be pulled apart). And division is fractions, no matter which division symbol is used — they all mean “divided by” & separate the numerator from the denominator. Do all of the operations indicated in the numerator, then do all of the operations indicated in the denominator, and finally divide the numerator by the denominator. Division has to go LAST. The Order of Operations as PEMDAS is incorrect for division statements — also known as fractions.
Comment by Dee — February 18, 2024 @ 4:34 pm
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i am very sorry but this article is very very wrong and i have no clue why you are trying everything to falsely make the equation =1 when in fact when you input it in a calculator, it gives you 16 as YOU showed. multiplication and division have the same priority thus you the number you are multiplying the parenthesis is actually 8/2 which is 4. in real physics you will never get an accurate answer using your methods and i m sorry for people that read this to find a proper answer.
Comment by tudorlasus — February 19, 2024 @ 1:34 pm
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It’s really quite a simple, third-grade solution to get the correct answer of 1. The obelus divides the expression into a dividend (term on the left) and a divisor (monomial term on the right). These equate to the numerator (dividend) and denominator (divisor) in fractional terms. Therefore, the correct way to interpret the expression is 8 ÷ 2(2+2) = 8 ÷ 8 = 8/8 = 1. Easy peasy lemon squeezy.
The problem seems to be that the computer programmers who set it up as a straight-across-Order-of-Operations (as PEMDAS) statement obviously missed that day in school when the Basic Algebra teacher went over what a “term” is, gave examples of how to recognize a monomial & how to divide one monomial by another monomial.
In the statement 8÷2(2+2) , the “8” can be factored out as “2(2+2),” making the statement…
2(2+2) ÷ 2(2+2)
Replace what’s inside both sets of parentheses in the statement with the variable “x” — in other words…
“The simplest way of viewing division is in terms of quotition and partition: from the quotition perspective, 20 / 5 means the number of 5s that must be added to get 20. In terms of partition, 20 / 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that “twenty divided by five is equal to four”. This is denoted as 20 / 5 = 4, or
20
___ = 4
5
In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.”
“Notation
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, “a divided by b” can written as:
a
__
b
which can also be read out loud as “divide a by b” or “a over b“. A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), as follows:
a/b
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the division sign (÷, also known as obelus though the term has additional meanings), common in arithmetic, in this manner:
a ÷ b”
Bottom line: Division is fractions.
Given that x is not zero, 2x divided by 2x equals 1, no matter which division notation is used.
Comment by Dee — February 21, 2024 @ 12:48 pm
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Thank you for your comment. I’m glad you made the point about how calculators and most computers treat the expression “as a straight-across-Order-of-Operations (as PEMDAS) statement.” Playing around in Wolfram’s platform seems to highlight this kind of functionality. I think I added the examples of how Wolfram handles two slightly different ways of writing out the expression in text form. In Wolfram, if I use the text phrase “eight divided by two times the sum of two plus two,” Wolfram’s platform really has no way of distinguishing the phrase “two times” as somehow intrinsically linked by juxtaposition to (2 + 2). It treats the word “times” as a written operator and subsequently solves the expression from left to right as if there were no juxtaposition. We might be able to imply that juxtaposition by the way we say the phrase or inflect our voice, but Wolfram’s language model doesn’t appear to be that sophisticated at this point.
However, if I type out “eight divided by twice the sum of two plus two,” then Wolfram’s language model treats “twice” as an adverb and, even though “twice” means “two times,” the language model then recognizes the phrase after “divided by” as the whole divisor, that is, it treats it as its own term or monomial. The absence of the word “times” causes the model (apparently) to think less linearly and recognize there is an intrinsic juxtaposed or implied connection there. It acts the same way with the word “thrice,” but when I tried fourfold, fivefold, etc., it stopped recognizing the intrinsic connection.
Scott
PS This post has been my most viewed post by a 5 to 1 margin since last June. Are you aware of any large groups or conferences where this article or this topic is being discussed? Thank you.
I am not aware of another site that is currently discussing this issue – I wish there was.
It really does seem to be a programming issue. On several math teaching sites & online calculators, when I type in “2x/2x,” it automatically converts the statement into a top-and-bottom fraction, with “2x” on top & “2x” on the bottom, showing the quotient of 1. When “x=(2+2),” that makes the same mathematical statement”2(2+2)/2(2+2),” and the very same calculator tells me that the result is now 16.
Here are a couple of those calculators that gave two different answers to what is the same statement, when x=(2+2):
[ first input as 2x/2x and then as 2(2+2)/2(2+2) or 8/2(2+2) ]
It boils down to understanding that division can be expressed as a fraction. With a fraction bar, everything “North” of the division symbol is the numerator & everything “South” of the division symbol is the denominator. In a horizontally written division statement (with an obelus or solidus), everything “West” of the division symbol is the is the numerator & everything “East” of the division symbol is the denominator (unless otherwise indicated with parentheses). It’s as if you wrote the top-and-bottom fraction on a sheet of paper & then rotated it 90 degrees to the left, in terms of directional position of the numerator & denominator, relative to the division symbol.
Some computer programmers remembered that day in school when monomial division was covered & programmed the calculator accordingly. Others forgot it altogether. And yet another group of programmers remembered the part with the coefficient & variable together being one term with a single value, but did not account for finding out the actual value of “x” & plugging in the numbers — which ought to give you the same result.
A certain group of calculator programmers need some refresher math tutoring.
“Any division of monomials can also be expressed as a fraction:
8x3 y2 z
8x3 y2 z ÷ 2x2y = __________ = 4xyz
2x2y
—————
Note that the original monomial division statement “8x3 y2 z ÷ 2x2y” uses a division sign (obelus) & there are no parentheses anywhere in the statement.
That being the case, 2x divided by 2x can be written horizontally as…
2x ÷ 2x or as 2x / 2x
…which is synonymous with the top-and-bottom fraction…
2x
___
2x
…and given that “x” does not equal zero, the statement has a quotient of 1, no matter which division symbol is used.
Any calculator that delivers a different result was programmed by someone who did not fully understand how to divide monomials (as a fraction — with no parentheses necessary). The statement is dividing one monomial by another monomial, with each monomial having a single value which is the PRODUCT of the coefficient multiplied by the variable.
In the monomial division statement “2x ÷ 2x,” x = (2+2)
Plug in the value of “x” & solve.
Comment by Dee — February 23, 2024 @ 3:25 pm
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Addendum:
Here’s another online calculator that gives the quotient of 1, whether input as…
…because it’s the exact same statement, no matter which division symbol is used.
Comment by Dee — February 22, 2024 @ 10:04 am
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I have had near arguments with people over this topic. In most cases, I facetiously admit defeat. You know the old saying: Argue with an idiot and they will bring you down to their level and beat you with experience.
I’m glad to see your depth of knowledge, based on mathematical laws, truly explain this concept. It’s hard to grasp for those who didn’t move beyond Algebra 1. I’m not being condescending; there is really a clear difference in mathematical education though. As you’ve stated, and I remember learning, the ‘math properties’ must be upheld first and foremost before submitting to ‘PEMDAS’ or other such acronyms used for basic math.
Most people have never even heard of the terms ‘obelus’ or ‘solidus’. So, how can they even follow, understand, or interpret a basic math problem? I’ve even tried a simple approach to explain these internet brainbusters. How can 8 ÷ 2(2 + 2) = 16 vs = 1? The numerator is 8. The denominator is 2(2 + 2) which is also 8, based on the distributive property, which is a Law! That computation = 1. To derive an answer of 16, you’re somehow suggesting that you’re reading the problem as 8 ÷ 0.5 which does equal 16. But how did you get there? I’ll wait for that explanation…
LOL…Perhaps this hotly contested topic is the crux of perplexion that’s prohibited NASA from returning to the moon. You know, because in 1969, this answer was indeed 1. Now, it’s 16. Or 9, for the other circulating stumper of 6 ÷ …yeah, you know the rest. (By the way, an answer of 9 for the preceding can only be achieved if 6 is divided by two thirds aka 0.667.) Haha!
On a personal note, I have to give credit to where it’s due. We all learn from someone. I’ve cherished the opportunity to learn Calc/Logic/Mechanics/E&M (and the prerequisite courses) in high school, taught by two great professors. They happened to be a married couple that passionately presented higher level mathematics and physics to a small handful of capable students. That was nearly 30 years ago. I’m forever grateful!
Scott, thank you for posting this. It reassures me that humanity still has a fighting chance…
~Alexis
Comment by Alexis — February 27, 2024 @ 7:07 am
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Thank you for reading, Alexis. I’m glad to have restored your hope in humanity! :-) I’m approaching 10K views for this article in the 10 months since I first posted it. That seems pretty significant for a post that’s completely off-topic from the theme of my blog. I wish I could be a fly on the wall whenever a math class or other group discusses my article. –Scott
Indeed, Alexis, the quotient of the statement 8 ÷ 2(2 + 2) is, was, and always will be 1, once it is understood that it’s the monomial division statement:
2x ÷ 2x
…when x = (2+2)
Comment by Dee — February 28, 2024 @ 2:22 pm
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I agree with your explanation and really enjoyed reading this article. I have been called ignorant and other names because my answer was 1. 2 seconds to see the problem and got 1 for my answer. I was brilliant in math to the point I’ve had teachers apologize for saying I was wrong when they discovered the text book was wrong. Not saying I’m a genuis but I see math as enjoyment, I’ve read up on math, laws rules, the people on Facebook that arguing about it being 16 will not read into that to understand how it’s 1 instead 16, most on there that get 1 do it by accident but a few have actually mention the rule you used to get 1 as well. Glad to see your post and I’ll read more of your articles for sure
Comment by Josh Jones — March 9, 2024 @ 8:19 pm
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Thank you for reading. I know where you’re coming from. I was doing algebra in 6th grade in a self-paced class (in a public school, no less!). When I got to 7th grade, they wouldn’t put me in an algebra class, so I got stuck in general math and was begging the teacher for more work. There’s a new one on the Math Challenge group on FB now I’ve been responding to. Every time, I hone my arguments just a little bit more. The “new” (to me, anyway) problem is 60 ÷ 5(1 + (1 + 1)). Of course, that equals 4. Peace to you, and don’t get too grumpy about that lost hour of sleep tonight (if you’re in America, that is).
Scott
First, take care of the innermost set of parentheses: (1+1) = 2
Now it’s…
60 ÷ 5(1 + 1(2))
…which brings up the question of whether what’s now inside the remaining parentheses is (2(2)) or if it is 1 + 1x [ i.e. x= (2) ] , with the 1 immediately to the left of the inner parentheses as the coefficient of what is in the innermost parentheses.
Personally, I believe that the coefficient to the left of what’s in the innermost parentheses, makes the statement…
I typed in the problem using a slash & it made it into the top-and-bottom fraction…
60
______________
5(1+1(1+1))
…which, according to that calculator, ultimately yields the quotient of 4. Apparently, it interprets the coefficient of what’s in the innermost parentheses is 1 (not the sum of “1+1” as I thought it was) — plus one after that.
Yes, the “1 + 1” before the parentheses would only be summed first if it was also in parentheses. But since the second “1” is juxtaposed to the parenthesis, then that should be multiplied first. Thank you for the link.
Scott
The statement is one monomial divided by another monomial. By definition, a monomial is one term, so it never needs to be encased in parentheses to comprehend that it has the single value of the PRODUCT of the coefficient multiplied by the variable (factor). Therefore, the coefficient can’t be ripped away from the variable (factor) & used in some other operation before calculating the value of the monomial itself.
Those who want to believe that the statement equals 36 need to stop depending on calculators which may have been programmed by someone who missed that day in 9th grade Basic Algebra class, when the teacher covered what a term is & how to divide one monomial by another monomial — as a top-and-bottom FRACTION. When any division statement is read aloud, the words, “divided by,” separate the numerator from the denominator, regardless of which division symbol is used (obelus, solidus or vinculum). Therefore, the statement
After you said that in a horizontally written division statement, the numerator is everything to the left of the division symbol (obelus or solidus) & the denominator is everything to the right of the division symbol, someone brought up a statement something along the lines of 4/2/4, questioning where the numerator is & where the denominator is in that multiple division statement. In the case of 4/2/4, he answer is either one half or it’s 8. depending on whether the numerator is just 4 or if the numerator is four-halves.
After contemplating this issue, it is clear that in a horizontally written statement which uses multiple division symbols, parentheses would indeed be necessary to delineate the numerator & denominator, respectively. However, in a horizontally written division statement containing only a single division symbol, the numerator is everything to the left of the division symbol & the denominator is everything to the right of the division symbol.
Dee: If we’re talking about using an obelus, then a problem like 6 ÷ 3 ÷ 4 should be worked left to right, per PEMDAS. The obelus, solidus, and vinculum group what they encounter to the next extant sign (+-x÷) not in a parenthetical construct but the obelus by itself does NOT represent the same relationship that a true fraction does. However, if you write the expression using the solidus, then you have an issue of how to group the terms, since the solidus typically serves the same function as the vinculum in that it groups what comes after. So if you write the expression 6/3/4, you have a genuinely ambiguous expression which would lead to confusion about which answer is correct. Is it ½ or 8? It’s up to the interpretation of the reader. However, if you write it like this: 6 ÷ ¾ or 6/¾, where there is an attempt to distinguish the true denominator, then you have a little more clarity. The problem arises, then, about the role of PEMDAS at this point. Why? Because strict PEMDAS would say you follow the order of the signs, so it’s no different than 6 ÷ 3 ÷ 4 at that point (½). But most of us learned when faced with a problem written in such a way that it’s clear we’re dividing by a fraction, PEMDAS is suspended. We first invert and then change the obelus to a multiplication sign. That’s NOT a PEMDAS step! So 6 ÷ ¾ or 6/¾ becomes 6 x 4 ÷ 3, or 8. Here’s the inconsistency that proves juxtaposed multiplication does take priority: The strict PEMDASian would say 6 ÷ 3(4) is THE SAME AS 6 ÷ ¾! It’s obvious to the naked eye that the two expressions are in fact NOT equal and not intended to communicate the same value. The solidus is a grouping symbol just like the parentheses, so there is an extra NON-PEMDAS step taken to solve the division by a fraction problem. In the same way, then, when looking at 6 ÷ 3(4), we take the extra step of applying the parentheses around the 3(4) to account for the grouping of the vinculum or solidus.
Scott
In response to your reply that, “If we’re talking about using an obelus, then a problem like 6 ÷ 3 ÷ 4 should be worked left to right, per PEMDAS,” I disagree. The obelus and solidus are synonymous & therefore interchangeable.
The fraction slash ⟨ ⁄⟩ is used between two numbers to indicate a fraction or ratio. Such formatting developed as a way to write the horizontal fraction bar on a single line of text. …This notation is known as an online, solidus”
“Division
The division slash ⟨ ∕⟩, equivalent to the division sign ⟨ ÷⟩, may be used between two numbers to indicate division. For example, 23 ÷ 43 can also be written as 23 ∕ 43. This use developed from the fraction slash in the late 18th or early 19th century”
—————-
With that being the case, your horizontally written statement of 6 ÷ 3 ÷ 4 is exactly the same as 6/3/4. And as you point out, that statement is ambiguous. Therefore, parentheses would have to be installed to indicate whether it was (6/3)/4 or 6/(3/4).
also from Wikipedia’s “Slash” page:
“Nowadays fractions, unlike inline division, are often given using smaller numbers, superscript, and subscript (e.g., 23⁄43).”
So if your statement was written with the denominator indicated by the superscript and subscript fraction, as 6 ÷ ¾ or as 6 / ¾ , then the numerator & denominator would be clear.
I prefer Wolfram to Wikipedia. He explains how the solidus is interpreted in written notation as opposed to how it works in computer language without parentheses. The main issue in the end in my mind is why are people pushing poorly written, ambiguous expressions to try to prove a point about PEMDAS? I always come back to to the fact that most formal equations for real-world values are written unambiguously, the whole PEMDAS debate is mostly irrelevant. https://www.wolframalpha.com/input?i=solidus&assumption=%7B%22C%22%2C+%22solidus%22%7D+-%3E+%7B%22MathWorld%22%7D
Let me just say that I am thoroughly enjoying our conversation on this subject, even though we slightly disagree on some points.
I actually don’t agree with your assertion that statements like 8 ÷ 2(2 + 2) are “poorly written.” It’s as clear as day, once one realizes that it’s the monomial division of 2x divided by 2x, with x = (2 + 2). The statement has a quotient of 1 — since anything other than zero which is divided by itself equals 1.
As for the Wolfram explanation of, “He explains how the solidus is interpreted in written notation as opposed to how it works in computer language without parentheses,” that just proves my point about some computer programmers having been absent on the day in 9th grade Basic Algebra class, when the teacher covered what a “term” is & how to divide one monomial by another monomial. Other programmers seem to have been present that day & wrote their calculator programs to account for that kind of monomial division — which is clearly understood sans additional parentheses.
What is interesting is that, since Wolfram uses a language model, if you type in “Eight divided by twice the sum of two plus two,” it returns the answer 1, but if you change the “twice” to “two times,” it returns the answer 16. So he does seem to have some of that built in, but when it keys on the word “times,” there are no other contextual clues to interpret that as anything but simple multiplication. It works for “thrice” as well, but once you get to “fourfold” and beyond, it doesn’t work. I don’t think I tried “quadruple.”
The fact that Wolfram’s online calculator (and others) yields two different answers to the same division statement with numerical input vs. language input, points to the computer programmer not fully understanding & accounting for the underlying concept of dividing one monomial by another monomial.
Implied multiplication by juxtaposition means that the viewer of the mathematical statement is looking at a monomial — all you have to do is replace whatever is in parentheses with a variable such as “x,” to see it clearly.
What some programmers have failed to recognize is that a monomial such as “2x” has, by definition, a single value which is the PRODUCT of the coefficient multiplied by a variable or variables. Put another way, the monomial “2x” holds the total value of two “x’s,” which can also be written as…
2x = [ x + x ]
It’s not a computer language problem — it’s a computer programmer problem. Some computer programmers did get it 100% right & wrote their calculator program to account for how to properly execute monomial division.
The people who insist that PEMDAS is the ONLY way to correctly calculate 8 ÷ 2(2 + 2), should stop depending on online calculators to do their thinking for them. Instead, they should start using their own brain to reason out what it is that they’re actually looking at. In this case, what they’re looking at is one monomial being divided by another monomial:
I really do appreciate your feedback. Considering others’ feedback always helps me sharpen my own thinking and leads me to more concise ways to express my thoughts and arguments. When I say “language,” what I’m really talking about is linguistics. Merriam-Webster defines it as “the study of human speech including the units, nature, structure, and modification of language.” I take “speech” to mean the written word as well as the spoken word, especially since as a preacher I’ve gotten into the habit of writing out my sermons so I can make more intentional use of my language as opposed to speaking extemporaneously. And in the context of this article, I don’t just mean words alone, but any symbols or figures that we use to communicate, calculate, or cantillate (how’s THAT for an alliteration!): numbers, punctuation, “character” words (e.g., ampersand, &), mathematical and scientific symbols, proofreading symbols, and even music notation.
All of these elements of language, and linguistics more broadly, have their place in their appropriate contexts, and they are subject to their own respective set of rules for putting them together in a coherent form that communicates the message and meaning we intend subject to the rules and conventions of their respective contexts. When someone composes a musical score, the main melody or tune is subject to certain patterns that follow the chords that underlie the melody. If the tune doesn’t match the chords, it sounds, well, discordant. The notes of the melody, harmony, or even a descant are not strictly random. They typically have some relationship with the chord, and often playing a note that doesn’t exactly fit the chord prefigures a change in the chord or even a change in the key signature. Intentional discordancy is not without significance either, as it could communicate chaos or irrationality.
When we write a sentence, we generally expect a subject and verb to be close together and to arrange direct and indirect objects appropriately with any modifiers or prepositions, and so forth. For example, consider the difference between the three sentences, which have the exact same words.
1. I eat fish only on Friday.
2. I eat only fish on Friday.
3. I only eat fish on Friday.
Sentence #1 is truly ambiguous, because the placement of “only” can be taken either way. Is it “Fish is the only thing I eat on Friday” (akin to Sentence #2) or “Friday is the only day I eat fish” (akin to Sentence #3)? Does that sound familiar in the context of this post? More on that in a bit.
In the blog post, I make reference to the relationship between the definite article, noun, and adjective in a Greek adjectival phrase. The position of (or absence of) the definite article impacts how the phrase can be interpreted. I’ll use transliterated words to demonstrate.
1. kalos logos [beautiful word]
2. ho kalos logos OR logos ho kalos [the beautiful word]
3. ho logos kalos OR kalos ho logos [the word is beautiful]
In Greek, Phrase #1, which has no definite article (the indefinite article “a” can fairly be implied absent other contextual clues), would be considered ambiguous by itself. We would need contextual clues to know whether it means “a beautiful word” or “a word is beautiful.” (Greeks do not have to use a form of the copulative verb “to be” if that is the only verb in the sentence.) In Phrase #2, the definite article precedes the adjective, which means the adjective is attributive, that is, it directly modifies the noun. It doesn’t matter if the noun is first or last; it’s attributive either way. Phrase #3 has a predicate construction. This means that the noun is the subject of a sentence, and the adjective would come after the verb in that sentence. In this case, it typically doesn’t matter where the adjective is, although there may be a nuanced difference one way or the other.
Given those three examples (music, English adverb placement, and Greek definite article placement), I think anyone who’s reading this is starting to see the bigger picture of how linguistics influences mathematics as well, especially in the context of the expression at hand. So let me use the expression in the same way I used the sample phrases above:
A. 8 ÷ 2(2 + 2) = 1 (in Dee’s and my worldview) or 16 (in the competing worldview)
B. (8 ÷ 2)(2 + 2) = 16 (in both worldviews)
C. 8 ÷ (2(2 + 2)) = 1 (in both worldviews)
Expression A seems unambiguous form the perspective of one’s worldview then. But are both worldviews equally valid? We can make arguments from our respective worldviews to try to convince others, but it is very difficult to convince one to change their worldview without a powerful defining event that shakes their worldview to the core. Otherwise, we’re comfortable with our ways. I happen to think that several of the arguments I’ve made to support my worldview are quite devastating to the competing worldview, but alas! there has been very little evidence of any change of heart among them.
Just like the position of adverbs and definite articles, so then is the generous use of parentheses needed to clearly avoid the ambiguity of the given expression. But let me make yet another appeal here for the case that the given expression, in light of my demonstration here, is not really ambiguous at all. The juxtaposition of the 2 to (2 + 2) is akin to Phrase #2 in my Greek examples above. The attachment between the two places them in an attributive relationship (the 2 is the definite article; the (2 + 2) is the adjective). The 2 directly modifies the (2 + 2) by telling us how many of them we need to divide by and keeps the monomial on one side of obelus WITHOUT an extant multiplication sign. In other words, it isn’t separated from its cofactor by the “action” of the obelus. There is no need for the extant multiplication sign because the relationship is clearly defined. If one were to place a multiplication sign between the 2 and (2 + 2), that would emphasize that the 2 and (2 + 2) are NOT cofactors and sever the relationship between them. This would make the expression like Greek Phrase #3 above, where the modifier is divorced from the modified and dragged kicking and screaming all alone into the action of the obelus. That which appeared to modify the (2 + 2) now modifies the 8. The implications of the expression change by adding the multiplication sign. Additionally, in the case of Greek Phrase #3, if we would add the implied copulative verb where it is not technically needed, that would also place emphasis on the verb and suggest a more nuanced meaning. (This also happens with Greek verbs; most Greek verb forms have an ending that tells you what “person” [1st, 2nd, 3rd, or I/we; you/you; he, she, it/they] is the subject of the verb. If there is no subject accompanying the verb, the corresponding pronoun is implied [“He eats”]. If a Greek pronoun is used as the subject, that implies emphasis [“He himself eats”],)
This may seem kind of heady to some, but I hope I’ve made my position a little easier to understand. My worldview and what I consider the strength of my arguments here and elsewhere, along with a ton of historical evidence, do convince me that the given expression is unambiguous and has no need of extra parentheses to understand it. For those who think writing ambiguous expressions is somehow educational and instructive when you know there are those who see through your ruse, I declare that you have met your match in me. Game over. Checkmate!
Thank you for enlightening me about grammatical construction in the Greek language — fascinating. At one time, I did speak & read Hebrew, which seems like it might have some similarities, but that was quite a while ago — I would need a refresher course!
I totally see what you’re saying regarding the syntax issue, with relation to spoken or written language (words) & I agree with you on that score.
On the subject of mathematical expressions akin to 8 ÷ 2(2 + 2), I would like to share this word problem example of the reason that the coefficient cannot be detached from the variable (factor):
We’ll go back to the original internet example I first encountered in 2011:
48 ÷ 2(9 + 3)
Let’s say I own a diner, and just as the breakfast special was about to end, 2 different groups of a dozen people each, walked in & were seated at separate tables. All of those customers ordered eggs. I look in the restaurant’s refrigerator & see that I have 4 dozen eggs left. If each customer receives an equal number of eggs, how many eggs does each customer get?
4 dozen eggs divided by 2 dozen customers
4 dozen ÷ 2 dozen
(also written as: 4 dozen / 2 dozen)
Viewing it in that context, it is apparent that “4 dozen” is a single quantity of eggs (48 eggs) & “2 dozen” is the single quantity of customers (24 customers). In other words, the 4 in “4 dozen” & the 2 in “2 dozen” cannot be separated from the factor (“dozen”) and used in another operation in the statement.
That’s a great example. I’ve created similar ones for other problems. I majored in Hebrew in seminary, but much of my work lately has been in Greek, as is indicated by the name of the blog. Thank you for reading!
I deliberately chose to use the word “dozen” because we are all accustomed to buying eggs in that “unit,” which is a box of 12 eggs. It’s easy to understand that when there are 4 full standard packages of eggs, there are 48 eggs in all. That mental image of “4 dozen eggs,” highlights the reason that the coefficient of 4 cannot be separated from its factor of “dozen.” Since multiplication is just a fast way of doing addition, the value of the term “4 dozen” is actually…
1 dozen + 1 dozen + 1 dozen + 1 dozen
Also, everyone can easily picture two individual groups of a dozen people, with each group seated as a “unit” of 12 customers at two separate tables in a restaurant, understanding that there are 24 total customers now seated in the restaurant, who are all ordering eggs for breakfast.
The use of “dozen” makes it easy to understand that…
4 dozen divided by 2 dozen = 2
…which can be numerically calculated as…
4(12) / 2(12) =
48 / 24 = 2
…or calculated by canceling out the like factor of “dozen,” leaving the statement as…
4 / 2
…which, of course, also equals 2.
And if the word “dozen” is replaced with a variable such as “x,” then the statement is…
4x / 2x
…which is one monomial being divided by another monomial
[ with x=(9+3) or x=(9+3) }
Implied multiplication by juxtaposition indicates that a term is a monomial — never needing parentheses around it, any more than “4 dozen” needs to be completely encased in a set of parentheses to be understood as one term with a single value. Therefore, the “4” in the term “4 dozen,” or the “2” in the term “2 dozen” cannot be detached and used in some other operation in the statement before calculating the total value of the monomial, first.
Yes, we are in agreement — words like “dozen” definitely clarify what quantities in the statement are actually being divided.
It’s nice to find someone else who is using some brain power to reason out what the statement means, rather than depending on some online calculator to do the thinking, or using rote memorization of a concept which does not fully comprehend one monomial being divided by another monomial (i.e. The Order of Operations as PEMDAS).
Did you see my latest post on in this? I combined one of my comments back to you with another I’d posted FB a few days earlier. I posted it on Saturday.
The issue that could arise with using language such as “twice” or “thrice,” is that it can still be seen as indicating multiplication, rather than seen as repeated additions (which is what multiplication actually is, in its most basic form). In other words, “twice” or “thrice” will make some people insert an explicit multiplication sign between the coefficient & the factor, thus making it appear that the division should be done with the coefficient first, before multiplying the factor. The proposition of a quantity of eggs being divided by a quantity of diner customers might be a better illustration of one monomial being divided by another monomial — most people can easily conceive of 4 dozen eggs being evenly portioned amongst 2 dozen people as…
(12 + 12 + 12 + 12) ÷ (12 + 12)
…which is…
48 ÷ 24
…which equals 2 eggs per diner customer.
The coefficient of “4” in “4 dozen” & the coefficient of “2” in “2 dozen actually “disappear” when the indicated number of additions is written out.
“Dozen” could lead to multiplication as well, because it’s a quantity by itself, even though it “cancels out” in your breakfast example. But “twice,” “thrice,” etc. are not strictly quantities by themselves, and this is where the linguistic aspect comes into play. “Twice” is an adverb. We can’t just say “8 divided by twice.” “Twice” needs more information to make a complete adverbial phrase that should be treated as a unit: “twice the sum of two plus two.” Mentally we would still do that multiplication, or recognize that when we say it like that, we should place a vinculum over or parentheses around 2(2 + 2) because the adverbial phrase is a unit.
Yes, it’s possible that “dozen” might lead to multiplication, as well. My experience with using the word “dozen,” though, is that people can easily visualize what “4 dozen” eggs looks like, on a shelf in the refrigerator — they see it as 4 “units” of 12 eggs each, as…
1 dozen + 1 dozen + 1 dozen + 1 dozen
rather than seeing it as…
4 * 12
It’s a subtle difference in how people view it — but it drives home the point that the coefficient (in this case, “4”) cannot be “ripped away” from the factor (“dozen”) & used in some other operation in a larger mathematical statement. The coefficient actually “disappears” when the indicated additions are written out. In the case of “4 dozen,” the coefficient of “4” doesn’t actually exist as a number unto itself — it just tells you how many times to add a quantity to itself, in much the same way that an exponent tells you how many times to multiply the base quantity by itself.
I don’t know if this has already been asked (too many comments to read through).
You stated:
Thus 4m2 ÷ 2m = 4m2/2m, not (4m2/2)m.”
This equation (original to the authors’ text) has the same basic form of Expression 1, with the only difference being all real numbers are used in Expression 1. I’m guessing that all of you agree that the expression to the immediate right of the equal sign in the example above is the correct way to interpret the expression on the Left. And of course, the expression on the right simplifies down to simply 2m.
Please justify how the expression simplifies to 2m ‘using all real numbers’?
Do ‘all real numbers’ not include ZERO?And last I heard, 0/0 is undefined.
Comment by Andrew Wilhelm — April 1, 2024 @ 8:48 am
| Reply
Thank you for reading. Expression 1 is a specific example of an expression in a similar format to the one with the variable m. I’m saying Expression 1 uses real numbers instead of variables. Of course, the expression quoted from the source would be true for m = any nonzero real number. The main point is that the juxtaposed 2m is handled as a single, inseparable value, so the 2(2+2) must be treated as a single inseparable value as well.
Sunday Morning Greek Blog 
Fascinating Scott, thank you!
Comment by Blue Collar Theologian — April 29, 2023 @ 1:48 am |
Interesting connection to Greek.
Comment by Michael Wilson — April 29, 2023 @ 6:35 am |
I love analyzing fractals!
Comment by Scott Stocking — April 29, 2023 @ 8:08 am |
I’m tired and this seems like you nerd out (and that’s a good thing)
Comment by SLIMJIM — April 30, 2023 @ 9:17 am |
Oh, yeah. I can get pretty nerdy sometimes. I wrote a 25-page paper about the variant readings in Acts 2:42 involve whether there should be a comma, or the word “and,” or nothing.
Comment by Scott Stocking — April 30, 2023 @ 1:36 pm |
Thats so awesome
Comment by SLIMJIM — April 30, 2023 @ 2:15 pm
https://sundaymorninggreekblog.com/2011/06/26/the-nature-of-%e2%80%9cthe-fellowship%e2%80%9d-%ce%ba%ce%bf%ce%b9%ce%bd%cf%89%ce%bd%ce%b9%cc%81%ce%b1-koinonia-in-acts-242-excursus-on-acts-238/ Here is the link to the blog post that condenses my 25-page paper.
Comment by Scott Stocking — April 30, 2023 @ 3:14 pm
flour for 3 ppl when 2 ppl needed 6 cups of flour… So i need 5 less cups for more ppl?
The confusion with the equation comes in at the 6÷2(3) part… Universally you have to perform the operations in the parenthesis first…
But after that whether or not you perform the distributive property first depends on what is being asked by the creator of the question…
Since 6÷2(3) is literally the same as 6÷2*3. If I’m asking how many cups of flour do i need to add for 3 ppl instead of 2… Then I’m looking for you to use PEMDAS and go left to right when you see any numbers outside parenthesis
But if i want to know how many more servings 3 cups of flour will give me… Then i would use the distributive property… Which would tell me that 3 cups of flour will add 1 more serving…
In that case 6÷2(3) gives me 1
The problem with the equation is it’s poorly written. It’s unambiguous. It assumes you know what the author is asking. But we don’t. The correct answer all depends on what is being asked by the equation.
Comment by supamand — June 5, 2023 @ 3:31 am
I’m not following how you’re assigning the units to each form of the expression. Would you mind mapping that out for me each way? It seems you’re confusing units.
Comment by Scott Stocking — June 5, 2023 @ 7:24 am
I think I see what you’re saying here, but you don’t need to make the equation so complicated in this case. If 6 cups of flour are necessary for every 2 persons, then you have 6 cups/2 persons, or 3 cups/person. Then if you want to know how many cups of flour for three people, just multiply 3 cups/person by 3 persons and get 9 cups. By defining the units that way, you are correct in treating the 6/2 separately, and as such, that should be in parentheses since that is the ratio you need to determine how much flour for however many people, which is my point. If you use units to define the values in the expression, then it makes sense to organize the expression the way it fits the units.
I agree the question is ambiguous IF you just focus on operations and NOT on the linguistic clues in the syntax of the expression. I’m keen to the linguistics of the expression, partly because of my historical experience with similarly formatted expressions and partly because I can see the connection between the construction of implicit multiplication and the implicit multiplication of exponents. What I and others are saying (Stephen Worlfram is one who I think would agree with me) is that there’s more to PEMDAS than just the operations. There’s another level that must be discerned, and that we must teach people to discern, if there’s ever going to be any agreement. Still, the best way to avoid ambiguity is to group accordingly using parentheses.
Then there’s the whole issue of established formulas, as in the following example:
Jerry’s Custom Jars has an intelligent robotic machine that can apply a label or ribbon of any given height around the circumference of a round mason jar and cut it exactly so it fits on the jar with no gaps or overlaps. They have an order for 35 mason jars with a simple yellow ribbon completely around the middle of each jar with no gaps or overlaps for a local VFW dinner honoring veterans. The jars will be centerpieces at the tables. How many jars can Jerry completely wrap with 500″ of ribbon for jars with a 2″ radius in the middle?
I would write the equation as
500″ ÷ 2π(2)”/jar,
because I have an instinctive, historical understanding that 2πr is the circumference of a circle, so why would I need to bother putting parentheses around that, given that I know it represents a single unit or distinct formula? But because some people focus solely on the operation and not the syntax or linguistics of the formula or aren’t smart enough to recognize the circumference formula, I’m expected to apply parentheses. I would to make it unambiguous to those with a limited understanding of PEMDAS, but I wouldn’t need it myself.
So the true answer to the question is 39 jars with some ribbon left over. If someone works that from a perspective of operations ONLY and doesn’t discern the circumference formula, they’d get an answer of 1570 jars, which is clearly absurd given the parameters of the expression. Those of us who see it differently are unwittingly biased against the long history we have of seeing the deeper principles behind PEMDAS. Thank you for reading.
Comment by Scott Stocking — June 5, 2023 @ 5:12 pm
Here is the fractals post I had for Dr. Jason Lisle’s YouTube video on it.
Comment by koineroad — May 5, 2023 @ 12:47 pm |
You are truly an idiot. PEMDAS Is and has always been sufficient. The answer is and always has been 16. You probably failed a test in college and it pissed you off so bad you thought you would get on the Internet and lie to people and convince them that you’re not an idiot….. but you are. DUMBASS
Comment by Jason — June 27, 2023 @ 2:11 pm |
Jason, thank you for your feedback. I decided to approve your unprofessional and vile comment to showcase that many people like you are more emotional about PEMDAS than intellectual. So thank you for modeling that and making your position look extremely weak juxtaposed to my intellectual analysis of the issue. If you or anyone else can’t behave yourselves more professionally and politely in future comments, I’ll think twice about approving them. FYI, I was a straight-A student in math throughout junior high and high school, and I passed my two semesters of college calc and one semester of differential equations before switching majors. So I know math, and I know how I solved problems. If you want to offer a more constructive analysis of my position like Frank did below, I’d be happy to respond to that. Otherwise, I’ll let your comment serve as a warning of how not to behave in my blog. Peace to you.
Comment by Scott Stocking — June 27, 2023 @ 6:12 pm
[…] If you like this article, you may also like https://sundaymorninggreekblog.com/2023/04/28/8-%c3%b7-22-2-1-why-pemdas-alone-is-not-enough/ […]
Pingback by Toward an “Active” PEMDAS: Strengthening Its Theoretical Foundation | Sunday Morning Greek Blog — June 1, 2023 @ 10:00 pm |
Great conversation. I’m “Team Frank”.
Comment by MB — July 26, 2023 @ 7:20 am |
Thank you for weighing in. Please check out WolframAlpha’s definition of “solidus,” which documents that my way is taught in most textbooks. https://www.wolframalpha.com/input?i=solidus&assumption=%7B%22C%22%2C+%22solidus%22%7D+-%3E+%7B%22MathWorld%22%7D
Comment by Scott Stocking — July 26, 2023 @ 7:25 am
Expressions 1-8 are all fine. You start losing it on expression 9 though. While 2(2+2) is in fact 8, that is not the equation. In expression 7 you have a(b+c) = ab + bc which is true, but the a in this equation is 8/2, not 2. So you must multiply 8/2 by (2+2), giving an equivalent of 8/2*2 + 8/2*2, which is 8 + 8, which is indeed 16. There is nothing implied or anything about the denominator 2 and the sum (2+2). 2(2+2) is just a normal multiplication 2 times (2+2), so the term to multiply using the distributive property is 8/2. The only way to group the 2 to the 2+2 would be to use parenthesis, 8/(2(2+2)), this would equal 1.
Expression 11. 8/2m can be interpreted as either (8/2)m or 8/(2m). How can we know which it is? There has to be a way to distinguish ambiguities. Enter PEMDAS! We have an order of operations that we use to distinguish in which order the operations occur. In this case it is division and multiplication, left to right. So, 8/2m is 8 divided by 2, multiplied by m, or 4m not 4/m.
Expression 12. Your equation, 8/(2+2)2, is not equivalent to 8/2(2+2). You cannot just move a denominator. Dividing by 2 is equivalent to multiplying by 1/2 or 0.5. Rewriting this equation would be 8 * 0.5(2+2), then you can move it, 8*(2+2)0.5. This reduces to (8*2 + 8*2)0.5, or (16 + 16)0.5, or (32)0.5, or 16.
Addendum (added 5/14/23). I’m not sure what you’re trying to say here. 8*(1/2*(2+2)) is indeed 16, you can either add first, 8(1/2(4)), or 8(2), or 16, or distribute first, 8((1/2*2)+(1/2*2)), or 8(1+1), or 8(2), or 16. Those both check out to 16. I’m not sure what you’re doing to divide (2+2) by 2?
Still Not Convinced? (Added weekend of 5/20/23). I still don’t know what you’re trying to say here. To get -1 you need to multiply the 1/2 by -2, 1/2*-2 = -1. So you must then multiply the other side also by -2. So 16 * -2 is indeed equal to -32. To go further with this point, you got an answer of -2. -2 * -2 is 4, not the 1 you claim.
Making It Real. Using your first example, v= d/t with d = 36 miles and t = 6(2 + 2 + 2) hours. Here, t is the quantity of 6(2+2+2), so that has to be in parenthesis. So, the equation is not 36/6(2+2+2), the equation is, 36/(6(2+2+2)), which is equal to 1, not 36. The same applies to your other equations here as well.
So in the end, 8/2(2+2) is indeed equal to 16, not 1.
Comment by Frank — June 27, 2023 @ 2:08 pm |
Thank you for your response, Frank. Your tone represents the kind of dialogue I would expect on this topic, unlike Jason’s vitriol above. I will answer your response starting from the top.
You claim the obelus only refers to the number immediately after it, and you use the obelus to turn 8 ÷ 2 into an improper fractional coefficient of (2 + 2). Yet the obelus, and the solidus (/) for that matter, in printed math both typically represent a linear replacement for the vinculum (fraction bar), and thus both would take on the “grouping” function of the vinculum, that is, everything that comes after the obelus, especially since it is the sole extant symbol in the expression, goes “after” (i.e., under) the vinculum. As such, there is NO logical or technical explanation of why the obelus would be treated differently by only treating the first number after it as the divisor. The very form of the obelus itself demonstrates that its horizontal line represents the vinculum, while the dot above represents the dividend before it, and the dot below represents the divisor that comes after it.
As such, by kidnapping the 2 from its partner (2 + 2), you’re changing the fundamental nature of the relationship between the implicitly multiplied elements. You’re forcing a divorce and pitting one against the other in an improper fraction (8/2). Even the term “improper” fraction implies that something untoward is going on with your methodology. What you’re doing to the expression is immoral and deceptive, if unwittingly so.
Note that here, I am using the terms “dividend” and “divisor” intentionally for the obelus. The expression is not yet in fractional form at this point, so it is technically wrong to refer to the elements as “numerator” and “denominator.” And on a side note, the entire expression is one “term,” because the addition sign is within parentheses, and what’s in the parentheses is subject to multiplication. You can’t break out just what comes after the addition sign as a separate term. So the argument that some make that there’s more than one term here is disproven.
Now the other issue at play here is the “sacred” connection of implied multiplication, especially in print form. WolframAlpha acknowledges this in its definition of the “solidus”:
“Whereas in many textbooks, “a/bc” is intended to denote a/(bc), taken literally or evaluated in a symbolic mathematics languages (sic) such as the Wolfram Language, it means (a/b) × c.”
So Wolfram Alpha supports the fact that many of us learned from textbooks (especially before calculators were the norm) that the answer to the expression is 1, that the product after the obelus is the divisor as Lennes wrote in 1917. However, when the calculator and computer languages came along, they were not programmed with the intuition and linguistic instinct that many of us learned in print (there are exceptions among and within the brands). When people started plugging these expressions into such devices, the devices had no regard for the sacred connection represented by implicit multiplication. The devices treated the ENTIRE expression as a linear equation without regard to parentheses, unless they were intentionally added. In contrast, our (those of us who get the answer 1) intuition tells us the parentheses are implied around the WHOLE product after the obelus. As a result, some of you chose to allow the calculator or computer to do your thinking for you, and you lost or never developed the instinctive connection represented by implicit multiplication. As such, the literal PEMDAS you preach is inferior to the methods we learned by developing our intuition because it derives from a corrupted form of artificial intelligence (AI). That answers your concern about Expression 11.
Now, back to the expression at hand. Given that the 2(2 + 2) represents a sacred bond that cannot and should not be broken, I’m not doing anything wrong with the “denominator” as you call it. So when you say “You cannot just move a denominator” with respect to Expression 12, you’re failing to recognize the sacred connection of implicit multiplication. I’m not moving a denominator because, as I demonstrated above and supported with older and newer academically sound sources, the whole product after the obelus is the divisor, or in your words, the denominator. I’ve merely commuted the elements of the denominator.
As for the addendum on 5/14, my point there was to show how the sacred connection of implied multiplication is changed by the way strict PEMDAS disciples work the problem. I’ve already demonstrated that above, so I don’t need to revisit it here. The only thing I would add is that, to make explicit what our instincts tell us, we would rewrite it:
8
———
2(2 + 2)
If no parentheses are needed for that, then why would we need them for the obelus? The vinculum is a vertical form of implied division, just as juxtaposition to parentheses is a form of implied multiplication. Of course, the expression with the vinculum would be reduced to a proper fraction, or in this case, a whole number, 1.
The point about the negative sign was only to illustrate that when a negative sign appears before parentheses without an express coefficient, the signs of the terms within the parentheses are supposed to change. That is, the coefficient is an implied -1 multiplied through the parentheses. Your strict PEMDAS would treat that the same as the 2 after the obelus and get a completely different answer than the expression implies.
Finally, as for the “Making It Real” section, my point there is that, since the expression has context and I know what the terms before and after stand for, I don’t need to put parentheses around the divisor/denominator because I know what it stands for. If I were plugging it into a “dumb” calculator, then yes, I would need to put parentheses around the divisor/denominator for the calculator to work according to my instincts (and isn’t that the way we should use a calculator?). But some of us can still do math without a calculator, and some of us can even do it with a slide rule, if you know what that is, so we don’t always need the calculator. The same would go for any such established formulas. Revolutions of a wheel = d/2πr is one such example.
The bottom line here is that calculators need the parentheses for clarity. As humans who have a certain instinctual and linguistic understanding of the expressions we see in a math problem, we don’t always need those. We recognize a certain form and solve accordingly. Our way of thinking is the highway, because it depends on human intelligence and insight. The strict view of PEMDAS promoted by some is a capitulation to the flawed AI of the computer age and seems to lack the kind of reflective analysis and critical thinking I’ve demonstrated here.
Peace to you, and again, thank you for your polite response.
Scott
Comment by Scott Stocking — June 27, 2023 @ 10:16 pm |
Thanks for your reply. I don’t agree with the name-calling above. If you don’t agree with someone doesn’t mean you need to start calling names. Sometimes people don’t agree, and that’s ok.
With that said, I don’t think we are going to see eye to eye, but I’ll try anyway.
The majority of what we are saying is, is 8/2(2+2) equal to 8/(2(2+2)) or (8/2)(2+2). You’re saying the former, I’m saying the latter. I say the latter as that follows the order of operations. 8 should divided 2 before multiplying because dividing comes before multiplying in the order of operations of this equation, left to right. Since there is no grouping on the 2(2+2) it should just go left to right.
From what I understand of your justification, what you’re saying is that because 2(2+2) is an implied multiplication that the 2(2+2) is part of the denominator.
Now to your reply.
First, the obelus and the solidus are the same thing. There is no difference, they both divide in the same way. 2÷2*2 and 2/2*2 are the same statement. However, as you have shown yourself, the vinculum is different as it does imply grouping. For example,
2
—-
2*2
This is not the same as 2/2*2. An equal statement to the above would be 2/(2*2). The vinculum implies that since the 2*2 is under the bar that it is part of the denominator. Why does this imply grouping, because it is actually under the bar.
Also, you said, “As such, there is NO logical or technical explanation of why the obelus would be treated differently by only treating the first number after it as the divisor.” So let me ask you, what is 2/4/2?
Is it, (2/4)/2, or is it 2/(4/2), is it 1/4 or is it 1? According to your definition above (“everything that comes after the obelus”) this would be 1, but according to the order of operations this would be 1/4. If you say that 2/4/2 is 1 then I think we’re done here and you’re probably just trolling at this point. However, if you say that it’s 1/4, then that goes against what you said in your last reply.
Second, implied multiplication is exactly what it says and nothing else, two values are implied that they multiply. 2(2) is the same as 2*2. Just because 2(2) is after a division 2/2(2) does not mean that 2(2) is in the denominator. It literally just means that it multiplies, 2/2*2. 2/2(2) and 2/2*2 are the same statements, equal to 2 not 1/2.
Third, changing the value to be a negative doesn’t change how the statement evaluates. -(2+2) is the same as -1(2+2), so 8/-1(2+2) still is 8 divided by -1, times the quantity 2+2. Again, to change the equation to get a -1, you must multiply by -2. 1/2 * -2 is equal to -1. Then to get the equation to solve you also need to multiply the other side by -2, so -2 * 16 is equal to -32. Where is this a problem? You also still failed to mention how your equation doesn’t equate when you multiply by the -2. -2 * -2 is 4, not the original 1 that you said this equation was.
Fourth, your “Making it real” section you were trying to imply that your v=d/t would look like 36/6(2+2+2) and then try to prove that it is incorrect, but your assumption was wrong because you have to input the value of t into the equation of v=d/t but an equivalent of v=d/t is v=(d)/(t) which would make the substituted equation v=(36)/(6(2+2+2)) which is 1 not 36 like you implied. Substituting without the parenthesis is incorrect and gives you the wrong answer.
Fifth, you just admitted that we need parenthesis for clarity, but since you are smart (“certain instinctual and linguistic understanding of the expressions”) you know what everyone always intended?
8/2(2+2) is intentionally ambiguous just so people like you and I would debate it, but, there is still one clear answer when you follow the order of operations and that is 16.
Thank you for your time.
Comment by Frank — June 29, 2023 @ 11:06 am
Yeah, we probably won’t see eye-to-eye, but I appreciate the discussion. My sticking point is the implicit multiplication. I view that as inseparable, like a prefix on a word or an inseparable prefix on a German verb. I was taught to recognize that as an inseparable term. So the parentheses aren’t really an issue. In my mind, I’m trained to put implied parentheses around it, if you will. Many of us who get the answer 1 do that instinctively. And with one extant operational sign, that seals it for me. It’s just a simple A/B problem for me at that point.
As for your 2/4/2 example, I would actually agree with you that it’s 1/4. There are two extant operational signs, so I’d follow OOO at that point, in the absence of parentheses. I would rewrite it as multiplicative inverses and multiply to prove its 1/4 either way: (2/1) * (1/4) * (1/2). Apply the associative property either way, and you get the same answer.
Peace to you, and thank you for the comment.
Comment by Scott Stocking — June 29, 2023 @ 7:18 pm
Are we starting to see your arguments crack?
If 2/4/2 is 1/4 then “everything that comes after the obelus” is an incorrect way to look at an equation. Let me try another question, what is -4x^2? Is it (-4*x)^2, or is it -4*(x^2)? Lets say x is 2, is it -8^2 or 64, or is it -4*4 or -16? If you say it’s -16 then you’d have a hard sell on why you think your previous claim is true. If you claim the answer is 64, I’ll just say you’re wrong.
I am a person who always kept my old textbooks in case anything ever needed to be looked up again. This is the first time I think I’ve had to look up something in my old college math textbook. There is no mention whatsoever of Implied Multiplication (Or Juxtaposition). I wonder why that may be, do you have any reason why the author of a college textbook would leave out such an important rule?
Comment by Frank — July 3, 2023 @ 2:31 pm
No, not really. Although I admit I’m still working some kinks out. When I said everything after the obelus, I was referring specifically to that kind of expression, with one extant sign.
Someone else pointed out to me the -4x^2 issue, and I have to admit that the exponent takes precedence in that, so it’s (-1)(4)(x^2).
That doesn’t really change my position on grouping, but a I need to qualify it a bit. Grouping takes precedence over LTR MDAS with extant operational signs. But I would provide an order of precedence for grouped expressions as well. I define grouped expressions as anything without an operational sign that suggests some sort of function upon or between the numbers. It’s a linguistic structure. So as in PEMDAS, parentheses first, then exponents and factorials, then implied or proximate multiplication.
I’m also tempted to define a fraction with a vinculum as implied division as well, different from signed division with an obelus or a solidus. Although it’s a convenient way to express division, especially when you have polynomial expressions, I don’t think it’s intended to be a universal substitute for another form of a division sign. It’s kind of complicated to explain. I’ve already said I don’t think it’s right to assume that in 8 ÷ 2(2 + 2), the 8 ÷ 2 is a fractional coefficient of the expression. First of all, it would be an improper fraction, so that’s bad form for an expression. Second, the obelus is a simple operational sign, so I don’t think it’s rational to think it only applies to part of an implicitly multiplied element. That seems to imply some sophistication for the sign.
Again, thank you for the discussion. This has been an exercise in “thinking out loud,” which is how the scientific method should work. Peace to you.
Comment by Scott Stocking — July 3, 2023 @ 4:47 pm
Thanks for your reply. I have (I think) one last question for you involving implied multiplication. I don’t think I’ve even really heard about it growing up all the way through college. I’ve seen it, but it’s always been “just multiply” the numbers. So my question is, how is it not just multiply, why do we have this extra rule where it doesn’t seem like one should exist?
I’ll even go out on somewhat of a simple proof.
If there is a number x, such that a(b+c) = x and a*(b+c) = x, for any value a, b and c, then the x must be the same value for both equations. Thus meaning that a(b+c) = a*(b+c) for all numbers a, b and c.
Now, if you can show me any number that disproves the above statement, i.e. find any numbers for a, b and c where those equations are not equal and I will bow down and call you a mathematical god. If you cannot, how can you possibly say that they are different and have different rules?
Comment by Frank — July 3, 2023 @ 7:12 pm
I don’t have any of my high school textbooks, and my college calculus textbook doesn’t even have an obelus in it as far as I can tell. I think what’s going on in my mind is that I’m used to seeing a standardly written division problem (especially one with variables) written with the vinculum as such:
9a^2
——
3a
which of course when written that way everyone agrees the answer is 3a. But when someone rewrites that with an obelus and doesn’t use parentheses to “group” the denominator, my mind still sees the same problem with the same answer: 3a. The same goes with the problem at hand. I’m accustomed to two standardized forms that aren’t used in the expression. First, if someone wants to say 8÷2 is the coefficient of the expression, that’s definitely NOT a standard way to write a fractional coefficient, especially if you don’t put parentheses around it. If someone wants me to think that, they’d have to write the expression as follows:
8
— (2 + 2)
2
But that creates a related problem, in that a coefficient would also be in its simplest form, so you have an improper fraction used as a coefficient where a whole number would do. The improper fraction would be okay if there’s no way to simplify it to a whole number, as in the equation for the volume of a sphere:
4
— πr^3
3
The second standard I would expect is that if you want two of the three values in the expression to be multiplied together, you wouldn’t make them the first and last values in the expression. You’d write the expression in such a way to avoid any confusion as to what you expected to be multiplied together:
8(2 + 2)
———
2
So whether intentionally or not, the expression is an extremely poor example of a mathematics problem, and people shouldn’t be using such a poorly written expression to promote something that they expect to be a standard way of working equations. That being said, I’m not inclined to give props to the people who try to push a poorly written expression as the basis for a standard. I’ll fight it tooth and nail if I have to, because it’s bad pedagogy.
One final thing: PEMDAS is for the most part applied to hypothetical, that is, non-real-world equations. A good example is the formula for a pitcher’s earned run average (ERA), taken from the MLB Web site: “The formula for finding ERA is: 9 x earned runs / innings pitched.” Let i = innings pitched and R = earned runs. We wouldn’t write that equation 9 ÷ i(R), right? In your way of thinking, you’d get the right answer, of course, but it doesn’t fit the logic of how the stat is formally calculated and what its significance is. The best way to write it is 9R/i or
9R
—
i
My point here is that most standardized formulas for things we encounter daily (whether we see them or not) are written in such a way that you don’t have to worry about PEMDAS. They’re simplified enough that the “order” is inherent in the formula. And if that formula happens to come after a division sign (as in calculating the number of revolutions of a wheel over a certain distance) instead of under a vinculum, most people will know to keep the formula intact and not start playing PEMDAS games with it.
Again, thank you for the dialogue and feedback. It helps me clarify and solidify my position even more. Peace to you.
Scott
Comment by Scott Stocking — July 6, 2023 @ 10:35 pm
I agree with almost everything you’ve said here.
9a^2
——
3a
Is 3a. I’ve mentioned that in my previous post that the vinculum does imply grouping because you can clearly see that the 3a is under the bar. However, the above equation is not equal to 9a^2/3a. This equation would be,
9a^2
—- a
3
Which becomes 3a^2*a, or 3a^3. Although, I also do agree that the equation itself is silly and should just be written 9a^2(a)/3, or 9a^3/3, or 3a^3, but I disagree that it’s incorrect. In mathamatics it is part of the journey to get to the reduced state, rarely do you start with a reduced function.
For example, this statement, a-b where b is a-1 is equal to 1. You could have this statement, or you could just call it 1, but 1=1 doesn’t really mean anything whereas a-b where b is a-1 is equal to 1 implies something about a and b. This could be extrapilated out to a-b where b is a-c is equal to c.
My point is, just because something seems silly doesn’t mean it’s incorrect. We have the order of operations only to tell us in which order the operations are calculated in.
It sounds like maybe you’re starting to agree that 8/2(2+2) is 16 which is,
8
—(2+2)
2
whereas
8
——
2(2+2)
is 1.
I agree that 8/2(2+2) could be written differently, 8(2+2)/2, or 4(2+2) but that’s not what was given to us, but all three of these statements are the same and the only way to get there is to use the order of operations.
So hopefully you can see where the people who get 8/2(2+2) = 16 come from, and I think that is where I’m going to leave it.
Good day.
Comment by Frank — July 13, 2023 @ 11:28 am
This is all silly. The only thing and debate here is the order of operations. You mention an order of operations that I was unaware of, two numbers next to each other, without an operation symbol gets precedence over left to right. I am not aware of sexual. I don’t think it’s correct, but if it is, you’re right (1)and if it’s not, you’re wrong (16).
The math is the math, and that does not change. Order of operations is a language syntax issue. It’s scarcely different than how the placement of a comma can change the meaning of sentence.
Finally, your velocity example is completely wrong. I am extremely confident that anytime a variable is represented by a letter, that variable must be calculated in full and not broken up into pieces as you have done, which would be completely insane and make algebra, completely unworkable in any sense.
Comment by Rich, fellow math nerd — June 29, 2023 @ 5:49 pm |
Thank you for your response. The implicit multiplication represented by the right side of the expression is akin to a prefixed word in a regular sentence, or in German, a verb with an inseparable prefix. If you strip the prefix from the word and put it in a different place in the sentence, you change the meaning of the sentence. Same goes for the coefficient here. The coefficient is not intended to be separated by OOO. OOO was never intended to alter those fundamental relationships.
If you think my velocity equation is wrong, then please suggest a different way to construct it. Put your keystrokes where your statements suggest.
Thank you for your response. Peace.
Comment by Scott Stocking — June 29, 2023 @ 6:36 pm |
Thank you for explaining it.
Comment by Erin Kastenschmidt — September 14, 2023 @ 8:37 pm |
Glad you enjoyed it. I need to update it, because I’ve found several discipline-specific style manuals that further support a juxtaposed or implied multiplication expression after a sign of division should be taken as the whole divisor.
Comment by Scott Stocking — September 14, 2023 @ 8:41 pm |
Thank you.
Very clear explanation.
My only suggestions are about terms and factors, that haven’t been mentioned much. Factors multiplied are single terms.
8 is a single term.
8/2 is 2 terms.
Euler, Elements of Algebra
Page 34, para 89.
“when we multiply both its terms, or its numerator and denominator,”
8/2(3) is 2 terms, as is 8/2(2+2).
storyofmathematics.com
Definition
A term is any single number or variable. An expression contains two or more terms. If two numbers or variables or both
are multiplied by each other, we consider them as a single term.
“0:51 So in this example, you have three terms.
0:56 The first term is 2 times 3.
0:59 The second term is just the number 4.
1:01 And the third term is 7 times y.”
Khan Academy Terms, Factors and Coefficients Transcript.
So the expression is the term 8, divided by the term 2(4) or 2(2+2), to give the
answer of 1.
Comment by Doug Hendrie — October 30, 2023 @ 1:29 pm |
Thank you, Doug. I appreciate the references you cited. Glad you enjoyed the read.
Scott
Comment by Scott Stocking — November 3, 2023 @ 3:10 pm |
You’re wrong and willfully ignorant. You have misapplied the Properties and Axioms of math and you’re too mathematically incompetent to realize it…
Let me explain the Distributive Property… Using a similar expression 6÷2(1+2)= 9 not 1
The Distributive Property is a PROPERTY of Multiplication NOT Parentheses and not Parenthetical Implicit Multiplication. As such it has the same priority as Multiplication and Multiplication does not have priority over Division.
The Distributive Property is congruent with the Order of Operations it doesn’t supercede the Order of Operations… The Order of Operations work because of the Properties and Axioms of math not in spite of them…
The Distributive Property when fully applied is an act of ELIMINATING the need for parentheses by drawing the TERMS inside the parentheses out not by drawing factors in… If you can’t draw a factor in and get the same result as drawing the TERMS inside the parentheses out then you haven’t applied the Distributive Property correctly…
The Distributive Property does NOT change or cease to exist because of parenthetical implicit multiplication…
The axiom a(b+c)= ab+ac however the variable “a” represents the TERM or TERM value i.e Monomial factor of the TERM outside the parentheses not just a numeral next to the parentheses. In this case a = 6÷2 OR 3. People just automatically assume that “a” is a single numeral…
6÷2(1+2)= 6÷2×1+6÷2×2 Distributive Property
Parentheses removed…
6÷(2(1+2))= 6÷(2×1+2×2) Distributive Property.
Inner parentheses REMOVED
This can be further demonstrated using the vinculum….
6
——(1+2)= 6÷2(1+2)= 9
2
6
———— = 6÷(2(1+2))= 1
2(1+2)
A vinculum (horizontal fraction bar) serves as a grouping symbol. Neither the obelus or solidus serve as grouping symbols. The vinculum groups operations within the denominator and when written in an inline infix notation extra parentheses are required to maintain the grouping of operations within the denominator….
________
2(1+2) = (2(1+2)) two grouping symbols each
That over bar (vinculum) is a grouping symbol
_______ _________
2(1+2) = 2×1+2×2
(2(1+2))= (2×1+2×2)
Note that when applying the Distributive Property one grouping symbol was REMOVED from each notation…
6. 6
———— = 6÷(2(1+2)) = 6÷(2×1+2×2) = ————–
2(1+2) 2×1+2×2
If you choose to Distribute the 2 into the parentheses by itself you have to do one of two things. Either take the division symbol with it, as division is right side Distributive or change the division to multiplication by the reciprocal…
÷2= ×0.5
So… 6÷2(1+2)= 6(1÷2+2÷2) still equals 9
Or… 6÷2(1+2)= 6(0.5×1+0.5×2) still equals 9
Variables can represent more than just a numeral and it’s important to understand that when you replace a variable with a constant value or a set of operations that represent a constant value that you apply grouping symbols where called for by the Order of Operations and the basic rules and principles of math… example 6÷a does not have parentheses BUT a= 2+4 so 6÷a = 6÷(2+4) not 6÷2+4. BUT if a=2×3 and we have a÷2 we can write 2×3÷2 because we evaluate Multiplication and Division equally from left to right…
a(b+c)… a=12÷3, b= 2×3, c= 2^2 we have…
12÷3(2×3+2^2) = 4(6+4)= 4(10)= 40
ab+ac =
12÷3×2×3+12÷3×2^2=
4×2×3+4×4=
8×3+16=
24+16=
40. <<< same answer
What most people don't understand is that you can't factor a denominator without maintaining all operations of that factorization WITHIN a grouping symbol. You can factor out LIKE TERMS from an expanded expression… 6÷2×1+6÷2×2+6÷2×3^2-6÷2×4=
6÷2(1+2+3^2-4) as the LIKE TERM 6÷2 was factored out of the expanded expression.
I hope this helps you understand the issue a little better….
You're confusing and conflating an Algebraic Convention given to coefficients and variables that are directly prefixed and form a composite quantity by this convention to Parenthetical Implicit Multiplication. They are not the same thing…
a/bc = a/(bc) by Algebraic Convention
a/b(c)= ac/b by the Commutative and Distributive Properties…
a/b(c) = a×b⁻¹(c) NOW the expression is Associative…
Wolfram Alpha supports the fact that 8÷2(2+2)= 8÷2×2+8÷2×2 …. Wolfram supports the fact that 8÷2(2+2)= (2+2)8÷2. Wolfram supports the fact that 8÷2(2+2)= 8×2⁻¹(2+2) AND Wolframalpha supports the fact that…
8
——-(2+2) = 8÷2(2+2)
2
Wolframalpha supports 16
Mathway supports 16
Mathpapa supports 16
Tiger Algebra supports 16
Google supports 16
Symbolab supports 16
Desmos supports 16
Photomath supports 16
Comment by Richard — November 1, 2023 @ 9:21 pm |
Mr. Smith: I will approve your post because I believe in the First Amendment, but I suspect at least part of your post was plagiarized from Facebook. For now, I’ll give you the benefit of the doubt that you’re the Richard Smith from the Math Challenge page on Facebook where I documented that at least a portion of your response was copied from and pasted here. (Yes, it’s plagiarism if you quote yourself from another previously published source and don’t acknowledge the original source.)
Before I get into responding to your short-sighted view of the Distributive Property, I want to cite some historical evidence that the ambiguity of such expressions as are promulgated on social media was called out nearly 100 years ago in Florian Cajori’s seminal work, A History of Mathematical Notation (1928), section 242, on the “Order of Operations.”
“242 Order of Operations in terms containing both ÷ and ×. –If an arithmetical or algebraical term contains ÷ and ×, there is at present no agreement as to which sign shall be used first. ‘It is best to avoid such expressions.’ [M. A. Bailey, American Mental Arithmetic (New York, 1892), p. 41)].” After citing three textbook examples of the day that recommend different ways of addressing the issue (i.e., in order as they occur or multiplication first; one textbook says to put the divided numbers in parentheses), he notes that “an English committee [“The Report of the Committee on the Teaching of Arithmetic in Public Schools” Mathematical Gazette, Vol. VII (1917), p. 238. See also p. 296] recommends the use of brackets to avoid ambiguity in such cases.” Because such ambiguity still exists today, as evidenced by the rightful detractors from your position, your failure to avoid this historically documented ambiguity of the expression leaves the answer at best open to interpretation, and at worst, demonstrates you’re just trolling people with your convoluted explanation. I’m pretty sure there are very few fifth graders today who could speak in terms of “monomials” and your nuanced definition of “term.”
Note here that this applies to expressions where the multiplication symbol is extant. In section 238, in discussing Leibniz’ Notation, he equates
a:(bc) with a:bc
(the colon being the symbol of division for Leibniz), so it’s clear that, in a division problem where the denominator consists of two implicitly multiplied factors, such implicitly multiplied factors are considered the ENTIRE denominator, with or without parentheses around the factors, and has been that way for over 100 years. I have already demonstrated in Wolfram’s description of Solidus that such occurrences in standardized formulas that do not have the parentheses in written form need to typically be enclosed in parentheses when input into a calculator or online problem-solving format.
The less convoluted way to look at this, and one much easier for a fifth grader to understand, is that given there is only one operational sign, the obelus, what comes before the obelus is the dividend, and the entire expression after the obelus is the divisor. Without clarifying parentheses, the problem is open to such interpretation, because the ENTIRE expression is itself one “term” as you define it. You can’t on the one hand say “parentheses first” (which would bypass the distributive property) and then claim the distributive property somehow gives you two terms to work with. Additionally, you can’t, on the one hand, claim the vinculum is a grouping symbol, and then not group underneath it everything that comes after the obelus! That’s a glaring contradiction in what you attempted to demonstrate above. If the obelus and vinculum have different functions as you claim (one doesn’t group; the other does; please cite a credible primary source for such an assertion, because I don’t believe that assertion to be true), then replacing one with the other is NOT making an equivalent substitution, and you’re changing the nature of the expression in doing so. With obelus, it’s clear what is the dividend and what is the divisor. When you insert the vinculum, you split the divisor of the original expression!
Finally (for the time being, anyway), implicit multiplication should be considered in the same category with other expressions of implicit multiplication, namely the “exponents” step. What is an exponent if not a representation of implicit multiplication (3^3 = (3)(3)(3)); a factorial is also an expression of implicit multiplication (3!=(3)(2)(1)). So why shouldn’t 2(2+2) or, in your plagiarized example, 2(1+2), be considered at the exponent level as well? No one has ever given a decent explanation of why that shouldn’t be the case.
I’ll deal with your misguided conception of the distributive property at another time. Suffice it for now to say I’ve never ever seen the distributive property defined in such a way as to include an obelus. It’s only about multiplication and addition. Again, cite a credible, primary source if you can find one.
Peace,
Scott
Comment by Scott Stocking — November 3, 2023 @ 8:41 am |
Let me address your misconception of the Distributive Property, Mr. Smith. If the distributive property is defined in terms of a(b + c) = ab + ac, then if you’re substituting something into the variable that isn’t a single integer or a different single-letter variable, it would be customary to place the substituted expression in parentheses to indicate without a doubt the entire expression replaces the variable. In the case of your example, then, it’s not clear, and perhaps intentionally and maliciously ambiguous, to not put 6 ÷ 2 in parentheses to stand for the single variable “a” of the generic description of the Distributive property. It leaves the question open, as I stated above, about the role of the divisor. If your side wants to insist that the 2(1 + 2) must have parentheses around it to be understood as the whole divisor, then our side is equally justified to insist you must put parentheses around the 6 ÷ 2 to indicate it stands for a single entity before the parentheses. All your pedantic babbling about monomials and terms is irrelevant at that point. If you want to be clear and be clearly understood, use parentheses. If you want to be an Internet troll, continue on as is.
If we look at it as a simple dividend ÷ divisor problem, then we would say the divisor is in the form of the Distributive Property, which is then divided into the dividend (6), thus obtaining the answer of 1. But since you fail to follow standard procedure of placing parentheses around the 6 ÷ 2, you cannot claim unequivocally that that expression is the coefficient of what is in the parentheses. It’s a simple as that really, and it shows why your convoluted explanation lacks any theoretical validity. Your conception of the problem is misguided and just plain wrong. The answer to the example you use and the expression I discuss is 1 in both instances, indisputably and indubitably.
Let me also say something about the linguistics of the expression. Someone may look at your example and say “Six divided by twice the sum of (1 + 2)” and come up with the correct answer of 1. This is, in fact, how Wolfram solves the text version of the problem, 6/(2 (1 + 2)). It places the entire divisor (without parentheses) under the vinculum in graphic form. The form you see here with the solidus and extra parentheses is how it’s copied into WordPress (I copied the answer with the vinculum and directly pasted it here without any editorial intervention), because the Comments section here doesn’t handle graphics. So again, this is one more nail in the coffin of your argument.
Peace to you,
Scott
Comment by Scott Stocking — November 3, 2023 @ 10:09 am |
Hi Richard —
You used Mathway’s calculator & Symbo Lab’s calculator as examples to prove your stance that the answer is 16. Here’s a link to Mathway’s calculator:
https://www.mathway.com/Algebra
and to Symbo Lab’s calculator:
https://www.symbolab.com/solver/step-by-step/2x%5Cdiv2x?or=input
Type in 2x÷2x into the input box. You’ll get an answer of 1, even though a division sign (obelus) is used & the statement has no parentheses anywhere.
Now, in that statement of “2x÷2x,” x = (2+2). So let’s plug in the value of “x” in that statement:
2(2+2)÷2(2+2)
Now run that input through Mathway’s calculator & through Symbo Lab’s calculator. It delivers an answer of 16 in both cases.
Mathematically, can you please explain how that statement has a different value from “2x÷2x,” when all you did was plug in the value of “x”?
Comment by Dee — February 22, 2024 @ 11:25 am |
You ROCK, Dee! Thank you for all the examples you’ve provided! Peace.
Comment by Scott Stocking — February 22, 2024 @ 7:00 pm
You ROCK, too, Scott! Thanks for writing the original column — lots of good stuff there.
Let me know if you’re ever planning to come to NYC — it would be fun to meet up (you have my email via WordPress).
— Dee
Comment by Dee — February 23, 2024 @ 8:51 am
My wife and I would love to go to NYC at some point.
Comment by Scott Stocking — February 23, 2024 @ 1:37 pm
Fantastic! Let me know when & we’ll arrange to meet up. Lots of good restaurants in NYC, so bring your appetite!
Comment by Dee — February 23, 2024 @ 2:20 pm
Scott writes: “I do not deny the importance of PEMDAS, but the reality of the problem is, any basic math problem like this can only have one correct answer. It’s not and never a matter of personal interpretation. Otherwise, the foundations of mathematics would crumble into oblivion, and not even Common Core could save us (not that it ever did anyone any good). This is math; it doesn’t care about and is never affected by your feelings about it.”
PEMDAS alone can’t save us and never could. Think of PEMDAS as a basic set of training wheels and nothing more. It is a ruleset put into place at an early time in our lives when math simplicity was key, not completeness. PEMDAS is nothing more than a basic math helper tool, but one that has an incomplete understanding of mathematics.
Mathematics, like any computer language, has syntax rules. If that syntax isn’t absolutely clear, the math problem can’t be solved. No, not even one answer. Like failing the syntax check when writing C language, failing a syntax check when writing a math problem results in the same exact dilemma, there can be no answer because of a failed syntax check.
However, with 8 ÷ 2(2 + 2), people seem to wish to try their hand at attempting to solve this syntactically problematic equation anyway. There is only ONE logical way to solve this problem IF you choose to ignore the glaring RED syntax error and forge ahead anyway. PEMDAS was devised for use by students BEFORE they have an understanding of implied multiplication. Once multiplication by juxtaposition (implied multiplication) becomes understood during Algebra, a different more advanced ruleset (aka. style guide) comes into play. What ruleset is that?
That ruleset is that 8 ÷ 2(2 + 2) is the same as 8 / 2(2 + 2). Most algebra texts have style guides that state that any equation delimited by a slash (/) automatically moves what’s left of the slash into the numerator and what’s right of the slash into the denominator… all of it (unless there’s an operator that stops this). Thus, 8/2(2+2) is the same as the fraction…
8
——-
2(2+2)
where 8/2*(2+2) (using the explicit * operator) is understood as
8
– * (2+2)
2
The above proves why using the correct syntax is critical. The * halts the Algebra rule which applies to /.
PEMDAS doesn’t agree with Algebra text rules when an equation includes juxtaposition. That disagreement is mostly because PEMDAS isn’t even aware of multiplication by juxtaposition.
Thus, the assumed solving logic for the equation 8 ÷ 2(2 + 2) is to apply advanced math rules ONLY. Why? Simple, because it contains multiplication by juxtaposition, an advanced math Algebra concept. Thus, logic dictates that the basic grade school learned PEMDAS rules can’t and don’t apply. Thus, any answer conceived using PEMDAS is, indeed, incorrect because PEMDAS has no knowledge of multiplication by juxtaposition.
With all of this said, grade school math instructors WILL teach their students using PEMDAS and WILL interpret and thus solve this problem as though it were written (8 ÷ 2) * (2 + 2) using PEMDAS rules, even though the original version of this equation 8 ÷ 2(2 + 2) contains the more advanced Algebra concept, math by juxtaposition. Regardless, the syntax error still persists.
Comment by commorancy — November 5, 2023 @ 11:32 pm |
Thank you. I’ve seen similar responses to this problem that raise the issue of syntax from a computational language perspective as you have, and they always follow your line of thinking. Your point about PEMDAS being a beginning math paradigm is something I had just concluded in responding to Richard (see his comment below) on one of his Facebook posts the other day. It’s almost like PEMDAS is used as a justification for a poorly constructed expression or an excuse to give kids “busy work” in math class instead of fostering a little more critical and theoretical thinking about math that is sorely lacking from our modern education system these days. I’ve made the point several times on Facebook that most standard, real-world equations are written in such a way that PEMDAS or Order of Operations is really a moot point. The most anyone needs to know for those is the first two steps (parentheses and exponents); after that, the rest of the standardized formula/equation falls into place quite naturally, reading from left to right. As an editor and erstwhile linguist, I’ve come to appreciate the value of a concisely written “plain language” instruction or explanation. It occurs to me that perhaps math texts for grade schoolers should adopt that same principle as well instead of expecting the unsuspecting to know how to “juggle” the elements of an expression according to their strict (and misguided) application of PEMDAS/OOO.
Comment by Scott Stocking — November 6, 2023 @ 8:40 pm |
PEMDAS, BODMAS, BEDMAS and BIDMAS all have their place when learning early mathematics. The point in using these helpers is in simplifying the rationale of how to solve simple left-to-right formatted mathematical equations. These tools are used at a time when students are between 6 and 11 years of age. At these ages, teaching a more advanced concept like multiplication by juxtaposition would probably prove impossible. To begin throwing Algebra concepts at an 7 year old who can barely grasp what 6 ÷ 2 means would make for a challenging grade school situation. A few students may be able to grasp these more advanced math concepts this early, but many more would not. At that level of math understanding, these advanced math concepts also aren’t really needed yet. Trying to teach 7 year old students the rules of Algebra is tantamount to throwing a student into the deep end of a pool before they know how to properly swim.
PEMDAS is intended to cater to all students of all maturity levels. There’s nothing wrong with using PEMDAS (et al) as long as they’re used to teach students with the type of mathematical problems that are best designed for use with PEMDAS. When advanced mathematical concepts, such as juxtaposition become involved, PEMDAS more or less makes way for more advanced mathematical understanding and rules. Attempting to teach these more advanced rulesets to 6-11 year olds is more than likely to confuse them at time when they need clarity and simplicity to grasp the basics.
Instead, mathematics courses need to become more clear as we move through them. For example, Algebra class should explain that while PEMDAS is useful for left-to-right formatted equations, many non-linear written Alegbra problems cannot be solved using PEMDAS alone. This needs to become a point of clarity and discussion when beginning advanced math classes like Algebra, Trigonometry and Calculus. However, I guess by the time students reach to these advanced math classes, it is assumed that they will “learn the ropes” on their own and learn as they “go along”.
Comment by commorancy — November 6, 2023 @ 9:10 pm
That’s a valid solution. I did just read your blog article on the expression and really appreciate your perspective. I recently looked up the various journal style guides to see how they handle it, and found it exactly as you said. The professional journals give precedence to implied/juxtaposed multiplication.
Comment by Scott Stocking — November 6, 2023 @ 9:19 pm
Yep, which is why so many calculators also give juxtaposition this precedence. It also seems that the multiplication by juxtaposition rule before division dates back to at least the 1920s or possibly earlier… it seems before the PEMDAS teaching term was even coined. Just some history there.
Comment by commorancy — November 6, 2023 @ 9:51 pm |
I thought I was replying at the other level, but instead it made a new comment. Gotta love WordPress sometimes.
Comment by commorancy — November 6, 2023 @ 9:52 pm |
I cited some of that history in my response to Richard on this blog a couple days ago. I recently bought Florian Cajori’s book “History of Mathematical Notation” for a little “light” reading.
Comment by Scott Stocking — November 6, 2023 @ 9:55 pm |
Good find. I just found that Cajori’s book is available in a Kindle edition, but it’s ~$20, same as a used copy. I’ve just added it to my Amazon holiday list in case someone wants to gift me a copy. Otherwise, I’ll check out Half-Price Books soon and see if I can find a copy there. I’d like to read that book.
Comment by commorancy — November 6, 2023 @ 10:06 pm
Totally wrong.. its 16. always 16. wolfram alpha gets you 16 too.
Comment by Scott A pike — November 29, 2023 @ 7:11 am |
Wolfram Alpha is way more than just a homework help site. If you enter “eight divided by twice the sum of two plus two” in the input field, it returns the answer of “1”. Also look at its description of Solidus. It explains that juxtaposed multiplication after a sign of division requires parentheses to be properly understood, and gives some examples of real-world formulas to prove it.
Comment by Scott Stocking — November 29, 2023 @ 7:53 am |
[…] 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough […]
Pingback by SMGB Indices | Sunday Morning Greek Blog — December 3, 2023 @ 4:40 pm |
I have occasionally used
F=ma. Newtons 2nd law of motion.
Divide both sides by ma. Now
F/ma = 1. Assign F = 60 Newtons,
m=15 kilograms and a = 4 meters
per second per second.
Then
60/15(4). If the strict PEMDAS folks
don’t get an answer of 1 then I have no idea what the value represents or the units to assign to it.
Comment by Ray Morse — December 30, 2023 @ 1:04 pm |
Mathematicians are trained not to use implied multiplication. They write the text books used in their classes. They know how to write expressions and equations without using it. Engineers and Scientists however use the notation for simplifying their math. Math professors know that, but they don’t teach Engineering and science classes.
Comment by Ray Morse — December 30, 2023 @ 1:25 pm |
Let’s talk about monomials…
from Study. com math teaching site:
https://study.com/learn/lesson/monomial-examples-factors.html
FAQs: “How do you identify a monomial?
Monomials are the product of a coefficient, and a variable or variables.”
A monomial has a single value so it’s as if it already has parentheses around it, in much the same way an exponent with a base number never needs to be encased inside parentheses. In both cases, you’re told how many times to multiply a quantity.
The monomial “2x” means “x” taken two times:
2x = [x + x]
from Cue Math teaching website:
https://www.cuemath.com/algebra/dividing-monomials/
“Practice Questions on Dividing Monomials”
“Q.1. Divide. 15a^2b^3 ÷ 5b”
The correct answer is listed as:
3a^2b^2
That means they’re teaching students that even though a division sign (obelus) is used to indicate “divided by,” the statement should be treated as a top-and-bottom vertical fraction, with 15a^2b^3 as the numerator & 5b as the denominator — with NO PARENTHESES anywhere in the statement.
The “2” in “2x” is not a stand-alone number — it’s the coefficient in a monomial which tells you how many times to multiply a quantity (which is actually adding the quantity to itself), in much the same way that an exponent tells you how many times to multiply the base number by itself. Just as the exponent is “attached” to the base number, the coefficient of a monomial is “attached” to the variable (factor). Thus, the monomial division statement “2x divided by 2x” is: [ x + x ] ÷ [ x + x ]. Notice that the coefficient “disappears” when the statement is written out in its most basic form (as the indicated additions of the quantity). That proves, once and for all, that “peeling off” the coefficient of the monomial (the “2” in “2x”) & using it in some other operation is not valid.
In the monomial division statement “2x ÷ 2x,” if x equals 4 [expressed as (2+2) ], it is:
2(2+2) ÷ 2(2+2)
which is also written as
8 ÷ 2(2+2) or 8 / 2(2+2)
which is the same as…
8
______
2(2+2)
…which has a quotient of 1.
A monomial has one single value (not two separate values that can be pulled apart). And division is fractions, no matter which division symbol is used — they all mean “divided by” & separate the numerator from the denominator. Do all of the operations indicated in the numerator, then do all of the operations indicated in the denominator, and finally divide the numerator by the denominator. Division has to go LAST. The Order of Operations as PEMDAS is incorrect for division statements — also known as fractions.
Comment by Dee — February 18, 2024 @ 4:34 pm |
i am very sorry but this article is very very wrong and i have no clue why you are trying everything to falsely make the equation =1 when in fact when you input it in a calculator, it gives you 16 as YOU showed. multiplication and division have the same priority thus you the number you are multiplying the parenthesis is actually 8/2 which is 4. in real physics you will never get an accurate answer using your methods and i m sorry for people that read this to find a proper answer.
Comment by tudorlasus — February 19, 2024 @ 1:34 pm |
It’s really quite a simple, third-grade solution to get the correct answer of 1. The obelus divides the expression into a dividend (term on the left) and a divisor (monomial term on the right). These equate to the numerator (dividend) and denominator (divisor) in fractional terms. Therefore, the correct way to interpret the expression is 8 ÷ 2(2+2) = 8 ÷ 8 = 8/8 = 1. Easy peasy lemon squeezy.
Comment by Scott Stocking — February 19, 2024 @ 6:37 pm |
The problem seems to be that the computer programmers who set it up as a straight-across-Order-of-Operations (as PEMDAS) statement obviously missed that day in school when the Basic Algebra teacher went over what a “term” is, gave examples of how to recognize a monomial & how to divide one monomial by another monomial.
In the statement 8÷2(2+2) , the “8” can be factored out as “2(2+2),” making the statement…
2(2+2) ÷ 2(2+2)
Replace what’s inside both sets of parentheses in the statement with the variable “x” — in other words…
x = (2+2)
Now the statement is:
2x ÷ 2x
which can be written as
2x / 2x
or as
2x
___
2x
from Wikipedia’s page “Division (mathematics)”:
https://en.wikipedia.org/wiki/Division_(mathematics)
“The simplest way of viewing division is in terms of quotition and partition: from the quotition perspective, 20 / 5 means the number of 5s that must be added to get 20. In terms of partition, 20 / 5 means the size of each of 5 parts into which a set of size 20 is divided. For example, 20 apples divide into five groups of four apples, meaning that “twenty divided by five is equal to four”. This is denoted as 20 / 5 = 4, or
20
___ = 4
5
In the example, 20 is the dividend, 5 is the divisor, and 4 is the quotient.”
“Notation
Division is often shown in algebra and science by placing the dividend over the divisor with a horizontal line, also called a fraction bar, between them. For example, “a divided by b” can written as:
a
__
b
which can also be read out loud as “divide a by b” or “a over b“. A way to express division all on one line is to write the dividend (or numerator), then a slash, then the divisor (or denominator), as follows:
a/b
Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (typically called the numerator and denominator), and there is no implication that the division must be evaluated further. A second way to show division is to use the division sign (÷, also known as obelus though the term has additional meanings), common in arithmetic, in this manner:
a ÷ b”
Bottom line: Division is fractions.
Given that x is not zero, 2x divided by 2x equals 1, no matter which division notation is used.
Comment by Dee — February 21, 2024 @ 12:48 pm |
Thank you for your comment. I’m glad you made the point about how calculators and most computers treat the expression “as a straight-across-Order-of-Operations (as PEMDAS) statement.” Playing around in Wolfram’s platform seems to highlight this kind of functionality. I think I added the examples of how Wolfram handles two slightly different ways of writing out the expression in text form. In Wolfram, if I use the text phrase “eight divided by two times the sum of two plus two,” Wolfram’s platform really has no way of distinguishing the phrase “two times” as somehow intrinsically linked by juxtaposition to (2 + 2). It treats the word “times” as a written operator and subsequently solves the expression from left to right as if there were no juxtaposition. We might be able to imply that juxtaposition by the way we say the phrase or inflect our voice, but Wolfram’s language model doesn’t appear to be that sophisticated at this point.
However, if I type out “eight divided by twice the sum of two plus two,” then Wolfram’s language model treats “twice” as an adverb and, even though “twice” means “two times,” the language model then recognizes the phrase after “divided by” as the whole divisor, that is, it treats it as its own term or monomial. The absence of the word “times” causes the model (apparently) to think less linearly and recognize there is an intrinsic juxtaposed or implied connection there. It acts the same way with the word “thrice,” but when I tried fourfold, fivefold, etc., it stopped recognizing the intrinsic connection.
Scott
PS This post has been my most viewed post by a 5 to 1 margin since last June. Are you aware of any large groups or conferences where this article or this topic is being discussed? Thank you.
Comment by Scott Stocking — February 21, 2024 @ 10:29 pm
I am not aware of another site that is currently discussing this issue – I wish there was.
It really does seem to be a programming issue. On several math teaching sites & online calculators, when I type in “2x/2x,” it automatically converts the statement into a top-and-bottom fraction, with “2x” on top & “2x” on the bottom, showing the quotient of 1. When “x=(2+2),” that makes the same mathematical statement”2(2+2)/2(2+2),” and the very same calculator tells me that the result is now 16.
Here are a couple of those calculators that gave two different answers to what is the same statement, when x=(2+2):
[ first input as 2x/2x and then as 2(2+2)/2(2+2) or 8/2(2+2) ]
https://www.mathway.com/Algebra
https://www.symbolab.com/solver/trigonometric-simplification-calculator/2x%20%5Cdiv%202x?or=input
…and these got a quotient of 1 in both forms:
https://quickmath.com/webMathematica3/quickmath/equations/solve/basic.jsp#c=solve&v1=%255Cfrac%257B8%257D%257B2%255Cleft%25282%2B2%255Cright%2529%257D
https://www.mathpapa.com/algebra-calculator.html
It boils down to understanding that division can be expressed as a fraction. With a fraction bar, everything “North” of the division symbol is the numerator & everything “South” of the division symbol is the denominator. In a horizontally written division statement (with an obelus or solidus), everything “West” of the division symbol is the is the numerator & everything “East” of the division symbol is the denominator (unless otherwise indicated with parentheses). It’s as if you wrote the top-and-bottom fraction on a sheet of paper & then rotated it 90 degrees to the left, in terms of directional position of the numerator & denominator, relative to the division symbol.
Some computer programmers remembered that day in school when monomial division was covered & programmed the calculator accordingly. Others forgot it altogether. And yet another group of programmers remembered the part with the coefficient & variable together being one term with a single value, but did not account for finding out the actual value of “x” & plugging in the numbers — which ought to give you the same result.
A certain group of calculator programmers need some refresher math tutoring.
— Dee in NYC
Comment by Dee — February 22, 2024 @ 9:17 am
from Study .com:
https://study.com/learn/lesson/monomial-examples-factors.html
from FAQ’s:
“How do you identify a monomial?
Monomials are the product of a coefficient, and a variable or variables.”
————-
from Algebra Class. com:
https://www.algebra-class.com/dividing-monomials.html
“Dividing monomials“
“Remember: A division bar and fraction bar are synonymous!”
—————
from Algebra Practice Problems.com , which teaches young students how to work out math problems:
https://www.algebrapracticeproblems.com/dividing-monomials/
“Any division of monomials can also be expressed as a fraction:
8x3 y2 z
8x3 y2 z ÷ 2x2y = __________ = 4xyz
2x2y
—————
Note that the original monomial division statement “8x3 y2 z ÷ 2x2y” uses a division sign (obelus) & there are no parentheses anywhere in the statement.
That being the case, 2x divided by 2x can be written horizontally as…
2x ÷ 2x or as 2x / 2x
…which is synonymous with the top-and-bottom fraction…
2x
___
2x
…and given that “x” does not equal zero, the statement has a quotient of 1, no matter which division symbol is used.
Any calculator that delivers a different result was programmed by someone who did not fully understand how to divide monomials (as a fraction — with no parentheses necessary). The statement is dividing one monomial by another monomial, with each monomial having a single value which is the PRODUCT of the coefficient multiplied by the variable.
In the monomial division statement “2x ÷ 2x,” x = (2+2)
Plug in the value of “x” & solve.
Comment by Dee — February 23, 2024 @ 3:25 pm |
Addendum:
Here’s another online calculator that gives the quotient of 1, whether input as…
2x ÷ 2x
or as
2(2+2) ÷ 2(2+2)
or as
8 ÷ 2(2+2):
https://www.snapxam.com/solver?p=8%5Cdiv2%5Cleft(2%2B2%5Cright)&method=0
…because it’s the exact same statement, no matter which division symbol is used.
Comment by Dee — February 22, 2024 @ 10:04 am |
I have had near arguments with people over this topic. In most cases, I facetiously admit defeat. You know the old saying: Argue with an idiot and they will bring you down to their level and beat you with experience.
I’m glad to see your depth of knowledge, based on mathematical laws, truly explain this concept. It’s hard to grasp for those who didn’t move beyond Algebra 1. I’m not being condescending; there is really a clear difference in mathematical education though. As you’ve stated, and I remember learning, the ‘math properties’ must be upheld first and foremost before submitting to ‘PEMDAS’ or other such acronyms used for basic math.
Most people have never even heard of the terms ‘obelus’ or ‘solidus’. So, how can they even follow, understand, or interpret a basic math problem? I’ve even tried a simple approach to explain these internet brainbusters. How can 8 ÷ 2(2 + 2) = 16 vs = 1? The numerator is 8. The denominator is 2(2 + 2) which is also 8, based on the distributive property, which is a Law! That computation = 1. To derive an answer of 16, you’re somehow suggesting that you’re reading the problem as 8 ÷ 0.5 which does equal 16. But how did you get there? I’ll wait for that explanation…
LOL…Perhaps this hotly contested topic is the crux of perplexion that’s prohibited NASA from returning to the moon. You know, because in 1969, this answer was indeed 1. Now, it’s 16. Or 9, for the other circulating stumper of 6 ÷ …yeah, you know the rest. (By the way, an answer of 9 for the preceding can only be achieved if 6 is divided by two thirds aka 0.667.) Haha!
On a personal note, I have to give credit to where it’s due. We all learn from someone. I’ve cherished the opportunity to learn Calc/Logic/Mechanics/E&M (and the prerequisite courses) in high school, taught by two great professors. They happened to be a married couple that passionately presented higher level mathematics and physics to a small handful of capable students. That was nearly 30 years ago. I’m forever grateful!
Scott, thank you for posting this. It reassures me that humanity still has a fighting chance…
~Alexis
Comment by Alexis — February 27, 2024 @ 7:07 am |
Thank you for reading, Alexis. I’m glad to have restored your hope in humanity! :-) I’m approaching 10K views for this article in the 10 months since I first posted it. That seems pretty significant for a post that’s completely off-topic from the theme of my blog. I wish I could be a fly on the wall whenever a math class or other group discusses my article. –Scott
Comment by Scott Stocking — February 27, 2024 @ 6:17 pm |
Indeed, Alexis, the quotient of the statement 8 ÷ 2(2 + 2) is, was, and always will be 1, once it is understood that it’s the monomial division statement:
2x ÷ 2x
…when x = (2+2)
Comment by Dee — February 28, 2024 @ 2:22 pm |
I agree with your explanation and really enjoyed reading this article. I have been called ignorant and other names because my answer was 1. 2 seconds to see the problem and got 1 for my answer. I was brilliant in math to the point I’ve had teachers apologize for saying I was wrong when they discovered the text book was wrong. Not saying I’m a genuis but I see math as enjoyment, I’ve read up on math, laws rules, the people on Facebook that arguing about it being 16 will not read into that to understand how it’s 1 instead 16, most on there that get 1 do it by accident but a few have actually mention the rule you used to get 1 as well. Glad to see your post and I’ll read more of your articles for sure
Comment by Josh Jones — March 9, 2024 @ 8:19 pm |
Thank you for reading. I know where you’re coming from. I was doing algebra in 6th grade in a self-paced class (in a public school, no less!). When I got to 7th grade, they wouldn’t put me in an algebra class, so I got stuck in general math and was begging the teacher for more work. There’s a new one on the Math Challenge group on FB now I’ve been responding to. Every time, I hone my arguments just a little bit more. The “new” (to me, anyway) problem is 60 ÷ 5(1 + (1 + 1)). Of course, that equals 4. Peace to you, and don’t get too grumpy about that lost hour of sleep tonight (if you’re in America, that is).
Scott
Comment by Scott Stocking — March 9, 2024 @ 10:05 pm |
Hmm. Below is a link to an article from Oct. 2023 that says X (formerly known as Twitter) has the “new” problem as:
60 ÷ 5(1 + 1(1 + 1))
https://www.hindustantimes.com/trending/viral-brain-teaser-can-you-solve-this-maths-question-using-bodmas-101697993980364.html
…which I believe holds the quotient of 3.
Here’s why:
First, take care of the innermost set of parentheses: (1+1) = 2
Now it’s…
60 ÷ 5(1 + 1(2))
…which brings up the question of whether what’s now inside the remaining parentheses is (2(2)) or if it is 1 + 1x [ i.e. x= (2) ] , with the 1 immediately to the left of the inner parentheses as the coefficient of what is in the innermost parentheses.
Personally, I believe that the coefficient to the left of what’s in the innermost parentheses, makes the statement…
60 ÷ 5(1+1(1+1))
60 ÷ 5(1+1(2))
60 ÷ 5(2(2))
which is…
60 ÷ 5(4)
60 ÷ 20
You see what I mean?
Comment by Dee — March 11, 2024 @ 12:42 pm
Check out this online calculator on this “new” problem:
https://quickmath.com/webMathematica3/quickmath/equations/solve/basic.jsp#c=solve&v1=%255Cfrac%257B60%257D%257B5%255Cleft%25281%2B1%255Cleft%25281%2B1%255Cright%2529%255Cright%2529%257D
I typed in the problem using a slash & it made it into the top-and-bottom fraction…
60
______________
5(1+1(1+1))
…which, according to that calculator, ultimately yields the quotient of 4. Apparently, it interprets the coefficient of what’s in the innermost parentheses is 1 (not the sum of “1+1” as I thought it was) — plus one after that.
I stand corrected.
Comment by Dee — March 11, 2024 @ 3:21 pm
Yes, the “1 + 1” before the parentheses would only be summed first if it was also in parentheses. But since the second “1” is juxtaposed to the parenthesis, then that should be multiplied first. Thank you for the link.
Scott
Comment by Scott Stocking — March 11, 2024 @ 10:14 pm
I now realize that my first solution was incorrect, as you point out. Inside the outer parentheses, it’s 1 + 1x, with x = 2.
The statement 60 ÷ 5(1+1(1+1)) is 20b ÷ 5b, with b = (1 + 1 (1+1)), or written as b = 3.
5b is “b” taken 5 times, which is…
5b = b + b + b + b + b
which in this case is…
5b = (1 + 1 (1+1)) + (1 + 1 (1+1)) + (1 + 1 (1+1)) + (1 + 1 (1+1)) + (1 + 1 (1+1))
5b = 3 + 3 + 3 + 3 + 3
Either way, 5b = 15
The statement is one monomial divided by another monomial. By definition, a monomial is one term, so it never needs to be encased in parentheses to comprehend that it has the single value of the PRODUCT of the coefficient multiplied by the variable (factor). Therefore, the coefficient can’t be ripped away from the variable (factor) & used in some other operation before calculating the value of the monomial itself.
Those who want to believe that the statement equals 36 need to stop depending on calculators which may have been programmed by someone who missed that day in 9th grade Basic Algebra class, when the teacher covered what a term is & how to divide one monomial by another monomial — as a top-and-bottom FRACTION. When any division statement is read aloud, the words, “divided by,” separate the numerator from the denominator, regardless of which division symbol is used (obelus, solidus or vinculum). Therefore, the statement
60 ÷ 5(1+1(1+1))
is properly written as the fraction…
60
______________
5(1+1(1+1))
…which has a quotient of 4.
Comment by Dee — March 12, 2024 @ 10:22 am
Scott —
After you said that in a horizontally written division statement, the numerator is everything to the left of the division symbol (obelus or solidus) & the denominator is everything to the right of the division symbol, someone brought up a statement something along the lines of 4/2/4, questioning where the numerator is & where the denominator is in that multiple division statement. In the case of 4/2/4, he answer is either one half or it’s 8. depending on whether the numerator is just 4 or if the numerator is four-halves.
After contemplating this issue, it is clear that in a horizontally written statement which uses multiple division symbols, parentheses would indeed be necessary to delineate the numerator & denominator, respectively. However, in a horizontally written division statement containing only a single division symbol, the numerator is everything to the left of the division symbol & the denominator is everything to the right of the division symbol.
— Dee
Comment by Dee — March 14, 2024 @ 8:53 am |
Dee: If we’re talking about using an obelus, then a problem like 6 ÷ 3 ÷ 4 should be worked left to right, per PEMDAS. The obelus, solidus, and vinculum group what they encounter to the next extant sign (+-x÷) not in a parenthetical construct but the obelus by itself does NOT represent the same relationship that a true fraction does. However, if you write the expression using the solidus, then you have an issue of how to group the terms, since the solidus typically serves the same function as the vinculum in that it groups what comes after. So if you write the expression 6/3/4, you have a genuinely ambiguous expression which would lead to confusion about which answer is correct. Is it ½ or 8? It’s up to the interpretation of the reader. However, if you write it like this: 6 ÷ ¾ or 6/¾, where there is an attempt to distinguish the true denominator, then you have a little more clarity. The problem arises, then, about the role of PEMDAS at this point. Why? Because strict PEMDAS would say you follow the order of the signs, so it’s no different than 6 ÷ 3 ÷ 4 at that point (½). But most of us learned when faced with a problem written in such a way that it’s clear we’re dividing by a fraction, PEMDAS is suspended. We first invert and then change the obelus to a multiplication sign. That’s NOT a PEMDAS step! So 6 ÷ ¾ or 6/¾ becomes 6 x 4 ÷ 3, or 8. Here’s the inconsistency that proves juxtaposed multiplication does take priority: The strict PEMDASian would say 6 ÷ 3(4) is THE SAME AS 6 ÷ ¾! It’s obvious to the naked eye that the two expressions are in fact NOT equal and not intended to communicate the same value. The solidus is a grouping symbol just like the parentheses, so there is an extra NON-PEMDAS step taken to solve the division by a fraction problem. In the same way, then, when looking at 6 ÷ 3(4), we take the extra step of applying the parentheses around the 3(4) to account for the grouping of the vinculum or solidus.
Scott
Comment by Scott Stocking — March 14, 2024 @ 11:53 am |
In response to your reply that, “If we’re talking about using an obelus, then a problem like 6 ÷ 3 ÷ 4 should be worked left to right, per PEMDAS,” I disagree. The obelus and solidus are synonymous & therefore interchangeable.
from Wikipedia’s “Slash” page:
https://en.wikipedia.org/wiki/Slash_(punctuation)#:~:text=of%20a%20ring.-,Division,18th%20or%20early%2019th%20century.
“Mathematics“
“Fractions
The fraction slash ⟨ ⁄⟩ is used between two numbers to indicate a fraction or ratio. Such formatting developed as a way to write the horizontal fraction bar on a single line of text. …This notation is known as an online, solidus”
“Division
The division slash ⟨ ∕⟩, equivalent to the division sign ⟨ ÷⟩, may be used between two numbers to indicate division. For example, 23 ÷ 43 can also be written as 23 ∕ 43. This use developed from the fraction slash in the late 18th or early 19th century”
—————-
With that being the case, your horizontally written statement of 6 ÷ 3 ÷ 4 is exactly the same as 6/3/4. And as you point out, that statement is ambiguous. Therefore, parentheses would have to be installed to indicate whether it was (6/3)/4 or 6/(3/4).
also from Wikipedia’s “Slash” page:
“Nowadays fractions, unlike inline division, are often given using smaller numbers, superscript, and subscript (e.g., 23⁄43).”
So if your statement was written with the denominator indicated by the superscript and subscript fraction, as 6 ÷ ¾ or as 6 / ¾ , then the numerator & denominator would be clear.
Comment by Dee — March 14, 2024 @ 4:52 pm
I prefer Wolfram to Wikipedia. He explains how the solidus is interpreted in written notation as opposed to how it works in computer language without parentheses. The main issue in the end in my mind is why are people pushing poorly written, ambiguous expressions to try to prove a point about PEMDAS? I always come back to to the fact that most formal equations for real-world values are written unambiguously, the whole PEMDAS debate is mostly irrelevant. https://www.wolframalpha.com/input?i=solidus&assumption=%7B%22C%22%2C+%22solidus%22%7D+-%3E+%7B%22MathWorld%22%7D
Comment by Scott Stocking — March 14, 2024 @ 5:14 pm
Scott —
Let me just say that I am thoroughly enjoying our conversation on this subject, even though we slightly disagree on some points.
I actually don’t agree with your assertion that statements like 8 ÷ 2(2 + 2) are “poorly written.” It’s as clear as day, once one realizes that it’s the monomial division of 2x divided by 2x, with x = (2 + 2). The statement has a quotient of 1 — since anything other than zero which is divided by itself equals 1.
As for the Wolfram explanation of, “He explains how the solidus is interpreted in written notation as opposed to how it works in computer language without parentheses,” that just proves my point about some computer programmers having been absent on the day in 9th grade Basic Algebra class, when the teacher covered what a “term” is & how to divide one monomial by another monomial. Other programmers seem to have been present that day & wrote their calculator programs to account for that kind of monomial division — which is clearly understood sans additional parentheses.
— Dee
Comment by Dee — March 14, 2024 @ 5:46 pm
What is interesting is that, since Wolfram uses a language model, if you type in “Eight divided by twice the sum of two plus two,” it returns the answer 1, but if you change the “twice” to “two times,” it returns the answer 16. So he does seem to have some of that built in, but when it keys on the word “times,” there are no other contextual clues to interpret that as anything but simple multiplication. It works for “thrice” as well, but once you get to “fourfold” and beyond, it doesn’t work. I don’t think I tried “quadruple.”
Comment by Scott Stocking — March 14, 2024 @ 5:54 pm
The fact that Wolfram’s online calculator (and others) yields two different answers to the same division statement with numerical input vs. language input, points to the computer programmer not fully understanding & accounting for the underlying concept of dividing one monomial by another monomial.
Implied multiplication by juxtaposition means that the viewer of the mathematical statement is looking at a monomial — all you have to do is replace whatever is in parentheses with a variable such as “x,” to see it clearly.
What some programmers have failed to recognize is that a monomial such as “2x” has, by definition, a single value which is the PRODUCT of the coefficient multiplied by a variable or variables. Put another way, the monomial “2x” holds the total value of two “x’s,” which can also be written as…
2x = [ x + x ]
It’s not a computer language problem — it’s a computer programmer problem. Some computer programmers did get it 100% right & wrote their calculator program to account for how to properly execute monomial division.
The people who insist that PEMDAS is the ONLY way to correctly calculate 8 ÷ 2(2 + 2), should stop depending on online calculators to do their thinking for them. Instead, they should start using their own brain to reason out what it is that they’re actually looking at. In this case, what they’re looking at is one monomial being divided by another monomial:
2x / 2x
…which is the same as the fraction…
2x
___
2x
with x = (2+2)
Comment by Dee — March 15, 2024 @ 1:35 pm
I really do appreciate your feedback. Considering others’ feedback always helps me sharpen my own thinking and leads me to more concise ways to express my thoughts and arguments. When I say “language,” what I’m really talking about is linguistics. Merriam-Webster defines it as “the study of human speech including the units, nature, structure, and modification of language.” I take “speech” to mean the written word as well as the spoken word, especially since as a preacher I’ve gotten into the habit of writing out my sermons so I can make more intentional use of my language as opposed to speaking extemporaneously. And in the context of this article, I don’t just mean words alone, but any symbols or figures that we use to communicate, calculate, or cantillate (how’s THAT for an alliteration!): numbers, punctuation, “character” words (e.g., ampersand, &), mathematical and scientific symbols, proofreading symbols, and even music notation.
All of these elements of language, and linguistics more broadly, have their place in their appropriate contexts, and they are subject to their own respective set of rules for putting them together in a coherent form that communicates the message and meaning we intend subject to the rules and conventions of their respective contexts. When someone composes a musical score, the main melody or tune is subject to certain patterns that follow the chords that underlie the melody. If the tune doesn’t match the chords, it sounds, well, discordant. The notes of the melody, harmony, or even a descant are not strictly random. They typically have some relationship with the chord, and often playing a note that doesn’t exactly fit the chord prefigures a change in the chord or even a change in the key signature. Intentional discordancy is not without significance either, as it could communicate chaos or irrationality.
When we write a sentence, we generally expect a subject and verb to be close together and to arrange direct and indirect objects appropriately with any modifiers or prepositions, and so forth. For example, consider the difference between the three sentences, which have the exact same words.
1. I eat fish only on Friday.
2. I eat only fish on Friday.
3. I only eat fish on Friday.
Sentence #1 is truly ambiguous, because the placement of “only” can be taken either way. Is it “Fish is the only thing I eat on Friday” (akin to Sentence #2) or “Friday is the only day I eat fish” (akin to Sentence #3)? Does that sound familiar in the context of this post? More on that in a bit.
In the blog post, I make reference to the relationship between the definite article, noun, and adjective in a Greek adjectival phrase. The position of (or absence of) the definite article impacts how the phrase can be interpreted. I’ll use transliterated words to demonstrate.
1. kalos logos [beautiful word]
2. ho kalos logos OR logos ho kalos [the beautiful word]
3. ho logos kalos OR kalos ho logos [the word is beautiful]
In Greek, Phrase #1, which has no definite article (the indefinite article “a” can fairly be implied absent other contextual clues), would be considered ambiguous by itself. We would need contextual clues to know whether it means “a beautiful word” or “a word is beautiful.” (Greeks do not have to use a form of the copulative verb “to be” if that is the only verb in the sentence.) In Phrase #2, the definite article precedes the adjective, which means the adjective is attributive, that is, it directly modifies the noun. It doesn’t matter if the noun is first or last; it’s attributive either way. Phrase #3 has a predicate construction. This means that the noun is the subject of a sentence, and the adjective would come after the verb in that sentence. In this case, it typically doesn’t matter where the adjective is, although there may be a nuanced difference one way or the other.
Given those three examples (music, English adverb placement, and Greek definite article placement), I think anyone who’s reading this is starting to see the bigger picture of how linguistics influences mathematics as well, especially in the context of the expression at hand. So let me use the expression in the same way I used the sample phrases above:
A. 8 ÷ 2(2 + 2) = 1 (in Dee’s and my worldview) or 16 (in the competing worldview)
B. (8 ÷ 2)(2 + 2) = 16 (in both worldviews)
C. 8 ÷ (2(2 + 2)) = 1 (in both worldviews)
Expression A seems unambiguous form the perspective of one’s worldview then. But are both worldviews equally valid? We can make arguments from our respective worldviews to try to convince others, but it is very difficult to convince one to change their worldview without a powerful defining event that shakes their worldview to the core. Otherwise, we’re comfortable with our ways. I happen to think that several of the arguments I’ve made to support my worldview are quite devastating to the competing worldview, but alas! there has been very little evidence of any change of heart among them.
Just like the position of adverbs and definite articles, so then is the generous use of parentheses needed to clearly avoid the ambiguity of the given expression. But let me make yet another appeal here for the case that the given expression, in light of my demonstration here, is not really ambiguous at all. The juxtaposition of the 2 to (2 + 2) is akin to Phrase #2 in my Greek examples above. The attachment between the two places them in an attributive relationship (the 2 is the definite article; the (2 + 2) is the adjective). The 2 directly modifies the (2 + 2) by telling us how many of them we need to divide by and keeps the monomial on one side of obelus WITHOUT an extant multiplication sign. In other words, it isn’t separated from its cofactor by the “action” of the obelus. There is no need for the extant multiplication sign because the relationship is clearly defined. If one were to place a multiplication sign between the 2 and (2 + 2), that would emphasize that the 2 and (2 + 2) are NOT cofactors and sever the relationship between them. This would make the expression like Greek Phrase #3 above, where the modifier is divorced from the modified and dragged kicking and screaming all alone into the action of the obelus. That which appeared to modify the (2 + 2) now modifies the 8. The implications of the expression change by adding the multiplication sign. Additionally, in the case of Greek Phrase #3, if we would add the implied copulative verb where it is not technically needed, that would also place emphasis on the verb and suggest a more nuanced meaning. (This also happens with Greek verbs; most Greek verb forms have an ending that tells you what “person” [1st, 2nd, 3rd, or I/we; you/you; he, she, it/they] is the subject of the verb. If there is no subject accompanying the verb, the corresponding pronoun is implied [“He eats”]. If a Greek pronoun is used as the subject, that implies emphasis [“He himself eats”],)
This may seem kind of heady to some, but I hope I’ve made my position a little easier to understand. My worldview and what I consider the strength of my arguments here and elsewhere, along with a ton of historical evidence, do convince me that the given expression is unambiguous and has no need of extra parentheses to understand it. For those who think writing ambiguous expressions is somehow educational and instructive when you know there are those who see through your ruse, I declare that you have met your match in me. Game over. Checkmate!
Scott
Comment by Scott Stocking — March 15, 2024 @ 6:26 pm
Thank you for enlightening me about grammatical construction in the Greek language — fascinating. At one time, I did speak & read Hebrew, which seems like it might have some similarities, but that was quite a while ago — I would need a refresher course!
I totally see what you’re saying regarding the syntax issue, with relation to spoken or written language (words) & I agree with you on that score.
On the subject of mathematical expressions akin to 8 ÷ 2(2 + 2), I would like to share this word problem example of the reason that the coefficient cannot be detached from the variable (factor):
We’ll go back to the original internet example I first encountered in 2011:
48 ÷ 2(9 + 3)
Let’s say I own a diner, and just as the breakfast special was about to end, 2 different groups of a dozen people each, walked in & were seated at separate tables. All of those customers ordered eggs. I look in the restaurant’s refrigerator & see that I have 4 dozen eggs left. If each customer receives an equal number of eggs, how many eggs does each customer get?
4 dozen eggs divided by 2 dozen customers
4 dozen ÷ 2 dozen
(also written as: 4 dozen / 2 dozen)
Viewing it in that context, it is apparent that “4 dozen” is a single quantity of eggs (48 eggs) & “2 dozen” is the single quantity of customers (24 customers). In other words, the 4 in “4 dozen” & the 2 in “2 dozen” cannot be separated from the factor (“dozen”) and used in another operation in the statement.
Numerically, the statement is…
4(12) ÷ 2(12)
…or…
4(9+3) ÷ 2(9+3)
…which is the monomial division of…
4x ÷ 2x
…when x=12 or x= (9+3).
What are your thoughts on this illustration?
Comment by Dee — March 16, 2024 @ 1:53 pm
That’s a great example. I’ve created similar ones for other problems. I majored in Hebrew in seminary, but much of my work lately has been in Greek, as is indicated by the name of the blog. Thank you for reading!
Comment by Scott Stocking — March 16, 2024 @ 1:59 pm
I deliberately chose to use the word “dozen” because we are all accustomed to buying eggs in that “unit,” which is a box of 12 eggs. It’s easy to understand that when there are 4 full standard packages of eggs, there are 48 eggs in all. That mental image of “4 dozen eggs,” highlights the reason that the coefficient of 4 cannot be separated from its factor of “dozen.” Since multiplication is just a fast way of doing addition, the value of the term “4 dozen” is actually…
1 dozen + 1 dozen + 1 dozen + 1 dozen
Also, everyone can easily picture two individual groups of a dozen people, with each group seated as a “unit” of 12 customers at two separate tables in a restaurant, understanding that there are 24 total customers now seated in the restaurant, who are all ordering eggs for breakfast.
The use of “dozen” makes it easy to understand that…
4 dozen divided by 2 dozen = 2
…which can be numerically calculated as…
4(12) / 2(12) =
48 / 24 = 2
…or calculated by canceling out the like factor of “dozen,” leaving the statement as…
4 / 2
…which, of course, also equals 2.
And if the word “dozen” is replaced with a variable such as “x,” then the statement is…
4x / 2x
…which is one monomial being divided by another monomial
[ with x=(9+3) or x=(9+3) }
Implied multiplication by juxtaposition indicates that a term is a monomial — never needing parentheses around it, any more than “4 dozen” needs to be completely encased in a set of parentheses to be understood as one term with a single value. Therefore, the “4” in the term “4 dozen,” or the “2” in the term “2 dozen” cannot be detached and used in some other operation in the statement before calculating the total value of the monomial, first.
Comment by Dee — March 18, 2024 @ 10:17 am
Exactly! That’s my point elsewhere about using other “count” adjectives or adverbs (twice, thrice, etc.). Thank you!
Comment by Scott Stocking — March 18, 2024 @ 10:20 am
Yes, we are in agreement — words like “dozen” definitely clarify what quantities in the statement are actually being divided.
It’s nice to find someone else who is using some brain power to reason out what the statement means, rather than depending on some online calculator to do the thinking, or using rote memorization of a concept which does not fully comprehend one monomial being divided by another monomial (i.e. The Order of Operations as PEMDAS).
Comment by Dee — March 18, 2024 @ 10:31 am
Did you see my latest post on in this? I combined one of my comments back to you with another I’d posted FB a few days earlier. I posted it on Saturday.
Comment by Scott Stocking — March 18, 2024 @ 10:37 am
The issue that could arise with using language such as “twice” or “thrice,” is that it can still be seen as indicating multiplication, rather than seen as repeated additions (which is what multiplication actually is, in its most basic form). In other words, “twice” or “thrice” will make some people insert an explicit multiplication sign between the coefficient & the factor, thus making it appear that the division should be done with the coefficient first, before multiplying the factor. The proposition of a quantity of eggs being divided by a quantity of diner customers might be a better illustration of one monomial being divided by another monomial — most people can easily conceive of 4 dozen eggs being evenly portioned amongst 2 dozen people as…
(12 + 12 + 12 + 12) ÷ (12 + 12)
…which is…
48 ÷ 24
…which equals 2 eggs per diner customer.
The coefficient of “4” in “4 dozen” & the coefficient of “2” in “2 dozen actually “disappear” when the indicated number of additions is written out.
Comment by Dee — March 18, 2024 @ 2:16 pm
“Dozen” could lead to multiplication as well, because it’s a quantity by itself, even though it “cancels out” in your breakfast example. But “twice,” “thrice,” etc. are not strictly quantities by themselves, and this is where the linguistic aspect comes into play. “Twice” is an adverb. We can’t just say “8 divided by twice.” “Twice” needs more information to make a complete adverbial phrase that should be treated as a unit: “twice the sum of two plus two.” Mentally we would still do that multiplication, or recognize that when we say it like that, we should place a vinculum over or parentheses around 2(2 + 2) because the adverbial phrase is a unit.
Comment by Scott Stocking — March 18, 2024 @ 10:00 pm
Yes, it’s possible that “dozen” might lead to multiplication, as well. My experience with using the word “dozen,” though, is that people can easily visualize what “4 dozen” eggs looks like, on a shelf in the refrigerator — they see it as 4 “units” of 12 eggs each, as…
1 dozen + 1 dozen + 1 dozen + 1 dozen
rather than seeing it as…
4 * 12
It’s a subtle difference in how people view it — but it drives home the point that the coefficient (in this case, “4”) cannot be “ripped away” from the factor (“dozen”) & used in some other operation in a larger mathematical statement. The coefficient actually “disappears” when the indicated additions are written out. In the case of “4 dozen,” the coefficient of “4” doesn’t actually exist as a number unto itself — it just tells you how many times to add a quantity to itself, in much the same way that an exponent tells you how many times to multiply the base quantity by itself.
Comment by Dee — March 19, 2024 @ 10:49 am
Correction:
Should have been…
[ with x=12 or x=(9+3) }
Comment by Dee — March 18, 2024 @ 10:20 am
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I don’t know if this has already been asked (too many comments to read through).
You stated:
Thus 4m2 ÷ 2m = 4m2/2m, not (4m2/2)m.”
Please justify how the expression simplifies to 2m ‘using all real numbers’?
Do ‘all real numbers’ not include ZERO?And last I heard, 0/0 is undefined.
Comment by Andrew Wilhelm — April 1, 2024 @ 8:48 am |
Thank you for reading. Expression 1 is a specific example of an expression in a similar format to the one with the variable m. I’m saying Expression 1 uses real numbers instead of variables. Of course, the expression quoted from the source would be true for m = any nonzero real number. The main point is that the juxtaposed 2m is handled as a single, inseparable value, so the 2(2+2) must be treated as a single inseparable value as well.
Comment by Scott Stocking — April 1, 2024 @ 9:25 am |
[…] 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough […]
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