Basic mathematics should not have any ambiguity. It’s time to declare juxtaposition as a property of mathematics (and our Arabic number system in general) and silence the ill-informed ambiguists!
The PEMDAS Convention:
Please
Excuse
Me
Doubting
A
Silly
Convention.
Harvard University math professor Oliver Knill has a wonderful site (Ambiguous PEMDAS) that documents many of the arguments against the PEMDAS convention (at least by the PEMDAS/Order of Operations legalists) of treating juxtaposed implicit multiplication with the same priority as multiplication with an extant operational sign. Of course, he’s referring to hotly debated, viral math expressions like:
8 ÷ 2(2 + 2) and 6 ÷ 2(1 + 2),
both of which equal 1 in my book, but to the underinformed PEMDAS/Order of Operations crowd, they equal 16 and 9 respectively.
Similarly, the Math Doctors site has an excellent article entitled Order of Operations: Historical Caveats – The Math Doctors in which they say that “nobody” made the rules for the order of operations and that the so-called “rules” are only descriptive. Doctor Peterson of the Math Doctors has been kind and gracious in answering many of my pointed questions and objections to the normalization of a PEMDAS convention that does not uniformly acknowledge the power of juxtaposition of numerals to imply certain mathematical operations, and for that I am grateful.
But all this begs the question: Why should there be ANY ambiguity in basic mathematics? Why do we have rules that are “descriptive” and not theoretically supported and sound? Isn’t mathematics a hard science (not Barbie’s “Math is hard!” but lacking or minimizing subjectivity), or at least the foundational backbone of the hard sciences that supports the objectivity needed for the hard sciences? Maybe most of the world doesn’t care about this argument about PEMDAS, but I see the admission or acknowledgment or tolerance (or whatever synonym) of ambiguity as a weakness of the discipline. The refusal to acknowledge the inconsistency of rejecting the binding power of juxtaposition in the 2(2 + 2) or 2(1 + 2) part of the expression flies in the face of what is otherwise inherent, intuitive, and obvious about the way the Arabic number system in its current form is constructed.
I agree with the Math Doctors that “nobody” made the rules of Order of Operations and that the “rules” of Order of Operations are descriptive (as opposed to theoretical, which seems to be the implication), but should we really allow such a laissez-faire, passive convention to be immune to such criticisms as I have raised in my writings and videos?
See, for example:
8 ÷ 2(2 + 2) = 1: A Discussion with ChatGPT on the Implications of Juxtaposition in Mathematics
Stocking’s Order: Implicit Constructions Correct the Misapplication of Order of Operations
My original critique, ever expanded and updated, 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough
My Rumble videos:
Stocking’s Order: Juxtapositional Grouping Is Foundational
Stocking’s Order; or Why 8 ÷ 2(2 + 2) = 1).
If a mathematical convention has no clear origins or has no theoretical background, then either someone should propose a theoretical background to support it or promulgate a rule based on demonstrated, objective properties and principles (as I have done above) that rightly and fairly critique the convention to bring it into compliance with the rest of the discipline. In other words, “SOMEONE MAKE A LEGITIMATE RULE AND ENFORCE IT!” Lay down the law instead of allowing this abuse of Order of Operations to tarnish the otherwise marvelous discipline of mathematics!
I understand that part of this comes from linear computer algorithms that take operations as they come. But that is an archaic algorithm now when compared the complexity of language models we have with artificial intelligence (AI). If a language model can figure out the parts of speech of a sentence, then certainly the math algorithms can be overhauled to recognize juxtaposition. The fact that this hasn’t happened yet some 50 years after the advent of the hand-held calculator is astounding to me. We’re basically programming our computers with a primitive, elementary model of mathematics and not accounting for the nuances that arise in Algebra. The order of operations rules should not be different for algebra than they are for arithmetic. Algebra is more or less a theoretical description of arithmetic, so the rules of Algebra should prevail. They should be uniform throughout the field of mathematics.
In my articles above, I go into detail on the influence of juxtaposition in the Arabic number system. For those who haven’t read them yet, here’s a quick summary of the prevalence of juxtaposition and its function in creating a singular value that takes priority in processing.
A fraction is the product of one operand multiplied by the reciprocal of another nonzero operand of the same or different value.[1] The operand below the vinculum (and by definition grouped by the vinculum[2],[3]) should be rationalized, when possible. When looked at this way, then, you have a quantity vertically juxtaposed to a grouped quantity, just like in 8 ÷ 2(2 + 2) you have a quantity (2) horizontally juxtaposed to a grouped quantity. The only difference is the orientation of the juxtaposition.
Yet when dividing by a fraction, we do not pull the numerator/dividend away from the fraction and divide that first, then divide by the denominator/divisor. We treat the fraction as a single quantity, invert the fraction first, then multiply. In other words, the element with a value juxtaposed to another grouped value is addressed first, then the expression is worked (and it’s not insignificant that such a transformation puts multiplication before division, just like it is in PEMDAS). In the same way then, the implied multiplication in 2(2 + 2) MUST be addressed first to be consistent with its cognate function in the former example. The obelus is NOT a grouping symbol, so it is WRONG to treat it as such and give it some nonexistent power to ungroup a juxtaposed quantity.
It is the same way when multiplying or dividing by a mixed number. We do not undo the implied addition of the juxtaposition of a whole number and a fraction but must first convert the mixed number to an improper fraction, then apply the rules for fractions in the previous paragraph.
This is the undeniable reality of how our number system works. I fail to see how anyone can disprove or ignore this reality simply by citing an untested and subjective convention that isolates one of several similar forms and says, “We’re treating it differently because it’s easier than trying to teach it the correct way.” To that I say “Poppycock!”
It’s time to stop capitulating to ambiguity! Mathematics is not worthy of such capitulation. Have some courage and take a stand for what is demonstrably true! Mathematics has hard and fast rules and properties, and the property of juxtaposition should be elevated to the same level as the associative, distributive, and commutative properties. Need a definition? Here it is:
Juxtaposition Property: When two or more values are juxtaposed without any intervening free-standing operators (i.e., operators not included in a grouped value), the juxtaposed values are considered inseparable and must be given priority over extant signed operators in the Order of Operations, regardless of what the implied operators are.
Scott Stocking
My highly informed opinions are my own and are the product of my own research.
[1] To put it another way, a basic division problem using the obelus can be converted to a multiplication problem by inverting (i.e., taking the reciprocal of) the divisor and multiplying by the dividend.
[2] “2 : a straight horizontal mark placed over two or more members of a compound mathematical expression and equivalent to parentheses or brackets about them,” VINCULUM Definition & Meaning – Merriam-Webster, accessed 05/25/25.
[3] “A horizontal line placed above multiple quantities to indicate that they form a unit. vinculum – Wolfram|Alpha, accessed 05/25/25.
Sunday Morning Greek Blog 
[…] See also my latest article on the subject, 8 ÷ 2(2 + 2) = 1: End the Ambiguity! Make a Rule! | Sunday Morning Greek Blog. […]
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The rule already exists. Juxtaposition indicates the presence of a MONOMIAL, which is one “unit.”
In 5th Grade math students are taught that Division=Fraction & therefore Fraction=Division. A fraction can be written vertically or horizontally, thusly:
4
— = 4/2
2
Holding that concept in mind, the monomial 4a divided by the monomial 2a can be written thusly:
4a
—- = 4a/2a
2a
Plugging in the value of the variable when a=12, the expression becomes…
4(12)
——- = 2
2(12)
…because the numerator 48 divided by the denominator 24 equals 2.
And since Fraction=Division, that same expression can also be written horizontally as…
4(12)/2(12)
…because it’s the exact same fraction (the numerator 48 divided by the denominator 24).
And since the slash & the division sign are interchangeable (again, see 5th Grade math), the same fraction can be written as…
4a÷2a
…which still equals 2 when the variable “a” does not equal zero (i.e. the “like” variable can be cancelled out, leaving 4 divided by 2).
Anyone who disputes this can look up what a monomial is in any Basic Algebra textbook. Nowhere does it say that a monomial needs to have parentheses surrounding it in all cases in order for the monomial to be understood as a single “unit.” In fact, the definition of a monomial is ONE TERM with a SINGLE VALUE which is the PRODUCT of its factors..
Proof via the following word problem:
In this scenario, I own & run a diner which serves a Breakfast Special from early morning opening until 11 AM. Today, at 10:58 AM, the last of the Breakfast Special customers left the restaurant. But then, at 10:59 AM, two separate groups of a dozen people each come into the diner, wanting to get the Breakfast Special. The 2 separate groups are each seated at their own table. Each member of the 2 groups, consisting of a dozen people apiece, wants eggs. I go into the restaurant kitchen to see how many eggs I have left. When I open the refrigerator, I see that there are 4 full cartons of a dozen eggs each, left on the shelf.
Split evenly among the Breakfast Special customers in the diner, how many eggs does each customer get?
The proposition is a single quantity of eggs split evenly among a single quantity of customers:
4 dozen eggs divided by 2 dozen customers
4 dozen ÷ 2 dozen = ?
And since dozen=12, this is written numerically as…
4(12) ÷ 2(12) = ?
…which is exactly the same as…
4a ÷ 2a = ?
…when a=12.
~ ~ ~ ~ ~ ~ ~
There is only one correct answer to that word problem: Each customer gets 2 eggs. That’s because 4 dozen is a single “unit” of quantity & 2 dozen is another single “unit” of quantity. The math doesn’t change because it’s phrased as “dozen” instead of saying 12.
Comment by Dee R. — May 27, 2025 @ 9:45 am |
Hi, Dee. Nice to hear from you again. In algebra, it appears the “rule” is there, it’s never officially stated that I’ve ever seen. It’s more anecdotal: if you see this form, treat it this way.
As I was writing my latest article, it occurred to me that juxtaposition is more of a property than a rule, that is, it’s built into the fabric of the Arabic number system.
You’ve made the point about “dozen” in your previous comments. While I don’t think it’s a perfect analogy, it does have a nexus with my argument. “Dozen” is a unit measurement, so it’s not necessarily an outright multiplication problem. It is more of a qualitative modifier than a quantitative modifier, so it wouldn’t be subject to OOO the way the sixteeners want you to understand it.
I have 8 people who want a piece of fruit and a pair of bags each with 2 apples and 2 oranges. 8 ➗ 2(2 + 2) = 1 piece of fruit per person. The “pair of bags” linguistically represents the whole divisor, the implied parentheses.
Thank you for reading and for your contributions!
Scott
Comment by Scott Stocking — May 27, 2025 @ 11:47 am |
Scott —
You say, ” “Dozen” is a unit measurement, so it’s not necessarily an outright multiplication problem.”
Actually, it is an implied multiplication expression to say “4 dozen” or “2 dozen,” because that’s 4 twelves or 2 twelves, correctly written mathematically as 4(12) & 2(12), respectively.
Everyone is used to thinking of packs of eggs by the unit of “dozen,” so they can easily picture 4 full cartons of eggs in the refrigerator. They immediately understand that there is a single total quantity of eggs in that collection of 4 dozen, or put another way, “4 twelves” of eggs.
Once the concept of “dozen” is established, it’s easy for people to grasp that in order to solve the problem of how many eggs each diner customer will get, you first have to figure out how many total eggs there are & then figure out the total quantity of people who will be splitting that quantity of eggs. In the case of 4 dozen eggs, that’s 4 twelves of eggs which equals 48 individual eggs, written mathematically as 4(12)=48. In the case of 2 dozen customers, that’s 2 groups of twelve of people each which equals 24 individual people, written mathematically as 2(12)=24.
Written mathematically, the proposition of 4 dozen eggs divided by 2 dozen diner customers is worked out mathematically as…
4(12)/2(12)=48/24=2
Doing the implied multiplication first, before dividing, is basic mathematical procedure which everyone can readily understand.
— Dee
Comment by Dee R. — May 28, 2025 @ 12:59 pm
Thank you, Dee. When I say dozen is a unit of measurement, I’m saying it’s a plural quantity characterized as a single unit or singular noun if you will. Grammatically in American English, we would say “A dozen eggs is expensive.” So if the expression is 12(2 + 2), that’s a dozen fours, but it’s meant to be understood as a singular value because of the juxtaposition. It’s calculated by multiplication, but the multiplication isn’t meant to be separated out like the sixteeners want to do.
Scott
Comment by Scott Stocking — May 28, 2025 @ 1:39 pm
I agree that “dozen” is a “plural quantity characterized as a single unit or singular noun.” That is precisely the reason I used it in the diner customers/eggs word problem — because we have been conditioned to understand that a dozen eggs is a unit of 12 individual eggs & a dozen diner customers means there are 12 people in the group.
When there are 4 units of 12 eggs each, it is easy for everyone to understand that there are 48 total eggs, to be split evenly among 2 separate groups of a dozen people each (i.e. 12 people per group). People do that simple math in their heads & instantly understand that each diner customer gets 2 eggs in that scenario.
What a number of people don’t recall is how to write out the math for that word problem.
First, you need to calculate how many eggs are available:
Total number of eggs=4 dozen=4(12)=48 eggs
Next, you need to calculate how many total customers there are who will split that quantity of eggs:
Total number of customers=2 dozen=2(12)=24 customers [people]
Now divide 48 eggs by 24 people=48÷24=2
…so the original proposition of 4 dozen eggs divided by 2 dozen customers is correctly written as…
4(12)÷2(12)=
48÷24=2
…or as…
4(12)/2(12)
or as the top-and-bottom fraction of…
4(12)
——–
2(12)
In all cases, it’s the total number of eggs divided by the total number of people, which is 48 divided by 24.
The bottom line is that in that word problem, there is no way around doing the juxtaposition (implied multiplication) BEFORE performing the division.
Comment by Dee R. — May 30, 2025 @ 4:03 pm
Grateful for you being a math wizard
Comment by SLIMJIM — May 31, 2025 @ 11:48 am |
Hi Scott,
After seeing Dave Peterson’s post on The Math Doctors website, “Implied Multiplication3: You Can’t Prove It,” I commented, sending him my word problem of 4 dozen eggs ÷ 2 dozen diner customers, to illustrate the rule that the implied multiplications MUST be done first, before the division (i.e. to determine the total value of what is being divided by the total value of what it is being divided by). I also provided proof of what is taught in schools, via links to a bunch of math tutoring websites designed to assist Algebra 1 students learn the material covered in the classroom, so they can pass their tests (on which they will not be allowed to use a computer calculator program). Those teaching websites instruct students on the rules & procedures for “How to Divide By A Monomial.” Dr. Peterson replied that I had contributed nothing to the discussion & has now apparently blocked my comments from appearing on that Math Doctors site.
You seem to have a good rapport with Dr. Peterson, so I am asking you to pose the following question to him:
If all of those math tutoring websites, with a lesson entitled something along the lines of, “How to Divide By A Monomial,” are not teaching the actual mathematical RULES to execute division-by-a-monomial (and demonstrating those rules with solved examples using the proper procedures), then what is it that is being taught to young Algebra 1 students by classroom teachers who will ask those division-by-a-monomial questions on a test?
References:
Greene Math .com: “How to Divide a Polynomial by a Monomial:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
Example 1, Example 2, Example 3
eCampus Ontario: “Divide Monomials”
https://ecampusontario.pressbooks.pub/prehealthsciencesmath1/chapter/5-5-divide-monomials-2/
Example 5.5.27
Open Stax .org: “Divide Monomials”
https://openstax.org/books/elementary-algebra-2e/pages/6-5-divide-monomials
Example 6.72, Example 6.143, Example 6.144
Siavula .com: “DIVIDING ALGEBRAIC MONOMIALS:
https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressions-part-2/08-algebraic-expressions-part-2-02
Worked Example 8.2
…and there are more.
Another question for Dr. Peterson: Given that x≠0 and x=1x…
x÷x=1
1x÷1x=?
— Dee
Comment by Dee R. — June 30, 2025 @ 2:14 pm |
Thank you for reaching out again, Dee. I have made my “best appeal” to Dr. Peterson, and he was gracious enough to give me a thorough response. He and I both agree that the idea many of the sixteeners push that arithmetic has different rules than algebra is absurd and inconsistent within the whole of mathematics. The push now must be to get juxtapositional binding acknowledged and recognized as a mathematical property on the level of the distributive, associative, and commutative properties. Only then will those who believe as we do have the theoretical prowess to declare the sixteeners’ argument to be null and void. Now that I’m done with a heavy travel season I’ve had for the past two+ months, I can start to devote some time to make that push.
Peace,
Scott
Comment by Scott Stocking — June 30, 2025 @ 9:59 pm |
I commend your dogged determination on this issue.
In Dr. Peterson’s blog piece titled, “Implied Multiplication 3: You Can’t Prove It,” he asked for “evidence of what is actually (explicitly) taught in various places, rather than mere examples of what is done (which could easily be mistakes or accidental assumptions).” In response, I sent him explicit evidence of what is being taught in schools around the world (from sites based in the U.S., Canada, South Africa, and in India), via lessons on tutoring websites with a title along the lines of “How to Divide By A Monomial.” Those websites are designed to aid students in learning the material taught in the classroom, on which they will be tested, so those sites must mirror what is explicitly taught in school. As a result, students are taught the rules for how to perform those calculations, so when they are confronted with classroom exam questions in that vein, they will be able to arrive at the correct answer (without the aid of a computer calculator program).
On those math tutoring websites I linked to, all of the concepts and rules were explicitly defined, described and demonstrated, ultimately giving students solved examples which are totally on point (i.e. horizontally written expressions using an obelus & no parentheses around the monomial term to the right of the obelus), showing students how the clearly laid-out rules for division-by-a-monomial are applied. On several of those sample problems, students are explicitly instructed to, “Rewrite as a fraction,” explaining that division is in fact the same thing as a fraction, and the entire monomial term was then shown as the monomial denominator of the top-and-bottom fraction.
If that is not “evidence of what is actually (explicitly) taught in various places,” then what exactly is Dr. Peterson looking for?
Comment by Dee R. — July 1, 2025 @ 9:26 am |
Addendum:
Are elementary school students specifically taught not to insert an explicit multiplication sign in “400” & “200” as “4(100)” & “2(100),”even though that’s what they are?
Would an elementary school student actually solve 400÷200 using PEMDAS, as shown below?
400÷200=
4(100)÷2(100)
4*100÷2*100
4*100=400
400÷2=200
200*100=20,000
~ ~ ~ ~ ~ ~
There is no evidence of a lesson which specifically instructed those young students not to insert those implied multiplication signs, even though that is what they’re made up of. By the time those students reach the 5th grade & are first learning PEMDAS, they already understand that “400” & “200” are each a single quantity which cannot be broken up, and therefore, they cannot use each component separately as its own stand-alone number. So how did those young math students learn that concept?
Comment by Dee R. — July 1, 2025 @ 11:13 am |
To sum up this whole proposition, PEMDAS, as currently taught, is substantively incorrect for division expressions.
That’s because, as taught in the 5th grade & then referred to again in Algebra 1: Fraction=Division, and therefore, Division=Fraction. In calculating the value of a fraction, division must go LAST (after doing all of the operations in the numerator & then the denominator). PEMDAS has division done in linear sequence, along with multiplication, wherever it occurs from left-to-right, after doing what’s inside parentheses (which is also not entirely correct, since it omits the use of the Distributive Law) & calculating the values of quantities with exponents.
Every Algebra textbook & bona fide teaching website instructs students that a term can be “factorized.” So “8” can be factored out as: 2(2+2), making the expression 8÷2(2+2)…
2(2+2)÷2(2+2)
Replacing what is inside the parentheses with the variable “a,” makes it…
2a÷2a
Any Algebra 1 teacher will tell you that, given that a≠0, the quotient of 2a divided by 2a is 1, whether written with an obelus, a slash, or as the top-and-bottom fraction of 2a over 2a.
It’s really that simple.
Comment by Dee R. — July 1, 2025 @ 3:49 pm |
Thank you, Dee, and Happy Independence Day! As I’ve said numerous times, the idea that a fraction is identical to a division problem with an obelus just doesn’t hold water if you analyze the uses of fractions. For example, if I say I ate two-thirds of an apple pie, I’m not intending that to be a division problem. It’s a ratio indicating how much of the pie I ate, i.e., it’s a stand-alone value, or a monomial. We must consider as well that a fraction is more accurate than a decimal if the decimal value has a repeating pattern. Perhaps you’re familiar with the proof that 0.999999…. (repeating to infinity) equals 1.0. If 1/3 (0.333…..) + 2/3 (0.666…..) = 1.0, then the sum of the decimal equivalents must equal 1. Via mathematical proof, this is proven true. However, you never actually get to the point of being able to “carry the 1,” so the decimal equivalent is 10^(-∞) off.
Granted, such a difference in real measurement is not really humanly perceptible, but it does reveal a potential issue when talking about astronomical or microscopical measurements, for example. Even if the fraction is part of a larger expression or formula, it’s always possible that the denominator of the fraction will “cancel out” exactly as a common factor with another element in the expression, so the simplification process (in theory; we tend to do this when manually solving an expression, but you don’t always “see” that when working the expression electronically) is the “division,” but it may not be with the numerator of the fraction. The Internet sites you offer are trying to simplify things for elementary students, but I’m looking more at the theoretical nuances of such things, so those examples, while helpful in demonstration, aren’t enough to advance the theoretical aspect of my argument.
What is key to the monomial discussion, then, is the juxtapositional binding. In the expression 8 ÷ 2(2 + 2), then, for the sixteeners to claim that somehow the 8 ÷ 2 becomes a fractional coefficient is contrary to the concept of the fraction. The obelus doesn’t have the binding property inherent with the fraction construction, so to replace the obelus with a fraction bar and say “they’re grouped” while unbinding the 2 from the (2 + 2) defies the clearly indicated juxtapositional binding. They try to have it both ways, but for whatever reason, they’re not seeing that. (Personally, I think they got burned at some point thinking they were justified in adding the multiplication sign and they got corrected by someone who believes as we do, and instead of thinking it through, they hunkered down in their rebellion, but that’s another story.)
This is where the linguistic aspect comes into play. We’ve seen how Wolfram interprets “Eight divided by twice the sum of two plus two” as 1, but “Eight divided by two times the sum of two plus two” as 16. “Twice” and “two times” mean the same thing at face value, but “twice” binds to the “two plus two” as an adverbial modifier, while “two times” is treated as a subject-verb. The latter treats “two” as an object and “times” as a verb substituting for the multiplication sign (2 x (2 + 2)). “Twice” has the multiplication implied in the adverbial phrase “twice the sum of two plus two,” so this more accurately reflects the implied multiplication of 2(2 + 2). As such, 2(2 + 2) is the mathematical equivalent of an adverbial phrase intended to be taken as a semantic unit, just like a fraction by itself is not a division problem, but a singular or monomial value.
I know that seems technical, but we’re talking theory here, so of necessity I must be technical and make the connection to other disciplines to prove my point. Thank you for letting me think out loud in response to your comments. Keep them coming. You keep me sharp!
Scott
Comment by Scott Stocking — July 3, 2025 @ 5:10 pm |
Hi Scott —
I hope you had a good 4th of July. I had a lot of fun!
In any case, your assertion that a fraction is its own thing & not division is mathematically incorrect, not only according to what is taught in 5th grade math, but also according to Algebra 1 instruction which recaps & reviews the fact that all division notation is interchangeable. Previously, I provided links to Algebra instructional websites which say point-blank that Fraction=Division. Math tutoring sites go on to tell high school students that the fraction bar=the slash & the slash= the obelus. For example…
from Australian Association of Mathematics Teachers:
https://topdrawer.aamt.edu.au/Fractions/Big-ideas/Fractions-as-division
“Anyone who has studied secondary school mathematics would probably be comfortable with the convention of ‘a over b’ meaning ‘a divided by b.”
~ ~ ~ ~ ~ ~ ~
from Greene Math .com Algebra Lesson:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“Dividing Polynomials by Monomials”
“In this lesson, we will learn how to divide a polynomial by a monomial (polynomial with one term). In order to perform this action, we will think back on operations with fractions. Let’s think about the following problem: 12 ÷ 3 = 4
We can rewrite this problem using a fraction bar. A fraction bar represents the division of the numerator by the denominator. In our example, 12 is being divided by 3, this means in fractional form 12 is our numerator and 3 is our denominator:
12
— = 3
4 ”
“Dividing a Polynomial by a Monomial”” * Set up the division problem as a fraction.”
“Example 1 Find each quotient (4x^4 + 2x^3 + 32x^3) ÷ 8x^2
Step 1) Let’s set up the division problem using a fraction:
4x^4 + 2x^3 + 32x^3————————-
8x^2 ”
~ ~ ~ ~ ~ ~ ~ ~
from Siavula (South Africa-based tutoring site):
https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressions-part-2/08-algebraic-expressions-part-2-02
“Operations with algebraic expressions”
“”WORKED EXAMPLE 8.2: DIVIDING ALGEBRAIC MONOMIALS: QUESTIONSimplify the following expression: 24t^7 ÷ 4t^5 SOLUTION: Step 1: Rewrite the division as a fraction. NOTEWe can write division as a fraction: a ÷ b= a—b This question is written with a division symbol (÷) but this is the same as writing it as a fraction: 24t^7 ÷ 4t^5 = 24t^7——–4t^5 “~ ~ ~ ~ ~ ~ ~
This is not an “elementary school explanation,” extremely simplified for children to understand it. That’s THE explanation for what a fraction actually represents — DIVISION. When a fraction cannot be simplified any further (e.g. 2/3), that is simply considered unexecuted division. The fraction of two-thirds is two divided by three. It is sometimes convenient to think of a fraction as having its own stand-alone value, but technically, mathematically, it is still division of the numerator by the denominator.
For further clarity, here is a “Lesson Guide” from OER (Open Educational Resources) Commons .org:
“Comparing Numbers with Ratios”
https://oercommons.org/courseware/lesson/1022/overview#:~:text=Ratio%20of%20Egginess-,A%20ratio%20is%20a%20comparison%20of%20two%20numbers%20by%20division,computing%2035%20%C3%B7%207%20=%205.
“Review the definition of ratio: A ratio is a comparison of two numbers by division.
The value of a ratio is the quotient that results from dividing the two numbers. For example, the value of the ratio 35:7 is 5, which you find by computing 35 ÷ 7 = 5.”
*. * . * . * . * . *
Fraction (and Ratio)=Division
…and therefore…
Division=Fraction (or Ratio)
With that being the case, 8÷2(2+2)=
8/2(2+2)
…which is also correctly written as…
8
——–
2(2+2)
…which equals…
8
—
8
…which, of course, equals 1, according to what is actually being taught in schools all over the world.
— Dee
Comment by Dee R. — July 8, 2025 @ 10:59 am
What the “sixteeners” don’t seem to understand is that. linguistically speaking, multiplication is just a shorthand “abbreviation” for doing addition. In other words, “4 dozen,” mathematically written as 4(12), is actually, “Twelve taken four times,” which is mathematically written out as 12 + 12 + 12 + 12. When there are 4 dozen eggs, that is a total quantity of 48 eggs, whether you count them out one at a time, add four twelves together, or multiply 12 by 4 — it’s the same total quantity of eggs!
Comment by Dee R. — July 10, 2025 @ 1:37 pm
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Hi Scott,
The “sixteeners” have yet to show textbooks or math tutoring websites which present solved examples of a juxtaposition in expressions with the same construction as 8 ÷ 2(2 + 2), being pulled apart by inserting an explicit multiplication sign where there is none, and then treating each individual piece of the original juxtaposition as if it had no relationship to the other part(s) of that juxtaposition. Showing a computer calculator program is not sufficient, as it may only indicate a programming flaw. Otherwise, their claim that PEMDAS is THE way to solve is disproven by math tutoring websites which affirmatively show solved examples which treat juxtapositions as inseparable single terms as the entire denominator of the fraction that students are instructed to rewrite the division expression as.
The question for the “sixteeners” is:
Are you trying to claim that all of those teaching websites, designed to aid students in learning what is taught in their classroom so they can pass their tests, are all teaching it wrong? If so, where is your evidence of that?
— Dee
Comment by Dee R. — July 19, 2025 @ 11:14 am |