If you like this article, you may also like 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough or 8 ÷ 2(2 + 2) = 1, Part 2: A Defense of the Linguistic Argument.
I’ve gotten caught up in the PEMDAS/Order of Operations (Parentheses/Brackets, etc.; Exponents; Multiplication & Division [left-to-right]; Addition & Subtraction [left-to-right]) discussions on social media. I must admit it is frustrating to see how religiously some people cling to an imperfect and almost thoughtless understanding of PEMDAS. It is a convention that, as currently practiced, has no theoretical soundness and is clearly NOT intuitive. In linguistic terms, I would describe the current application of PEMDAS as passive, that is, it doesn’t seek to recognize certain theoretical and intuitive constructs in mathematics, and even undermines them, and as such represents a primitive view of mathematics.
By “primitive,” I mean that some of the proponents of a strict PEMDAS are childlike in its defense, resorting to emotion and circular arguments based on their assumptions. No one has offered any proof that what they believe about its inner workings is in fact how people have historically interpreted some of these expressions. Nor have they offered anything to counter the various arguments and peer-reviewed documentation I have offered except by returning to their assumption and repeatedly showing us how they think the target expression should be worked even as we question their premise. PEMDAS pays too much attention to the individual signs and symbols, but it lacks the sophistication to understand the “syntax” or intention of certain conventions of expression we have developed over the centuries.
PEMDAS as practiced today is like reading the synoptic gospels of the New Testament without reading the gospel of John, or like reading the New Testament without also reading the Old Testament to understand its foundations. If you’re not biblically minded, it’s like reading Shakespeare without a 16th century dictionary at hand. I aim to fill that void here and make PEMDAS something so theoretically reliable that we won’t need to have wasteful discussions in social media about who’s right and who needs to go back to elementary school. My purpose is to propose such a theoretical structure whereby we can agree on one rule to address the issue of these viral problems.
Properties
I want to begin with a discussion about the basic mathematical properties we all had to learn at the beginning of the semester in Algebra. Properties are demonstrable principles in mathematics that will help lay the foundation for what some have begun categorizing as “grouping.” But more on that later.
Associative and commutative properties are intuitive. By that I mean that it’s obvious to most that if we have the expression
a * b * c
it does not matter how we “group” the terms, that is, how we “associate” them within parentheses or how we “commute” terms in the order of presentation, we get the same result. Every permutation of the variables multiplied together yields the same result. They’re all cofactors of the product, after all, so you get the same result every time. As such the following expressions are ALL literal identities of each other.
(a * b) * c ≡ (a * b)c ≡ (ab)c ≡ a * (b * c) ≡ a(b * c) ≡ a(bc) ≡ c * (a * b) ≡ c(a * b) ≡ c(ab) ≡ b * (c * a) ≡ b(c * a) ≡ b(ca), etc.
Distributive property is intuitive as well because it’s the undoing of factoring:
a(b+c) ≡ (ab+ac)
That expression represents implied (or implicit) multiplication, and the result demonstrates that the distributive property is a form of grouping equivalent to putting parentheses or brackets around the entire expression. As a type of grouping, then, it belongs to the P in PEMDAS. In addition, I would argue that even when there is an explicit multiplication sign used for the distributive property:
a * (b+c) ≡ (ab + ac)
this does NOT negate or override the application of the distributive property, and the expression thus written is also considered grouped and should be treated as if it has parentheses or brackets around it. Why? Because we know that if two expressions are equal or identical to a third expression, the other two expressions are identical as well:
a(b+c) ≡ a * (b+c).
Since both expressions are identical, it is necessary to consider both as grouped, as implied multiplication by the action of the distributive property at least, and in the case of the first one, by the syntax of implied multiplication. As such, my first proposal to strengthen PEMDAS and place it on a sounder theoretical ground is this:
Proposition 1: Mathematical properties should be considered at the same level of grouping as parentheses in PEMDAS, and when so recognized, the human person working such expressions has both the responsibility and authority, by virtue of any applicable mathematical properties, to place such expressions within the appropriate form of brackets to indicate priority given to the calculation. In doing so, the human person working the expression should ensure that any units of measurement or counting associated with the expression are not affected by so organizing.
The Trouble With Computers and Calculators. Most calculators do not recognize grouping characterized by this implicit syntax (I know syntax is primarily a linguistic concept, but for math, it refers to the arrangement of the signs and symbols, including the numbers, of an expression and how those influence the interpretation of the expression). You typically must apply the parentheses or brackets around the expression to force it to do so. Without parentheses, calculators assume all else in the syntax of an expression is explicit.
Implicit Multiplication Generally
Having trouble accepting implied multiplication is a form of grouping? What is a number with an exponent if not implied multiplication? The term 33, or in calculator language, 3^3 is 3 * 3 * 3, but we don’t undo the exponent notation if a division sign comes before it. Why? Because we’ve given this form of implied multiplication priority over signed multiplication in PEMDAS. The implied multiplication is grouped into the simple expression, algebraically speaking, of an. If you do deconstruct the implicit multiplication of an exponent, you’ve obviously taken PEMDAS to an unacceptable extreme.
PEMDAS does not address what to do with a factorial expression.
4! = 4 * 3 * 2 * 1
Algebraically, this is expressed as
n! = nPr =n(n – 1)(n – 2)…(n – r + 1])
where nPr is the number of permutations (P) of n things taken r at a time and n = r.
The factorial is also considered a form of implied multiplication, then. However, since PEMDAS doesn’t explicitly deal with factorials in the order of operations, how should we treat it? Algebraically, the first term in the grouping doesn’t have parentheses around it, so if we come across a ÷ n!, are we going to treat that as (a/n) * (n – 1)! then? Absolutely not! I’ve already hinted at why. It’s considered a grouping of factors implicitly multiplied. Because it’s a grouping, then, we would also treat it as if it were either an exponent or within parentheses. If these two types of implicit multiplication (powers and factorials) are generally treated at the E level in PEMDAS, why shouldn’t the implicitly multiplied expressions a(b + c) or even ab for that matter, or any variation thereof as described above, be treated at least at the E level, if not the P level of PEMDAS?
This leads me to my second proposition and a subsequent corollary to strengthen PEMDAS and place it on a more solid theoretical ground:
Proposition 2: In the absence of properties, any terms or expressions using implied multiplication should be considered at the grouping level of Parentheses in PEMDAS, and when so recognized, the human person working such expressions has both the responsibility and authority, by virtue of the implied multiplication, to place such expressions within the appropriate form of brackets to indicate priority given to the calculation. In doing so, the human person working the expression should ensure that any units of measurement or counting associated with the expression are not affected by so organizing. [Added 7/6/23] Within the grouping level, exponents and factorials should be processed first, then any implied multiplication would follow. So 3x^2 (without parentheses) should be evaluated as 3(x^2) and NOT (3x)^2.
Corollary 1: Any expression using explicit or implicit grouping markers (parentheses, brackets, juxtaposition, etc.) should NOT be rearranged in such a way as to undo the intention of the syntax or to fit certain elements into the framework of a property. For example, an expression in the form a/(bc) (explicit grouping) or a/bc (implicit grouping) should not be undone and reworked to become (a/b) * c or (ac)/b. NOTE: Some on social media tried to say the distributive property worked as follows:
b ÷ a(x+y+z) = [b * (x+y+z)]/a.
It’s ridiculous to think that a coefficient (or cofactor) should become a divisor in this situation.[1]
Let me also add here a call to return to standards. If the ISO standards don’t recognize the obelus anymore, then let’s just stop using it altogether. We can retain references to it for historical purposes, but in formal education (including textbooks), edit it out. People have tried saying the obelus is different from the vinculum, but again, I’ve never seen any formal explanation as to why this is, because the obelus isn’t a standard! I’m using it in this article because that’s how the expressions appear in the social media threads.
Excursus on Implicit Functions—Added 08/29/23 (from a post I made on Facebook), and lightly edited on 04/01/24.
Exponential terms don’t require a standard operator sign (+-x÷/) but rely on juxtaposition of the exponent.
Roots do not require a standard operator sign but the juxtaposition of the superscripted power and value in relation to the radical or of a fractional exponent. In some settings, the radical is NOT fully evaluated unless the answer can be expressed as a rational number. If the not-fully-evaluated radical is in the denominator of a fraction, the fraction must be rewritten so the radical is only in the numerator.
Factorials don’t require a standard operator sign but rely on the juxtaposition of a punctuation mark.
Cosine, sine, tangent, etc. don’t require a standard operator sign, but rely on the position of the measurement next to an abbreviated form of the word, and in the case of inverse trig functions, the juxtaposition of a -1 “exponent.”
Logarithms don’t require a standard operator sign but rely on the juxtaposition of a subscripted base (as needed) next to the abbreviated form of the word and the position of the value after the word/base combination.
Absolute value does not require a standard operator sign but juxtaposes, or brackets, the value between two pipes.
A fraction written with a vinculum does not require a standard operator sign that typically disappears when the operation is performed. The vinculum typically only disappears when the fraction can be represented as a whole number (mixed numbers, which consist of a juxtaposed whole number and fraction [implicit addition!!], typically have to be converted to an improper fraction when working them in an expression) or when, during the evaluation of the expression, the denominator divides evenly into the numerator. The fraction is a vertical juxtaposition of the two values with the vinculum indicating the grouping in the denominator and the relationship with the numerator. This is implicit division without actually dividing the two values to create a decimal. A fraction is considered a single value and NOT necessarily a division problem in the context of an expression. [This paragraph expanded from original on 04/01/24.]
Now while some of these functions do require parentheses to group the values being evaluated, NONE of them have a standard operator sign extant outside of any values in parentheses. These are all implicit functions, that is, they essentially have an implied set of brackets (or parentheses or some other grouping symbol) around them so they’re taken as a unit, or operand if you will. We don’t break out the implied multiplication of the exponent or factorial and supply the signs when evaluating the expression. Neither do we break out the implied addition when dividing by a mixed number, except in the manipulation of the mixed number to an improper fraction. In my opinion, that belongs in the P (parentheses) step of PEMDAS.
In the same way, an expression such as 2(2+2) is an implicit notation (no operational sign outside of the parentheses), so it belongs in the category of implicit functions. Operator signs break up an expression into operands. All this talk about “terms” that some have promulgated in the social media pages is meaningless, especially since the given expression above is technically one term by their own definition. We can debunk the contention 8 ÷ 2 should be treated as a fractional coefficient to what’s in parentheses. The obelus indicates we have two operands: 8 and 2(2+2). As an implicit function, with presumed parentheses around it, the 2(2+2) stands alone as the divisor in the expression. It should NOT be separated.
By the way, here’s how Wolfram defines “operand”: “A mathematical object upon which an operator acts. For example, in the expression 1×2, the multiplication operator acts upon the operands 1 and 2.” Implicit functions should have priority over explicitly signed operations and would need to have its own priority in PEMDAS for the functions included in that category.
End excursus.
Creating the Theoretical Foundation for PEMDAS.
Are you following me so far? Because I’m about to make it even more interesting. PEMDAS is neither intuitive nor theoretical (at least as applied) because there’s no proof to show why any one operation should be given preference over another. It is a convention that has rather haphazardly come into being historically without, as far as I can tell, any serious discussion about its theoretical or practical validity. As such, it does NOT mathematically or logically carry the same weight as properties.
The “intuitive” thing for a person to do just seeing a complex expression for the first time would be to simply work left-to-right through the symbols because we learn to read left-to-right in most languages. So such a person would treat the following as if it were in a series of brackets:
2 * 3 + 4 ÷ 5 – 6 = –4
becomes
{[(2 * 3) + 4] ÷ 5} – 6 = –4.
We used to play a game like that in 5th & 6th grade math class, where the teacher would give a string of numbers and operators (occasionally they’d put some operation in parentheses to see if we were paying attention) and we’d have to keep up with the math by taking the total as each new operator-number combination was announced and perform that operation on the running total. There was no going back through the string and figuring out PEMDAS or whether the distributive property applied. Whoever spit out the correct answer first got to advance toward the front of the class.
When we’re talking about PEMDAS, then, we’re talking about pedagogical construct that, historically speaking, seems to have been formulated rather hastily and loosely without much theoretical forethought. First, PEMDAS doesn’t account for everything that could be happening in an expression. I used the example of the factorial expression above. I’ve never seen anything in any explanation of PEMDAS that suggests where such an expression would fall. The factorial is important, for example, for calculating permutations. Treating a factorial in the wrong level of PEMDAS can potentially create false expectations about the probability of some event, like drawing winning lottery numbers. Exponents seems to be the most logical place because the factorial is a form of grouping by implied multiplication just like exponents are.
Second, in practice, PEMDAS is decidedly passive. It leaves too much room for interpretation, especially when historically, people have considered terms with implied multiplication to be grouped, bound, and inseparable. The PEMDAS absolutists in the social media threads on such problems have no respect for that historical reality. They suggest that the older generation’s understanding of PEMDAS is outdated when the real problem is their methodology and lack of critical thinking skills. This lack of critical thinking is becoming increasingly prevalent in American education today. Instead of a passive PEMDAS, we need an active, robust PEMDAS that focuses more on intentional grouping and a more sophisticated definition of grouping based on the assumptions made in the syntax of algebra.
Third, strict PEMDAS, at least as practiced by the absolutists, is no better than a calculator, because it fails to consider the implications of implicit multiplication. A calculator may be able to perform calculations quickly, but it doesn’t have a “mind” like you and I do to analyze the syntax and context of an expression.
The Priority of Operational Signs. Now that we have addressed the grouping issue and the weak nature of PEMDAS as currently practiced (if the absolutists are to be believed, anyway), it’s time to address the basic functions of mathematics: multiplication and addition. Since multiplication and addition are the primary operations mathematics, division and subtraction are respectively subordinate to them. We all can agree, I suppose, that division is multiplication by the inverse of the divisor, and subtraction is addition performed on a negative number. So while we’re beefing up the order of operations with a more active and robust use of parentheses to ensure any expression is solved the way its creator intended, let’s get rid of the obelus and the slash (solidus) as well and make everything about multiplication and addition. After all, isn’t that what we do when we divide by a fraction? We manipulate the expression (invert and multiply) to perform multiplication first, so wouldn’t that argue for the priority of multiplication in that instance? Let’s use only the vinculum to indicate “division,” and let’s put parentheses around a fraction with a vinculum so there’s no question that the vinculum is grouped. Fractions are, after all, more accurate than decimals if the decimal equivalent is not finite.
Let’s change all subtraction expressions to adding negative numbers expressions. Let’s keep all expressions with radicals as exponent expressions as much as possible. Let’s focus more on ensuring the proper ORDER of operations using parentheses and brackets rather than assuming everyone treats every expression in a uniform way. Obviously, that’s not happening right now. We can’t have a system of math where people are coming up with two (or more) answers to the same expression because they’ve been taught different things about it or work from a different set of assumptions. We need to firm up PEMDAS so we can have a uniform understanding of what fits where and restore uniformity. Without that, we’re wasting precious time arguing about ambiguities. It’s time someone cleared those ambiguities up. That’s why I’m stepping forward here to offer a theoretical solution.
One more question: If multiplication and addition are the primary operations of mathematics with division and subtraction respectively subservient to them, what is the theoretical reason then that multiplication and division are treated equally left-to-right? Why wouldn’t multiplication and addition be given priority over division and subtraction? I’ve never seen an explanation for this. After all, there are some out there who have a multiplication-first view of PEMDAS. I’m not sure I can fault that view given that multiplication has formal properties associated with it while division does not.
Conclusion
This whole process of “thinking out loud” in the social media posts has been, in my mind, anyway, a helpful discussion. In the process, I have more firmly codified my own beliefs about PEMDAS, as I’ve done here, and have continued to do so over the past year. I am even more firmly convinced that those who haven’t gotten the same answer as me on the more popular manifestations of this PEMDAS debate have a lack of theoretical understanding that should be corrected. I hope that I have filled that gap and that I have convinced at least some of you that the answer is 1, not the square of the numbers in parentheses (see Demonstration Proofs 1 & 2 below). I offer my proofs below of the three most popular expressions as a demonstration of application to my principles.
I do think PEMDAS is a valuable pedagogical tool, when properly understood with solid theoretical underpinnings. Professor Oliver Knill has repeatedly said the only way to ensure a problem is worked the way one means to is to overuse, if necessary, parentheses and other groupings.[2] Otherwise, ambiguous problems remain open to interpretation, and we’ll never get any kind of uniformity. My attempt here is to promote a uniformity of understanding so that we only get one answer for each unique expression.
Demonstration Proofs.
Proof #1
Given
36 ÷ 6(2 + 2 + 2)
Grouping by distributive property and implied multiplication
36 ÷ [6(2 + 2 + 2)]
Multiplication by inverse and grouping under vinculum
1
36 * —————-
[6(2 + 2 + 2)]
Application of distributive property
1
36 * ———————–
[6(2) + 6(2) + 6(2)]
Multiply terms in brackets
1
36 * —————-
[12 + 12 + 12]
Add terms in brackets
1
36 * —-
36
Cancel common factors and simplify
1
QED
Proof #2
Given
8 ÷ 2(2 + 2)
Grouping by distributive property and implied multiplication
8 ÷ [2(2 + 2)]
Multiplication by inverse and grouping under vinculum
1
8 * ———–
[2(2 + 2)]
Application of distributive property
1
8 * —————-
[2(2) + 2(2)]
Multiply within brackets
1
8 * ——
[4+4]
Add terms in brackets
1
8 * —
8
Cancel common factors and simplify
1
QED
Proof #3 (From Professor Oliver Knill’s “Ambiguous PEMDAS” Web page. When presented with this problem, 58 of his 60 calculus students got the following result indicated in the proof. The other two treated (3y – 1) as the expression under the vinculum. NONE got the answer promoted by the PEMDAS absolutists.)[3]
Given
2x/3y – 1; x = 9, y = 2
Group by implicit multiplication and vinculum
(2x)
—— – 1
(3y)
Substitute
[2(9)]
——- – 1
[3(2)]
Multiply in brackets
18
—- – 1
6
Simplify
3 – 1
Subtract
2
[1] Professor Oliver Knill from Harvard (Ambiguous PEMDAS (harvard.edu)) cites Lennes, 1917. [No title given]: “When a mode of expression has become wide-spread, one may not change it at will. It is the business of the lexicographer and grammarian to record, not what he may think an expression should mean but what it is actually understood to mean by those who use it. The language of algebra contains certain idioms and in formulating the grammar of the language we must note them. For example that 9a2 ÷ 3a is understood to mean 3a and not 3a3 is such an idiom. The matter is not logical but historical” (emphasis in original). In scans from Lennes printed article, these statements appear: “A series of operations involving multiplication and division alone shall be performed in the order in which they occur from left to right.” However, the subheading in the very next paragraph says: “The Above Rule Contrary to Actual Usage,” and the author goes on to say: “It would, however, follow from this rule for carrying out multiplications and divisions in order from left to right, that
9a2 ÷ 3a = (9a2 ÷ 3) × a = 3a3.
But I have not been able to find a single instance where this is so interpreted. The fact is that the rule requiring the operations of multiplication and division to be carried out from left to right in all cases, is not followed by anyone. For example, in case an indicated product follows the sign ÷ the whole product is always used as divisor, except in the theoretical statement of the case” (emphasis in original).
The following section of the text has the subheading “The Established Usage” reconfirms that the obelus indeed means the entire product after it is intended to be the divisor, and not just the first term. (Who says the obelus has no grouping power!) He also goes on to say “All multiplications are to be performed first and the divisions next.”
I’ve also found similar analysis in a 1935 textbook Second-Year Algebra by Hawkes, Luby, & Touton, p. 19. That’s how I understood it the late 70s and early 80s in high school and college and in the early 2000s when I taught JH & HS math and algebra. I seriously doubt nearly 100 years of history (at least) of this understanding has changed in the past ten years or so.
[2] Ambiguous PEMDAS (harvard.edu); see also https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html
[3] Knill, Oliver. “Ambiguous PEMDAS.” Ambiguous PEMDAS (harvard.edu), retrieved May 29, 2023.
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Hi Scott —
It’s not just the way this concept was taught & understood many decades ago — here is how division is currently being taught in 5th grade math & how monomial division is currently being taught in Basic Algebra:
from Better Lesson .com
Common Core: Fifth Grade
Number and Operations — Fractions
https://teaching.betterlesson.com/browse/common_core/standard/272/ccss-math-content-5-nf-b-3-interpret-a-fraction-as-division-of-the-numerator-by-the-denominator-a-b-a-b-solve-word-problems-invo?from=standard_level1
“Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).”
~ ~ ~ ~ ~ ~ ~
Australian Association of Mathematics Teachers:
https://topdrawer.aamt.edu.au/Fractions/Big-ideas/Fractions-as-division
“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 Slyavula Technology Powered Learning:
https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressions-part-2/08-algebraic-expressions-part-2-02
Grade 8
Algebraic Expressions
“WORKED EXAMPLE 8.2
DIVIDING ALGEBRAIC MONOMIALS
Simplify the following expression: 24t^7 ÷ 4t^5
SOLUTION:
Step 1: Rewrite the division as a fraction
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
…= 6t^2 “
~ ~ ~ ~ ~ ~ ~
from ecampus Ontario:
https://ecampusontario.pressbooks.pub/prehealthsciencesmath1/chapter/5-6-divide-polynomials-2/
“Divide a Polynomial by a Monomial”
“Example 5.6.2
Find the quotient:
(18x^3 – 36x^2) ÷ 6x
Solution
Step 1: Rewrite as a fraction.
(18x^3 – 36x^2)
——————-
6x “
~ ~ ~ ~ ~ ~
from Cue Math:
https://www.cuemath.com/algebra/simplifying-expressions/
“Simplifying Expressions”
“Example:
Simplify the expression: 3/4x + y/2 (4x + 7). By using the distributive property, the given expression can be written as 3/4x + y/2 (4x) + y/2 (7).”
~ ~ ~ ~ ~ ~ ~
from OpenStax .com
https://openstax.org/books/elementary-algebra-2e/pages/6-6-divide-polynomials Elementary Algebra
“Divide Polynomials”
“Example 6.78
Find the quotient: (18x^3 – 36x^2) ÷ 6x
Solution
Rewrite as a fraction.
18x^3 – 36x^2
——————–
6x
…Simplify.
3x^2 – 6x “
~ ~ ~ ~ ~ ~ ~
from Mathway, A Chegg Service:
https://www.mathway.com/popular-problems/Algebra/1012959
“Enter a Problem”
” 2x ÷ 2x
Rewrite the division as a fraction
2x
—-
2x
… 1 “
* – * – * – * – * – *
There is only one possible correct answer to 6 ÷ 2(1+2), which can be written as 2x ÷ 2x when x=(1+2) or x=3, because the division sign (obelus) is interchangeable with the fraction bar (and the slash). Also, parentheses are clearly not necessary around a monomial, because it is understood to be one term with a single value (which is the PRODUCT of the coefficient multiplied by the variable or variables). Given that those are the acknowledged facts in the math teaching world, the expression 6 ÷ 2(1+2) is exactly the same as the top-and-bottom fraction…
6
——–
2(1+2)
or 2x ÷ 2x, which can be written as the top-and-bottom fraction
2x
—-
2x
…both of which equal 1.
— Dee
Comment by Dee — June 21, 2024 @ 4:31 pm |
In principle, I don’t find anything wrong with the fraction bar being a replacement for the obelus, especially when working with monomials. I appreciate that you found examples of that and included them above, especially since those contradict those who want to separate the cofactors of a monomial and corrupt the process.
I did find an interesting twist to the first step in the order of operations in a college algebra textbook I hadn’t seen anywhere else. Here’s what it says:
1. Perform any operations inside parentheses, or above or below a fraction bar. (!!!!)
2. Compute all indicated powers.
3. Perform all other multiplication operations and any division operations in the order in which they occur from left to right.
4. Perform additions and subtractions in order from left to right.
Irving Donovan & William Wooton, Elementary Algebra for College Students, 6th ed. Wiley, 1984.
Notice in #1 the inclusion of the fraction bar as a grouping symbol parallel to the function of parentheses! That’s the first I’ve ever seen that mentioned. So that means the fraction bar is much more than just a sign of division, as I’ve been saying all along. It’s a grouping symbol. This is where the ambiguity of the obelus needs to be corrected. The obelus is to the dividend and divisor what the fraction bar is to the numerator and denominator. The difference is, the fraction bar groups like a vinculum, both the numerator/dividend and denominator/divisor individually above and below respectively, but also both elements as a monomial. No parentheses needed.
If I had longer expressions in the numerator and denominator, however, in order for me to switch the fraction to a division problem with an obelus, that would require me to add the implied parentheses to indicate the monomial nature of the fraction. Yet others balk at adding implied parentheses around a monomial in the form of 2(1 + 2) or something similar after an obelus. Not sure what it will take to get some of the hardheads to understand that juxtaposition, whether it’s implied multiplication, a mixed number (implied addition), or a fraction (implied division), creates a monomial.
And in case I haven’t emphasized this enough, a fraction is more precise than a decimal when a repeating decimal is involved or when it can’t be simplified to a whole number or short finite decimal. Although the fraction may imply or suggest division, sometimes the division doesn’t have to take place, so I would argue it’s not just another form of writing a division problem. Sometimes the fraction or mixed number gives you all the information you need without converting it to a decimal.
Scott
Comment by Scott Stocking — June 24, 2024 @ 10:32 pm |
PEMDAS (or GEMDAS) is an incorrect methodology to solve monomial division expressions. That’s because Division is a Fraction (and vice versa), and to evaluate a fraction, you do all of the operations indicated in the numerator, then do all of the operations indicated in the denominator, and then finally divide the numerator by the denominator. In evaluating a fraction, division must go LAST!
The only real reason to require parentheses in a monomial division expression is if there is more than one division symbol in the expression, so it is clear in which order the divisions are to be performed.
Example:
a/b/c can be interpreted as…
(a/b)/c which can also be written as
a/b
—–
c
or it can be interpreted as…
a/(b/c) which can also be written as
a
—–
b/c
Written horizontally with an obelus or slash, brackets would be needed to clarify what the order of multiple divisions is actually intended to be.
When inputting a division expression into a calculator (to have the calculator solve the math problem for you), it is advisable to put parentheses around a monomial is because some computer programmers missed that day in school when the 5th grade teacher went over the fact that Division=Fractions (and vice versa) & also missed that day in school when the Basic Algebra teacher went over how to divide monomials — rewriting a horizontally written monomial division expression using an obelus (or slash) as a top-and-bottom fraction. Because they missed those days in school, those programmers did not take monomial division into account in their program. That’s the reason that some computer calculator programs need “extra help” to interpret what the correct fraction actually is.
When calculating, on your own (i.e. not using a calculator program), the value of a division expression with only one division symbol, division must go LAST — because, as 5th graders are taught, the division sign (obelus), the slash (solidus), and the fraction bar (vinculum) are synonymous with one another & are therefore interchangeable.
PEMDAS (or GEMDAS) needs to be changed to:
Parentheses & Juxtaposition
(i.e. the implied multiplication will completely resolve the parentheses, either by using the Distributive Property or doing the operation within the parentheses & then multiplying by the coefficient just outside the parentheses)
Exponents
Multiplication (explicit)
Addition/Subtraction
Division
…which can be remembered as…
“Please Just Excuse My Aunt Sally Dee.”
That will allow all math students to correctly solve division expressions (i.e. as the fractions that they are).
It will also end the debate as to internet division memes such as…
6 ÷ 2(1+2)
…because that can correctly be written as the top-and-bottom fraction…
6
———-
2(1+2)
or it can be viewed as
2x ÷ 2x, when x=(1+2) or x=3, which can also be written as the fraction…
2x
—-
2x
Either way, the quotient is 1, not “x squared.”
Let’s start teaching these concepts in a consistent way, from 5th grade math through Algebra!
Comment by Dee — June 27, 2024 @ 6:20 pm
Hi Scott.
Have you seen this February 2017 article from The American Mathematical Monthly?
“Discussions: Relating to the Order of Operations in Algebra“
by N.J. Lennes, University of Montana
https://www.jstor.org/stable/2972726?seq=1
from the article:
“2. Actual Usage
Actual Usage. The Above Rule [The order of operations as PEMDAS] Contrary to Actual Usage. The rule stated above is agreed to by practically all those writers on algebra who make any mention of the matter at all. Chrystal gives a detailed development and writers on elementary algebra have in general followed him. It would, however, follow from this rule for carrying out multiplications and divisions in order from left to right, that
9a^2 ÷ 3a = (9a^2 ÷ 3) X a
But I have not been able to find a single instance where this is so interpreted. The fact is that the rule requiring the operations of multiplication and division to be carried out from left to right in all cases, is not followed by anyone. For example, in case an indicated product follows the sign ÷ , the whole product is always used as divisor, except in the theoretical statement of the case.
Writers meet the situation in different ways:
(a) Some always use the fractional form to indicate division, this being equivalent to a symbol of aggregation. Thus, ab/cd = (ab) ÷ (cd)
(b) Some write out the words in full, thus: “divide this expression by that expression.
(c) Some use the sign ÷ to mean that the whole product, following the sign shall be the divisor. The most important exception to (c) occurs in the development of the theory in such a text as Chrystal. In Chrystal, ÷u X v is sometimes used to mean ( u)v. In such cases, however, the notation u X v and not uv is used.”
“The followers of Chrystal have too often blindly copied his theory, but have not taken pains, as he did, to avoid inconsistency in usage. Examples of such inconsistency in theory and usage could be multiplied ad infinitum. One text, which has been in very wide use, states (in developing the theory)
60 – 40 ÷ 5 X 3 – 20 = 60 – (40/5) X 3 – 20
[to be clear, (40/5) is shown in the text as the top-and-bottom fraction of 40 over 5]
but on the next page we read:
10bc ÷ 12a =
10bc
——–
12a “
“The Established Usage. When an indicated product follows the sign ÷ . the whole product is, by overwhelming preponderance of actual usage, to be regarded as the divisor. Hence, the’ true rule as to the order of operations when both multiplications and divisions are involved is not the one stated above, but the following:
All multiplications are to be performed first and the divisions next.
That is, 9a^2 ÷ 3a = 3a and not 3a^3.”
“Mathematical Idioms. It might be agreed that, for the sake of simplicity and logical coherence, the past tense of the verb to drink should be drinked, but even so, English speaking people would continue to say drank, and not drinked. Precisely, for the same reason, all who know anything about the language of algebra regard 9a^2 ÷ 3a as equal to 3a and not 3a^3, and, therefore, the rule just given is the correct one as determined by actual usage.”
~ ~ ~ ~ ~ ~ ~
And there you have it.
— Dee
Comment by Dee — June 28, 2024 @ 4:23 pm
Yes, I have seen this article and cite it in one of my blog posts. But it’s 1917 article, not a 2017 article! It demonstrates that this argument about juxtaposition = monomial has been going on for well over 100 years. The author concludes that the juxtaposition is an idiomatic (i.e., linguistic and syntactical) usage.
Scott
Comment by Scott Stocking — June 28, 2024 @ 10:32 pm
I realized that you had already cited it, after I sent the comment. My apologies.
Here’s more on the interchangeability of division symbols:
from Math .net:
https://www.math.net/fraction-bar#:~:text=The%20meaning%20of%20the%20bar,the%20denominator%20is%20the%20divisor.
“A fraction bar (also referred to as the vinculum) is a bar that separates the numerator and the denominator of a common fraction. As shown below, the numerator is written above the fraction bar and the denominator is written below the fraction bar.”
“Fraction bars can also be written in different orientations, such as 1/12, or ½. The meaning of the bar is the same regardless of how it is written. It separates the numerator and denominator, and also indicates division. In a division problem, the fraction bar is the division symbol while the numerator is the dividend and the denominator is the divisor.”
~ ~ ~ ~ ~ ~
from University of Cambridge, “enrich”:
https://nrich.maths.org/2515
“History of Fractions”
“The word fraction actually comes from the Latin “fractio” which means ‘a breaking down, a division’.”
“In India fractions were written very much like we do now, with one number (the numerator) above another (the denominator), but without a line.
…It was the Arabs who added the line (sometimes drawn horizontally, sometimes on a slant) which we now use to separate the numerator and denominator”
. * . * . * . * . * . *
The slash & obelus are the same as the fraction bar.
Fraction = Division
…and therefore…
Division = Fraction
If the fraction bar is considered a grouping symbol, then so are the slash & obelus. They are all one and the same!
Comment by Dee — July 1, 2024 @ 12:20 pm
I realized that you had already cited it, after I sent the comment. My apologies.
Here’s more on the interchangeability of division symbols:
from Math .net:
https://www.math.net/fraction-bar#:~:text=The%20meaning%20of%20the%20bar,the%20denominator%20is%20the%20divisor.
“A fraction bar (also referred to as the vinculum) is a bar that separates the numerator and the denominator of a common fraction. As shown below, the numerator is written above the fraction bar and the denominator is written below the fraction bar.”
“Fraction bars can also be written in different orientations, such as 1/12, or ½. The meaning of the bar is the same regardless of how it is written. It separates the numerator and denominator, and also indicates division. In a division problem, the fraction bar is the division symbol while the numerator is the dividend and the denominator is the divisor.”
~ ~ ~ ~ ~ ~
from University of Cambridge, “enrich”:
https://nrich.maths.org/2515
“History of Fractions”
“The word fraction actually comes from the Latin “fractio” which means ‘a breaking down, a division’.”
“In India fractions were written very much like we do now, with one number (the numerator) above another (the denominator), but without a line.
…It was the Arabs who added the line (sometimes drawn horizontally, sometimes on a slant) which we now use to separate the numerator and denominator”
. * . * . * . * . * . *
The slash & obelus are the same as the fraction bar.
Fraction = Division
…and therefore…
Division = Fraction
If the fraction bar is considered a grouping symbol, then so are the slash & obelus. They are all one and the same!
— Dee
Comment by Dee — July 2, 2024 @ 11:42 am |
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