This article was inspired by a response to a reader’s comment on one of my other PEMDAS/OOO articles. I thought the point I made was worthy enough to be a separate post.
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 of the sum of the fractions is 10^(-∞) off.
Granted, such a difference in real measurement is not 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 is the “division,” but it may not be with the numerator of the fraction. For example, if I have the expression 6 x (2/3), I “cancel” the common factor of three from the 6 (leaving 2) and the denominator 3 (leaving 1, thus a whole number), so I’m left with 2 x 2 = 4. Division is happening in the cancellation of the common factor, but it happens with an element outside of the fraction itself. Therefore, the fraction itself is NOT a division problem. There are other elements acting on the fraction.
What is key to the monomial discussion, then, is the juxtapositional binding. In the expression 8 ÷ 2(2 + 2), 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 combination. The latter treats “two” as the subject and “times” as the verb substituting for the multiplication sign, thus 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, with or without the external parentheses, just like a fraction by itself is not a division problem, but a singular or monomial value. Therefore, 8 ÷ 2(2 + 2) = 1. Period. End of debate.
In today’s educational environment, there is a lack of critical thinking development. Young people are trained in mathematics simply to get to the point of an answer, not to learn the theory behind mathematics. In other words, they’re trained like calculators, and in this instance, they’re trained to process a problem like a cheap calculator would solve it instead of taking a broader view of the linguistic aspects of any mathematical expression. It’s really no different than giving a non-English speaker an English text and English dictionary and asking them to translate without any concept of grammatical rules, syntax, idioms, etc. PEMDAS/OOO is a formula designed to follow how a calculator solves an expression, not how students of math have solved the expressions in the past and certainly not to make the paradigm consistent with algebra, which treats an expression like 8 ÷ 2a as 8 ÷ (2a) and NOT 8 ÷ 2 x a. The linguistic argument supports the algebraic view of such expressions and therefore supports the same view for arithmetic.
Scott Stocking
My views are my own.
Sunday Morning Greek Blog 
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Addendum:
More on ratios being division…
from Study .com:
https://study.com/academy/lesson/what-is-ratio-in-math-definition-lesson-quiz.html#:~:text=Ratios%20mean%20division.,part%2Dto%2Dwhole%20comparison.
“FAQ
Does ratio mean multiply?
Ratio does not mean multiply. Ratios can be written as fractions and the fraction bare means division. Ratios mean division. For instance the ratio 1 to 3 means 1 divided by 3.”
“What does a 1 to 3 ratio mean?
The ratio 1 to 3 is comparing 1 item to 3 items. This can be written as 1 to 3, 1:3, or 1/3, and it can be thought of as dividing 1 by 3.”
= = = = = = = =
from Thought Co .com:
https://www.thoughtco.com/what-is-ratio-definition-examples-2312529#:~:text=In%20mathematics%2C%20a%20ratio%20is,is%20dividing%20termed%20the%20consequent.
“Because ratios are simple division problems, you can also write them as fractions.”
Comment by Dee R. — July 8, 2025 @ 11:51 am |
I want to clarify about ratios. A ratio may be in the form of a fraction, but it is not always intended to represent a decimal value to be calculated. For example, if I have a recipe for a casserole that feeds 4 people but I need to make enough for 12, I would use a ratio. If the recipe calls for 6 eggs for 4 people, then I would use the ratio and the resulting proportion to calculate how many eggs I’d need for 12 people: 6 eggs/4 people = x eggs/12 people. We would cross multiply to solve: 6(12) = 4x. Only then would division come into play to solve for x: 72/4 = 18 eggs. To find the “constant of proportionality” here, we divide the 12 people by 4 people and get 3. Theoretically, then, every other ingredient in the recipe must be multiplied by 3 to find out how much you need to make the casserole for 12 people.
Often times a ratio is the comparison of two different units of measurement. Velocity is the ratio of distance traveled compared to the time it takes to travel the distance. Riding a bike 30 miles in 2 hours returns a ratio of 30/2 miles/hour, or 15 miles/hour, or in fractional form, 15 miles/1 hour. We can divide and simplify the number, but the units are not reducible. In reality, velocity is not just a fraction, because we have to know the ratio of the units in case we need to use that ratio to calculate how far we could go in a certain time or how long it would take to go a certain distance. The same would go for other real-world measurements like density (mass/volume) or number of rotations of a wheel (distance/circumference or D/2πr) and converting that to both your analog velocity display and your odometer advancement on your car. Some ratios, like phi (φ) are irrational constants and, when displayed as a fraction, cannot have the irrational root in the denominator.
Allow me to quote from my source mentioned in my other response, Basic Technical Mathematics with Calculus by Allyn J. Washington (1970): “A ratio of a number a to a number b (b ≠ 0) is the quotient a/b. Thus a fraction is a ratio.” But then as the author goes on to describe examples of ratios, he never once converts them to a decimal equivalent (or decimal approximation for that matter, if you get my drift). So even though it may represent division, it is often division never fully executed, at least not until an actual measurement of some sort is needed.
A notable exception to this is trigonometric functions. Sine, cosine, tangent, etc. all represent ratios of the length of one side of a right triangle to another. So in a right triangle with sides 3, 4, and 5 units long, the tangent of the larger angle is 4/3, while the tangent of the smaller angle, which is also the cotangent of the larger angle, is ¾. One is the inverse of the other. But because we’re working with a right triangle, the Pythagorean Theorem is at play. So when we try to solve for the length of the hypotenuse, for example, we often wind up with an irrational square root of a number. Because trigonometric ratios are used to calculate real-life measurements, it then usually becomes necessary to convert the ratio to a decimal to make the measurement easier to make.
The concept of the ratio, then, is not quite as simplistic as your sources make it out to be. I’m sure there are very smart people behind your sources, but keep in mind they are writing to audience that may have a limited mathematics background and may not be well versed in the theory. I’m trying to make a theoretical argument here, so I’m trying to use peer-reviewed sources and more technical literature than just a basic math or algebra text.
Note on my source: In the first line of the Preface, the author states the purpose of his textbook: “This book is intended primarily for students in technical and pre-engineering technology programs or where a coverage of basic mathematics is required.” He also cites a number of reviewers who assisted him with the text, so I would consider this book to be a peer-reviewed primary source.
Scott
Comment by Scott Stocking — July 8, 2025 @ 9:37 pm |
I appreciate your detailed references and examples regarding ratios. Yes, a ratio is a comparison, but the bottom line on ratios is that, at some point, it does boil down to division.
As stated at an earlier juncture, when ratio or a fraction cannot be simplified any further without converting it to a decimal, it is simply considered unexecuted division.
Comment by Dee R. — July 9, 2025 @ 10:53 am
Below is a perfect parallel example from a bona fide math tutoring website (designed to help young algebra students learn the material covered by their teacher, to be able to pass their classroom exams). Note that the original expression is written with an obelus, with no parentheses around “2a” to the right of the obelus, and yet, “2a” is kept together as the denominator of the corresponding top-and-bottom fraction.
from Math only Math .com:
https://www.math-only-math.com/division-of-polynomial-by-monomial.html
“Division of Polynomial by Monomial”
“For example: 4a3 – 10a2 + 5a ÷ 2a
Now the polynomials (4a3 – 10a2 + 5a) is written as numerator and the monomial (2a) is written as denominator.
4a3 – 10a2 + 5a
———————-
2a “
Comment by Dee R. — July 9, 2025 @ 4:05 pm
If the original expression (4a3 – 10a2 + 5a ÷ 2a) doesn’t have parentheses around the polynomial, then the fraction you present isn’t the correct way to interpret that expression as written. The only two terms to be divided are 5a/2a. In the fraction you present, the 2a would factor out of the first two terms, and the a out of the third term, so the final form would be 2a^2 – 5a + 5/2.
Scott
Comment by Scott Stocking — July 9, 2025 @ 7:29 pm
Scott Stocking https://public-api.wordpress.com/bar/?stat=groovemails-events&bin=wpcom_email_click&redirect_to=http%3A%2F%2Fsundaymorninggreekblog.wordpress.com&sr=1&signature=14594ff7252bf9d875d1e8837735fc12&user=259896690&_e=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&_z=z just commented on Stocking’s Order: 2(2 + 2) as a Singular Semantic Elem https://sundaymorninggreekblog.com/2025/07/05/stockings-order-22-2-as-a-singular-semantic-element/
You say, “If the original expression (4a3 – 10a2 + 5a ÷ 2a) doesn’t have parentheses around the polynomial, then the fraction you present isn’t the correct way to interpret that expression as written.”
I thought so, too, when I first read that section. However, when I went back & looked at it again, I realized that the first time we see the expression, it’s…
https://www.math-only-math.com/division-of-polynomial-by-monomial.html
“For example: 4a3 – 10a2 + 5a ÷ 2a”
Then, the lesson goes on to instruct the students on how to make a top-and-bottom fraction out of that…
“Now the polynomials (4a3 – 10a2 + 5a) is written as numerator and the monomial (2a) is written as denominator.”
All it does is identify (in parentheses ) which part constitutes the “polynomials” & which part constitutes the “monomial,” so that there will be no confusion as to what part goes where in the top-and-bottom fraction. Once the terms were clearly delineated for the math students, the lesson continues with…
“Therefore, we get
4a3 – 10a2 + 5a
Comment by Dee R. — July 10, 2025 @ 10:24 am
That still doesn’t make any sense, though, because generally students are taught to combine like terms in any polynomial to simplify, so they would still do the 5a ÷ 2a first absent the parentheses. I looked at that Web page, and that’s just a poorly written example. Later on, they clarify what they mean by saying “Divide (polynomial expression) by (monomial expression)” so you know what the elements are. But you would never see something like that in an actual math text where they would assume the student knows the polynomial part is intended to be the whole dividend/numerator without parentheses or some other grouping indicator before the obelus.
Scott
Comment by Scott Stocking — July 10, 2025 @ 4:16 pm
In response to your comment, “That still doesn’t make any sense, though, because generally students are taught to combine like terms in any polynomial to simplify, so they would still do the 5a ÷ 2a first absent the parentheses. I looked at that Web page, and that’s just a poorly written example,” I submit that the expression speaks for itself — IF one understands how to parse the expression — which is being demonstrated on the math tutoring website, GreeneMath .com. As has been clearly stated on a number of other math teaching websites, the obelus=slash & slash=fraction bar, because Fraction=Division, with the division symbol separating the numerator from the denominator.
If you have a piece of paper & a pencil, and you are asked to write the improper fraction of “Forty-eight twenty-fourths,” you will presumably write 48, then draw a horizontal line under that, and then write 24 under that line.
If you are then told to write the improper fraction of “Forty-eight twenty-fourths” on a single horizontal line, going left-to-right, you will presumably write 48, then a slash to the right of that, and then 24 to the right of the slash.
Same fraction of “Forty-eight twenty-fourths,” correctly written using two different division symbols.
In every 5th grade math textbook, it affirmatively tells students that the slash, the division sign (obelus) & the fraction bar are all the same thing — they all mean “divided by.”
“Forty-eight twenty-fourths” is exactly the same as “Forty-eight divided by twenty-four,” in that they calculate out to 2, using division.
Parentheses are not necessary for that expression of 4a^3 – 10a^2 + 5a ÷ 2a because it’s 4a^3 – 10a^2 + 5a / 2a, and the slash is in the same place as the fraction bar as far as separating the numerator from the denominator.
Want more proof? Take a piece of paper & write the expression 4a^3 – 10a^2 + 5a ÷ 2a on it. Now rotate the piece of paper 90 degrees to the right. The slash is now, for all intents & purposes, the fraction bar, with the numerator of “4a^3 – 10a^2 + 5a” above it & the denominator of “2a” below it (they’re just facing a different way). It’s the same fraction!
As you can see from that “rotate the paper” exercise, the correct way to parse a division expression written with a slash or an obelus is that everything to the left of the slash or obelus is the numerator & everything to the right of the obelus is the denominator, just as everything above the fraction bar is the numerator & everything below the fraction bar is the denominator — UNLESS OTHERWISE INDICATED WITH PARENTHESES!
In this example of 4a^3 – 10a^2 + 5a ÷ 2a, if the author wanted it to be interpreted as the difference of 4a^3 minus 10a^2 plus the quotient of 5a ÷ 2a, it would have to be written thusly:
(4a^3 – 10a^2) + (5a ÷ 2a)
If there are no parentheses, the way to parse the expression originally written as 4a^3 – 10a^2 + 5a, is as shown on the algebra tutoring website GreeneMath .com
4a^3 – 10a^2 + 5a
————————
2a
Comment by Dee R. — July 10, 2025 @ 5:12 pm
The obelus is a graphic icon which visually communicates that, “What you’re looking at is a fraction,” with the numerator to the left of the division symbol & the denominator to the right of the division symbol. The placement of the division symbol (obelus or slash) in a horizontally written division expression directly corresponds to what is above the fraction bar & what is below the fraction bar in a top-and-bottom fraction.
from Nemeth aph tech .org:
https://nemeth.aphtech.org/lesson3.4
“In print, a common way for fractions to be displayed is to have the numerator, the top number, above a horizontal fraction line and the denominator, the bottom number, below the line. However, fractions may also be displayed with the numerator the the left of a diagonal line, or slash, and the denominator to the right. Also in some cases, the numerator may be slightly raised and the denominator slightly lowered. Remember that a fraction is really only a division problem where the division has not taken place.”
“Remember that everything to the left of the fraction line is considered the numerator and everything to the right of the fraction line is the denominator.”
~ ~ ~ ~ ~ ~ ~
from Fiveable Library:
“Intermediate Algebra”
https://library.fiveable.me/key-terms/intermediate-algebra/%C3%B7
“Definition
The division symbol, also known as the obelus, is a mathematical operation that represents the division of one number by another. It is used to indicate that one quantity is to be divided by another, resulting in a quotient.”
“5 Must Know Facts For Your Next Test
1. The division symbol (÷) is used to indicate that the number or expression to the left of the symbol is to be divided by the number or expression to the right of the symbol.”
“4. Simplifying a rational expression often involves canceling common factors in the numerator and denominator to obtain the lowest possible terms.”
Comment by Dee R. — July 11, 2025 @ 11:15 am
Hi Scott,
Below is a link to a comment I ran across on PlusMaths (a site affiliated with Cambridge University in the UK), regarding an article entitled, “The PEMDAS Paradox,” that I think you will be interested in.:
https://plus.maths.org/content/pemdas-paradox?page=2#comment-12647
A rather long comment posted on 20 October 2025 by somebody called “Katyakitka” starts with:
“saying The Order of Operations is Interior of Parenthesis; Exponents; Multiplication and Division whichever comes First left to right and then Addition and Subtraction left to right is incomplete.
First it is the Order of Operations between Terms. We have dropped the part “Between Terms” over the years in our obsession to oversimplify or use less syllables.”
…and the comment ends with:
“Yes, Math has Grammar.”
You may want to reply to this person.
— Dee
Comment by Dee R. — December 18, 2025 @ 2:55 pm
You say, “The obelus doesn’t have the binding property inherent with the fraction construction….”
According to what is taught in 5th grade (that Fraction=Division) & then recapped & demonstrated in Algebra 1, your assertion is incorrect. For example…
from Greene Math .com:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“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 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 Slide Share .net (a Scribd company):
https://www.slideshare.net/slideshow/94-16609182/16609182
“Dividing by a monomial”
“Write the division as a fraction”
“Example 3
Divide 4x^3 + 8x^2 + 10x by 2x
SOLUTION
(4x^3 + 8x^2 + 10x)÷2x=
Write as a fraction
4x^3 + 8x^2 + 10x
———————–
2x “
~ ~ ~ ~ ~ ~ ~ ~
from Lumen Learning Courses:
https://courses.lumenlearning.com/uvu-introductoryalgebra/chapter/9-5-dividing-polynomials-by-a-monomial/
“Find the quotient 56x^5÷7x^2
Rewrite as a fraction.
56x^5
——–
7x^2
Answer 56x^5 ÷ 7x^2 = 8x^3 “
~ ~ ~ ~ ~ ~ ~ ~
from Flex Books ck12 .org:
https://flexbooks.ck12.org/cbook/ck-12-cbse-maths-class-8/section/12.4/primary/lesson/division-of-algebraic-expressions/
“12.4 Division of Algebraic Expressions”
Solved example 2:
“2) Divide 12a^2 ÷ 6ab
12a^2
——— = …
6ab
2a
—-
b “
~ ~ ~ ~ ~ ~ ~
from eCampus Ontario:
https://ecampusontario.pressbooks.pub/prehealthsciencesmath1/chapter/5-6-divide-polynomials-2/
“5.6 DIVIDE POLYNOMIALS”
“Example 5.6.2
Find the quotient: (18x^3 – 36x^2) ÷ 6x
Solution
Step 1: Rewrite as a fraction.
18x^3 – 36x^2
—————–
6x “
* . * . * . * . * . * . *
On tutoring websites & education websites (for teachers) all over the world, there are a myriad of examples of this juxtapositional “stickiness,” in which the original expression was written with an obelus & there are no parentheses around the monomial divisor, which is converted to a top-and-bottom fraction, in which the entire monomial is the denominator (i.e. NOT just the numerical coefficient). This is how the concept is currently being taught, everywhere on the planet: A monomial is ONE TERM with a SINGLE VALUE which is the PRODUCT of its factors. A monomial does NOT get split into its constituent parts before its total value is calculated!
When calculating 8÷2(2+2), the “8” can be “factorized” as 2(2+2), making the expression…
2(2+2)÷2(2+2)
When “(2+2)” is replaced with a variable such as “a,” the expression becomes…
2a÷2a
when a=(2+2) or a=4
…and as is demonstrated on all those Algebra 1 teaching websites, the resulting top-and-bottom fraction is…
2a
—-
2a
…which, of course, equals 1.
Comment by Dee R. — July 8, 2025 @ 2:08 pm |
Dee, thank you for commenting. I have no quarrel with the way algebra understands the obelus, even though many algebra texts now are abandoning the use of the obelus. (I have a source titled Basic Technical Mathematics with Calculus by Allyn J. Washington published in 1970 that does not appear to use the obelus at all.) Algebra recognizes juxtapositional binding when it comes to a form like 8 ÷ 2a = 4/a, NOT 4a. The problem is the sixteeners do NOT recognize the force of juxtapositional binding when it comes to 8 ÷ 2(2 + 2). They violate the rules of algebra, which is why they are wrong.
I agree with your latest post as well, that anyone would be hard pressed to find an expression in a quality mathematics or even arithmetic text that is written with an obelus the way the viral expression is written. Most scholars agree that the form is vague and leads to confusion, especially for those who graduated from arithmetic and moved on to algebra. PEMDAS/OOO rules are not supposed to be a justification for crafting a poor expression for the sole purpose of playing “Gotcha!” on their forums. Those who do that are the true Internet trolls.
Scott Stocking
Comment by Scott Stocking — July 8, 2025 @ 8:21 pm |
I have no doubt that there have been algebra textbooks which contain no division examples which use the obelus. With that said, however, I linked to a number of examples which are used on current math tutoring websites, to demonstrate how to perform division-by-a-monomial. The first thing the young algebra students are told is, “Rewrite as a fraction” — and then it is demonstrated how to do that.
I disagree with the notion that 8 ÷ 2(2 + 2) is “ambiguous.” Once it is understood what a monomial is, then it’s perfectly clear that 2(2 + 2) is a single term [i.e. it’s “2a” when a=(2+2) or a=4].
At the time that children are taught PEMDAS in 5th grade, the idea of juxtaposition is NOT mentioned or demonstrated — that comes later on in school. THAT needs to change. A good way to do that is to have those young math students solve the simple real-world word problem of 4 dozen eggs split evenly among 2 groups of a dozen diner customers each. That proposition of a single quantity of eggs divided by a single quantity of people clearly demonstrates that there can be single “units” composed of multiple elements — and that the individual components which make up each single quantity cannot be pulled apart & used separately in another operation as if it had nothing to do with the other part. In other words, the children need to be educated that the “4” in “4 dozen” is not a separate thing from the “dozen” — it’s 4 cartons of a dozen eggs each, which, if you count them up, comes to 48 eggs in all.
Here’s an exercise to show how poorly online calculators are programmed:
Go to Google and cut & paste the following into the search box:
2a÷2a when a=4
Search.
Google AI will return an answer of “1,” explaining that any non-zero quantity divided by itself equals 1.
Now leave “2a÷2a” in the search box, exactly as it was, and delete “when a=4,” and perform the search again. The AI now tells you that the answer is “a-squared.”
The machine read the exact same mathematical expression totally differently when it wasn’t told the specific value of the variable. That’s not “ambiguity” — that’s just poor programming.
Comment by Dee R. — July 9, 2025 @ 10:41 am
With regard to your comment, “anyone would be hard pressed to find an expression in a quality mathematics or even arithmetic text that is written with an obelus the way the viral expression is written,” there are a number of math tutoring websites which demonstrate how to divide-by-a-monomial, with solved examples written with an obelus & no parentheses around multiple juxtaposed elements immediately to the right of the obelus, which was then converted to a top-and-bottom fraction containing the entire term (i.e. the numerical coefficient & variable or variables) as the denominator — not just the number sitting to the immediate right of the obelus.
If the “sixteeners” are correct, then they should have no problem citing a bunch of similar solved examples in textbooks and/or on bona fide math tutoring websites, in which variables could represent what is inside the parentheses in the expression 8 ÷ 2(2 + 2) [i.e. 2a÷2a when a= (2+2) or a=4], showing the example solved using PEMDAS from left-to-right, ignoring the juxtaposition by inserting an explicit multiplication sign where there wasn’t any, and treating each piece of it as if it was a completely stand-alone quantity. Let’s see those solved examples which instruct young algebra students to use that PEMDAS methodology to solve it.
Comment by Dee R. — July 9, 2025 @ 11:11 am
The problem the sixteeners have is their paradigm says that algebra rules are different than arithmetic rules. Until they experience a paradigm shift, they will forever be guilty of fomenting the ambiguity. The ambiguity is the result failing to recognize juxtapositional binding. They’ve gotten lazy and developed a paradigm based on how basic calculators work, not on demonstrated historic application. Because so many accept PEMDAS/OOO uncritically and folks who know better were busy using the classical understanding of the form to build rocket ships, synchronous-orbit satellites, and space telescopes, the problem was essentially ignored.
Scott
Comment by Scott Stocking — July 9, 2025 @ 7:44 pm
You say, “The problem the sixteeners have is their paradigm says that algebra rules are different than arithmetic rules.”
I challenge the “sixteeners” to clearly articulate which specific algebraic rules are “different” & conflict with earlier rules, and ask them to present concrete evidence of those differences & conflicts as shown in algebra textbooks & on bona fide algebra tutoring websites which teach young students how to perform operations in algebra. I found only references which REINFORCE concepts taught earlier in school, such as this:
from Greene Math .com:
“How to Divide a Polynomial by a Monomial”
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“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 think back on operations with fractions. Let’s think about the following problem:
12 ÷ 3 = 4
We can re-write this problem using a fraction bar. A fraction 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
Comment by Dee R. — July 10, 2025 @ 11:03 am
As stated earlier, a constant such as “12,” or a variable by itself such as “x,” are each considered to be a monomial which has an “invisible” coefficient of “1,” which is written mathematically as 1(12) & 1x [i.e. 12=1(12) & x=1x]. Linguistically, that would be expressed as, “One twelve” & “One ‘x,’ ” respectively. That concept can be expanded to include the existence of multiple “units,” as in, “Two units of twelve” & “Two units of ‘x,’ “which, written mathematically, is 2(12) & 2x, respectively. When x=12, or “dozen,” those are the same — they both mean “Two units of a dozen each” or “Two units of twelve each,” which equals 24.
Comment by Dee R. — July 9, 2025 @ 12:33 pm
The proof I just posted a little while ago uses the Transitive Property of Equality (a fundamental mathematical LAW) to demonstrate how to interpret 8÷2(1+3) correctly — as the term “8” factored out first as 2(4) and then as 2(1+3), and therefore “8” and “2(1+3)” are are both the exact same monomial. And any non-zero quantity divided by itself equals 1.
Comment by Dee R. — September 18, 2025 @ 4:35 pm
Remember that a constant such as “5,” or a variable by itself such as “a,” are each a monomial which is understood to have a coefficient of 1, making those monomial terms “1(5)” and “1a,” respectively. Linguistically speaking, that would be, “One five,” and, “One ‘a,’ ” respectively.
I challenge all of the “sixteeners” to find Algebra textbooks or bona fide teaching websites (NOT a calculator program & NOT some random YouTube, TikTok, or other social media poster who may not actually be a teacher), in which a solved example which is written with an obelus & has juxtaposition immediately to the right of the obelus (with no parentheses around it), in which young algebra students are specifically instructed to insert an explicit multiplication sign where there was none, and the solution is demonstrated in those textbooks & online tutoring sites as going strictly left-to-right, doing the multiplication & divisions as they come up (according to PEMDAS), which supports their contention that 8÷2(2+2)=16.
I have already provided numerous examples that classroom teachers (on sites which help teachers design a lesson plan for “Division by a Monomial”) & tutoring websites treat a monomial term such as “2a” as having a single value — not broken apart by inserting an explicit multiplication sign between component parts of “2” & “a,” and treated as if those were two completely separate terms that have nothing to do with each other.
Comment by Dee R. — July 8, 2025 @ 3:28 pm |
[…] Stocking’s Order: 2(2 + 2) as a Singular Semantic Element | Sunday Morning Greek Blog July 5, 2025 […]
Pingback by SMGB Indices | Sunday Morning Greek Blog — July 14, 2025 @ 4:35 pm |
Let’s work this from a different angle…
8 ÷ 8=
8
—–
8
This concept is confirmed by Common Core 5th grade mathematics:
https://www.thecorestandards.org/Math/Content/5/NF/#:~:text=Interpret%20a%20fraction%20as%20division,rice%20should%20each%20person%20get
“Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).”
…and by
Greene Math .com:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“…we will think back on operations with fractions. Let’s think about the following problem: 12 ÷ 3 = 4 We can re-write this problem using a fraction bar. A fraction 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—– = 4 “3
~ ~ ~ ~ ~ ~ ~ ~
So we have
8
—–
8
We can factor out each 8 as 2(4), making the fraction…
2(4)
——-
2(4)
First, take care of the operation indicated in the numerator: 2(4)=8.
Now the expression is…
8
——–
2(4)
Now write this fraction horizontally with a slash in place of the fraction bar:
8/2(4)
The operation indicated in the denominator of the fraction must be done before dividing the numerator by the denominator:
2(4)=8
The “4” in parentheses can also be converted to (2+2), making the denominator…
2(2+2)
Rewrite the horizontal fraction with 8 as the numerator & 2(2+2) as the denominator:
8/2(2+2)
…which can also be correctly written as…
8÷2(2+2)
Now the horizontal fraction is…
8/8
…which equals 1.
Backtrack 8/2(2+2) and it comes out to 8 over 8, which equals 1. And as every 5th grade math textbook tells students, Fraction=Division, so 8/2(2+2) can be written with an obelus in place of the slash, as…
8÷2(2+2)
Comment by Dee R. — July 15, 2025 @ 1:55 pm |
Let’s work this from a different angle…
8 ÷ 8=
8
—–
8
This concept is confirmed by Common Core 5th grade mathematics:
https://www.thecorestandards.org/Math/Content/5/NF/#:~:text=Interpret%20a%20fraction%20as%20division,rice%20should%20each%20person%20get
“Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b).”
…and by
Greene Math .com:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“…we will think back on operations with fractions. Let’s think about the following problem: 12 ÷ 3 = 4 We can re-write this problem using a fraction bar. A fraction 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—– = 4 “3
~ ~ ~ ~ ~ ~ ~ ~
So we have
8
—–
8
We can factor out 8 as 2(4), making the fraction…
2(4)
——-
2(4)
First, take care of the operation indicated in the numerator: 2(4)=8.
Now the expression is…
8
——–
2(4)
Now write this fraction horizontally with a slash in place of the fraction bar:
8/2(4)
The operation indicated in the denominator must be done before dividing the numerator by the denominator:
2(4)=8
The “4” in parentheses can also be converted to (2+2), making the denominator…
2(2+2)
Rewrite the horizontal fraction with 8 as the numerator & 2(2+2) as the denominator:
8/2(2+2)
Now the horizontal fraction is…
8/8
…which equals 1.
Backtrack 8/2(2+2) and it comes out to 8 over 8, which equals 1. And as every 5th grade math textbook tells students, Fraction=Division, so 8/2(2+2) can be written with an obelus in place of the slash, as…
8÷2(2+2)
Comment by Dee R. — July 15, 2025 @ 1:59 pm |
This rightly describes how you and I think about the expression at hand, but it does not offer a theoretical proof of why we believe that way. Let me take a different tactic on this. How does a computer know that a collection of numerals should be interpreted either as an ordinal number or an ID number? Each individual character has its own Unicode identifier (typically base 16/hexadecimal for those writing code at least). When a computer looks at a number, it will typically default to considering a set of juxtaposed numbers as an ordinal number. However, when you start to use variables in code, you have to identify what type of data fits in that variable. If I want a variable to be considered an ordinal, I would write “Dim x as Number” or “Dim x as Integer.” If I want it to be considered as simply an ID number that shouldn’t be considered an ordinal number subject to mathematical operations, I would say “Dim X as Text” or “Dim X as String.” Excel’s cell formatting does something similar. When I tell the computer I want a juxtaposed set of numerical digits (or even a single digit) to be considered as an ordinal number, it applies a built-in place-value paradigm so it knows that each digit is multiplied by a power of ten and the order of the numbers and relationship to a decimal (implied or extant) point and added together. There are no extant operational signs indicating that’s how the computer understands the numbers.
It’s the same with language models. The computer sees a juxtaposed collection of letters separated by a space and considers each collection a “word.” It compares the word to a built-in dictionary so it can discern meaning, even to the point of examining the words around it understand how the word is used in context.
If a computer can recognize juxtaposed letters as words and juxtaposed numerals as ordinal numbers, then it can (or should be able to) recognize that a juxtaposed multiplication form (like 2(2 + 2)) is a single numeric value. The same should be applied to fractions as well. Mixed numbers might be a bit trickier–they may still require parentheses with an extant addition sign. Set the paradigm to recognize that two numbers juxtaposed to and on either side of a solidus/slash represent a unique value. Establishing such a paradigm would minimize the need for an extra set of parentheses and reduce file size.
Scott
Comment by Scott Stocking — July 15, 2025 @ 9:23 pm |
Your question of, “How does a computer know that a collection of numerals should be interpreted either as an ordinal number or an ID number?” brings to mind that computers do not rule the mathematical world. Computers need to be reprogrammed to reflect how humans have interpreted mathematical expressions for hundreds of years.
Every algebra textbook & tutoring website instruct students that a monomial is ONE TERM with a SINGLE VALUE which is THE PRODUCT of its factors. Examples of monomials are given, such as “2x,” “5y,” or “10ab^2.” Note that those monomials are not shown in parentheses. On a number of teaching websites, division-by-a-monial expressions is explained & demonstrated with the original expression written with an obelus, and the students are instructed on how to simplify & solve:
“Step 1) Rewrite as a fraction.”
Then, the teaching webstie shows exactly how to do that:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“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 ”
https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressions-part-2/08-algebraic-expressions-part-2-02
“WORKED EXAMPLE 8.2DIVIDING 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 ”
And there are many more online examples of this lesson, instructing students on how to divide by a monomial — using an obelus & no parentheses around the monomial denominator.
Computer programmers across-the-board need to wake up & write calculator programs that interpret monomial division expressions the way it has always been taught & is still currently being taught to students all over the world: It’s understood that 4 dozen eggs split evenly among 2 dozen diner customers is “4 dozen ÷ 2 dozen,” in which it is well-understood that “4 dozen” is a single quantity & “2 dozen” is another single quantity. It’s not 4…times…12, & 2…times…12, as if everything is its own separate entity. It’s 4a ÷ 2a when a=12.
Computer programmers need to review how to plug in the value of a variable once it is known, and write a program to interpret juxtaposition in that way.
Comment by Dee R. — July 17, 2025 @ 3:51 pm
Hi Scott,
Here’s an online calculator which parses the expression the same way, whether input as 8÷2(2+2) , as 2(2+2)÷2(2+2), as 2(4)÷2(4), or as 2a÷2a:
https://scanmath.com/calculator/step-by-step-calculator/8%5Cdiv2%5Cleft(2%2B2%5Cright)
Each of those inputs still equals 1.
In other words, this online calculator does not use PEMDAS the way it is being interpreted by the “sixteeners.” Juxtaposition is treated as being ONE TERM (a monomial) with a single value which is the PRODUCT OF ITS FACTORS.
— Dee R.
Comment by Dee R. — July 22, 2025 @ 3:27 pm
It is obvious that the Order of Operations as PEMDAS is incomplete (i.e. it omits implicit multiplication by juxtaposition in the “Parentheses” step) & is substantively incorrect when division is indicated in a linear expression, because as any 5th grader can tell you (via Common Core Curriculum), Fraction=Division, and therefore, Division=Fraction — which means that DIVISION MUST GO LAST!
Here’s more proof — go to Google Search and copy & paste the following question into the search box:
Is the expression 4a÷2a when a=12 the same thing as 4(12)÷2(12)?
Then search.
The Google AI Overview answer is, “Yes, the expression 4a÷2a when a=12 is the same as 4(12)÷2(12).”
And it goes on to give a step-by-step explanation as to the reason that that is true.
from the Google AI Overview response:
“Understanding the Expression: The expression 4a÷2a involves the variable ‘a’. When a specific value is given for ‘a’, you substitute that value into the expression.
Substitution: In this case, a=12. So, wherever you see ‘a’, you replace it with 12.
4a becomes 4×12 or 4(12).
2a becomes 2×12 or 2(12).
The Resulting Calculation: Therefore, 4a÷2a when a=12 becomes 4(12)÷2(12).
Calculating the value:
4(12) = 48
2(12) = 24
48 ÷ 24 = 2
So both expressions evaluate to 2.”
Comment by Dee R. — August 1, 2025 @ 11:57 am
Hi Scott,
Given that d, a, x, b, c & (b+c)≠0:
d=ax
so…
d÷d=ax÷ax
and…
d÷d=d÷ax
x=(b+c)
so…
ax÷ax=a(b+c)÷a(b+c)
which means that…
ax=a(b+c)
and because d=ax & ax=a(b+c)…
d=a(b+c)
so…
ax÷ax=a(b+c)÷a(b+c)
and…
d÷d=d÷a(b+c)
d÷d=1
so…
a(b+c)÷a(b+c)=1
and because d÷d=1 & d=a(b+c)…
d÷a(b+c)=1
. * . * . * . * . * . *
Now let’s plug in the values of the variables:
d=8
a=2
x=4
b=1
c=3
d÷d=8÷8=1
ax÷ax=2(4)÷2(4)=8÷8=1
a(b+c)÷a(b+c)=2(1+3)÷2(1+3)=2(4)÷2(4)=8÷8=1
and because d=ax & ax=a(b+c), then d=a(b+c), so…
d÷d=d÷a(b+c)=
8÷2(1+3)=
8÷2(4)=
8÷8=1
Factoring out a term is a real thing in math — which does not change the original term’s value. When factoring out the monomial term “24” as 2(12), which is the monomial 2a when a=12 or a=(9+3), the total value of the term remains exactly the same — 24.
DEFINITION: A monomial is one INSEPARABLE term (formed by implied multiplication by juxtaposition), holding a single combined total value of the PRODUCT of its factors.
— Dee
Comment by Dee R. — September 18, 2025 @ 4:11 pm
Below is a perfect parallel example from a bona fide math tutoring website (designed to help young algebra students learn the material covered by their teacher, to be able to pass their classroom exams). Note that the original expression is written with an obelus, with no parentheses around “2a” to the right of the obelus, and yet, “2a” is kept together as the denominator of the corresponding top-and-bottom fraction.
from Math only Math .com:
https://www.math-only-math.com/division-of-polynomial-by-monomial.html
“Division of Polynomial by Monomial”
“For example: 4a3 – 10a2 + 5a ÷ 2a
Now the polynomials (4a3 – 10a2 + 5a) is written as numerator and the monomial (2a) is written as denominator.
4a3 – 10a2 + 5a
Comment by D Rosenberg — September 1, 2025 @ 9:18 am |
Dee: I’ve been crazy busy this past month, so sorry for not having responded. The “Math Only Math” Web site you cite in your most recent comment is suspect on a couple different levels.
First, the expression is written without any indication of grouping. In context, it seems the assumption is that any polynomial written in the form ax^n + bx^(n-1) + cx^(n-2) + … should be treated as a grouped element. But that doesn’t fly in the PEMDAS/OOO world at all since addition and subtraction come after division and multiplication, and it’s never explicitly defined as such in the lesson. If the polynomial expression is to be considered grouped, it would need to have parentheses around it or a vinculum over it when written inline like that. Most standardized math texts would make that crystal clear to avoid any ambiguity.
Second, the author(s) of the Web site apparently use the plural term “polynomials” to refer to the individual terms in the (singular) polynomial expression. The correct terminology for the individual elements of the polynomial is “terms,” so the lack of proper terminology presents an additional level of ambiguity to the learner.
Third, if you look at the answer they give at the end of the example, the answer is incorrect. They say the answer is 4a^2 − 5a + 5/2, but 4a^3/2a = 2a^2. The poor editing and quality control on this site makes it one I would not trust to defend any mathematical argument.
As for Dr. Peterson on the Math Doctors site: I have a great deal of respect for him, and we have had an extended private conversation about my theories, but he continues to think I’m on a fool’s errand trying to push my views. That’s okay–I should probably expect that. But I’m too far into this now to be dissuaded. As I have indicated in earlier responses, these Web sites are not primary sources, so while they have value for demonstrating how the material is taught, they are not theoretical enough to carry any weight with me or I would presume with Dr. Peterson as well.
Have a great Labor Day!
Scott
Comment by Scott Stocking — September 1, 2025 @ 10:35 am |
Scott,
No worries about response time — I just appreciate your responding to what I wrote.
I see what you’re saying about that website being “suspect.” Here’s something else I came across that may be of interest to you:
1904 edition of “Algebra an Elementary Text-Book” by Chrystal:
chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://djm.cc/library/Algebra_Elementary_Text-Book_Part_I_Chrystal_edited.pdf
See pages 13, 14 & 15:
“For Multiplication – Law of Distribution” and “Division for Purposes of Algebra”
“Another notatation for a quotient is very often used, namely
a
—–
b
or a/b. As this is the notation of fractions and already has a meaning attached to it in the case where a and b are integers, …Hence we conclude that
a
—-
b
is operationally equivalent to a÷b in the case where a and b are integers.”
“Quotient and Fractions”
“…there is no obstacle to regarding it [fraction] as an alternative notation for a÷b.”
~ ~ ~ ~ ~ ~
There is more information there on the Distributive Law that I did not post here, and goes on to talk a lot about division being the inverse of multiplication — but worth reading.
— Dee
Comment by Dee R. — September 1, 2025 @ 1:52 pm
Hi Scott,
Given that d, a, x, b, c & (b+c)≠0:
d=ax
so…
d÷d=ax÷ax
x=(b+c)
so…
ax÷ax=a(b+c)÷a(b+c)
Therfore…
d÷d=ax÷ax=a(b+c)÷a(b+c)
d÷d=1
so…
ax÷ax=1
and…
a(b+c)÷a(b+c)=1
Now let’s plug in the values of the variables:
d=8
a=2
x=4
b=1
c=3
d÷d=8÷8=1
ax÷ax=2(4)÷2(4)=8÷8=1
a(b+c)÷a(b+c)=2(1+3)÷2(1+3)=2(4)÷2(4)=8÷8=1
Factoring out a term is a real thing in math — which does not change the original term’s value.
Comment by Dee R. — September 18, 2025 @ 9:45 am
Hi Scott,
Here’s a question for Dr. Peterson, as well as the ‘sixteeners’:
Before performing the division in the expression 4a ÷ 2a, how many terms are there?
~ ~ ~ ~ ~ ~ ~
Below is a link to some reference material on the subject, from a math tutoring website:
“Dividing Terms in Algebra”
https://www.mathematics-monster.com/lessons/how_to_divide_terms_in_algebra.html#:~:text=The%20term%20being%20divided%20(4a,when%20we%20divide%20the%20terms.
— Dee
Comment by Dee R. — September 3, 2025 @ 12:23 pm
Hi Scott,
Here’s a question for Dr. Peterson, as well as the ‘sixteeners’:
Before performing the division in the expression 4a ÷ 2a, how many terms are there?
~ ~ ~ ~ ~ ~ ~
Below is a link to some reference material on the subject, from a math tutoring website:
“Dividing Terms in Algebra”
https://www.mathematics-monster.com/lessons/how_to_divide_terms_in_algebra.html#:~:text=The%20term%20being%20divided%20(4a,when%20we%20divide%20the%20terms.
— Dee
Comment by Dee R. — September 3, 2025 @ 12:25 pm
Hi Scott,
I hope everything is OK with you. You can reply to this whenever you find a free minute — no worries.
On topic…
Google this question:
When a=(9+3), does 4a÷2a=4(9+3)÷2(9+3)?
The AI answer comes up as “Yes,” with the same quotient of 2 for both sides of the equation, even though they each use an obelus as the division symbol & there are no parentheses around “4a,” “2a,” 4(9+3), or 2(9+3). That’s because 4(9+3) & 2(9+3) are just substituting the value of the variable “a” into the expression, which does not effect how to calculate the monomial division of 4a÷2a.
4(9+3)=48
…which, before performing the implied multiplication of the monomial “2a” which is 2(9+3) after substituting in the value of the variable on the other side of the division sign, makes the interim step…
48÷2(9+3)
…which is the internet division meme from 2011.
— Dee
Comment by Dee R. — September 8, 2025 @ 3:37 pm
Hi Scott,
Consider this…
Given that no variable equals 0 & (b+c) also does not equal 0, simplify the following expression:
a(b+c)÷a(b+c)
Is there any algebra teacher on the planet who would tell students to use PEMDAS to solve, versus telling students to use the Distributive Law to solve a(b+c)÷a(b+c), expanding parentheses to form (ab+ac)÷(ab+ac), resulting in a quotient of 1?
— Dee
Comment by Dee R. — September 15, 2025 @ 1:58 pm
The proof I just posted a little while ago uses the Transitive Property of Equality (a fundamental mathematical LAW) to demonstrate how to interpret 8÷2(1+3) correctly — as the term “8” factored out first as 2(4) and then as 2(1+3), and therefore “8” and “2(1+3)” are are both the exact same monomial. And any non-zero quantity divided by itself equals 1.
Comment by Dee R. — September 18, 2025 @ 5:08 pm |
Hi Scott,
Happy Thanksgiving! I hope all is well with you.
On topic: Consider this…
A monomial is one term holding a single value which is the product of its factors. A constant is a monomial (with a coefficient of 1).
Also, note the The Substitution Property of Equality [also called “Leibniz’s Law“].
from Study .com:
https://study.com/academy/lesson/substitution-property-of-equality-definition-examples.html#:~:text=%7B%20y%20=%202%20x%20+%203,in%20the%20study%20of%20mathematics.
“The substitution property of equality, one of the eight properties of equality, states that if x = y, then x can be substituted in for y in any equation, and y can be substituted for x in any equation.”
“If we did not use this property in algebra, we would not be able to plug in known values for variables into mathematical expressions and equations.”
“Let’s say that we have 5x and we know that x = 5. We can use the substitution property of equality to plug in the value of x into 5x:
So we would get 5 x 5, which is 25″
~ ~ ~ ~ ~ ~ ~ ~
Additionally, 5th graders learn that Fraction=Division, thus Division=Fraction & that is reiterated later on in algebra:
“Dividing by Monomials”
https://openstax.org/books/algebra-1/pages/6-3-1-dividing-by-monomials
Look at the division expression -56a^6b^3÷9a^4b^2
“1. Rewrite the division expression as a fraction.
Fraction form:
-56a^6b^3
—————–
9a^4b^2 “
– – – – – – – – – – –
Note that there were no parentheses around 9a^4b^2, yet it was demonstrated for young math students as being the entire denominator — not just the coefficient of “9” which is immediately to the right of the obelus.
~ ~ ~ ~ ~ ~ ~
Because of the substitution property of equality [“Leibniz’s Law“], 48÷24 can be substituted for its factored forms…
48=4(12)
24=2(12)
Thus…
48÷24=4(12)÷2(12)=2
And since 4(12)=48, then the expression can alternatively be presented as…
48÷2(12)
…which can be rewritten as the fraction
48
——–
2(12)
…all of which also equals 2, just as the originally written version of the expression 48÷24.
The substitution property of equality means that…
48÷2(12)=48÷2(9+3)
…which still equals 2.
Another illustration of the substitution property of equality:
4x÷2x when x=12
Substitute in 12 for x:
4x÷2x=4(12)÷2(12)=48÷24=2
…and when x=12, 4x÷2x can be written as the fraction
4x
—- =
2x
Substituting in the value of x:
4(12)
—— =
2(12)
48
—— =
2(12)
48
—— = 2
24
The Order of Operations as PEMDAS is never to taught to 4th or 5th graders specifically demonstrating how to “process” implied multiplication by juxtaposition because they haven’t learned about that yet & elementary school teachers don’t want to confuse the issue with explicit multiplication. When students later learn about implied multiplication by juxtaposition, they are explicitly instructed that that implied multiplication by juxtaposition constitutes a monomial, defined as one inseparable “unit” (even if composed of multiple elements) holding a single value which is the product of its factors.
All of this conclusively proves that PEMDAS is an incorrect methodology to use to simplify a horizontally written division expression for several reasons:
1. Constants are monomials & a monomial can be factored out, still maintaining its original value.
2. Parentheses are not necessary around a monomial to be understood as being one term with a single value which is the product of its factors, therefore, for all intents & purposes, a monomial has “implied parentheses” around it. Remembering that multiplication is just a fast way to do addition, using the example of the monomial 5x, it holds a single value which is 5 x’s (or “x” taken 5 times): 5x=x+x+x+x+x. When x=5, 5x=5+5+5+5+5=25
3. The “Law of Substitution” ensures that equivalent replacements (e.g. factoring out a term) maintain mathematical balance and integrity.
4. Division=Fraction (see “Remember: A division bar and fraction bar are synonymous!” on Algebra Class website https://www.algebra-class.com/dividing-monomials.html ), so a monomial division such as 48÷2(9+3) can be rewritten as the fraction 48 over 2(9+3), with the entire implied-multiplication-by-juxtaposition monomial term to the right of the obelus as the whole denominator — not just its coefficient.
That sums up how implied-multiplication-by-juxtaposition must be handled in calculations — as a single combined value/entity.
— Dee
Comment by Dee R. — November 27, 2025 @ 1:38 pm |
Happy Thanksgiving to you, Dee. I do understand all this. I haven’t had much time to devote to this topic for a while, but I have started some preliminary work on a possible publication. Probably will tackle it in earnest after the holidays.
Scott
Comment by Scott Stocking — November 27, 2025 @ 9:51 pm |
Yes, it’s a busy time now, so I understand that your focus is on other matters.
The crux of my comment was about Liebniz’s Law, as regards the validity of substitution in mathematics. Factoring out a term falls into that category of “substitution for an identical value.” So working backwards from 8÷2(2+2):
8 can be factored out as 8=2(2+2), so 2(2+2) can be substituted for 8, making the expression 2(2+2)÷2(2+2). Now substitute the letter-variable “b” for what is inside the parentheses [i.e. b=(2+2) or b=4] making it 2b÷2b. Now cancel out the like-factor of “b,” leaving 2÷2 which equals 1, or substitute back in the value of “b” thusly:
2b÷2b when b=(2+2) or b=4
2(4)÷2(4)=8÷8=1
The monomial term 2(2+2)=2(4)=8, making the original expression 8÷2(2+2)=8÷8=1, according to Liebniz’s Law as applied in mathematics as the Substitution Property of Equality.
Enjoy the holiday season!
Comment by Dee R. — November 28, 2025 @ 2:11 pm
That’s all well and good, but since those who think the answer is 16 don’t care whether the sign is there or not, they would still read it as 2 x 4 ÷ 2 x 4 and get 16. I don’t think such an argument would convince them otherwise, regardless of how we see the logic of it. One of the conclusions I’ve come to as I’ve reflected on this in my hiatus from writing about the subject is that MDAS part of PEMDAS only applies when the actual signs are present. When juxtaposed form are used, there’s a special meaning.
For example, when we use the decimal system, we can only have exact precision when we have a terminating decimal. That’s because our decimal system is base 10. However, when we need precision (especially for very large numbers) that involve rational fractional values that do note terminate, we need to use fractions. By using fractions, we change the “base” (in a sense) from 10^-1 to d^-1 where d is the denominator of the fraction. So instead of the fraction 2/3 being limited to the number of 6’s your calculator will recognize after the decimal, with 0.666…. being 6(10^-1) + 6(10^-2) + 6(10^-3) + … + 6(10^-∞), we have simply 2x(3^-1).
The same can be said of the inverse of the division function represented by the fraction, the form in question 2(2 + 2). The value in parentheses when its power is not negative (i.e., not a fraction) serves the same function. It is 2(4^1) as different way of expressing the number “8.” This may seem like a bit of semantics, but it is important where the value inside of the parentheses may be variable in the application of the expression. The bottom line is that what’s good for fractions (juxtaposed, implied division) is good for juxtaposed implied multiplication. I believe this is “lost knowledge” in an age where we have a paucity of critical thinking when it comes to hard sciences in favor of the softened, sloppy, and lazy approach many young people are exposed to. There’s no critical analysis of HOW we got to these forms and what may have been the motivation to develop them.
Scott
Scott
Comment by Scott Stocking — November 28, 2025 @ 9:42 pm
Liebniz’s Law, applied as the substitution property of equality in math, always takes precedence over a simple convention such as PEMDAS.
Every algebra textbook & bona fide teaching website tells students that a monomial term an be factored out; in this case, the monomial constant “8” can be factored out as two fours, written mathematically as 2(4). So when a=2 & b=4, 8=2(4)=ab. Thus, ab÷ab=2(4)÷2(4)=8÷8=1 — that’s just substituting in IDENTICAL values, in accordance with Liebniz’s Law (i.e. the substitution property of equality in math).
A law ALWAYS trumps application of a convention. A convention cannot be applied if it violates a LAW. Inserting an explicit multiplication sign & treating each factor as a separate entity having nothing to do with its other factors is in direct violation of the substitution property of equality — it makes the monomial constant “8” NOT equal to its factored form of “ab” when a=2 & b=4. Therefore, using PEMDAS is incorrect.
A law is a law is a law — it MUST be followed. A convention may be used if either there is no law which covers the situation or the convention does NOT conflict with any law. In the case of 8÷2(4), Liebniz’s Law of identical values substitution most definitely applies because implied multiplication by juxtaposition is how a monomial is formed (one inseparable term holding a single value which is the product of its factors), so it trumps the convention of PEMDAS.
The law MUST be obeyed!
Comment by Dee R. — December 1, 2025 @ 1:41 pm
Hi Scott,
Here’s something else that is relevant to Liebniz’s Law/Substitution Property of Equality, from Maths Teacher.com Australia:
https://www.mathsteacher.com.au/year8/ch04_algebra/01_pron/pron.htm
“Year 8 Interactive Maths”
“A pronumeral is a letter that is used to represent a number (or numeral) in a problem.”
~ ~ ~ ~ ~ ~ ~
That is total confirmation that the letter represents a number, which means the process of substitution can go back & forth from letter to number and back again — because they are “identicals,” which are indiscernible from one another (i.e. they share all properties), with the equals sign goings in both directions.
Thus, when z=8:
z÷xy=8÷xy
…which means “8 divided by the product of xy,” because xy is a monomial (one “unit” holding the value of the product of its factors). Therefore, when x=2 & b=4:
xy=2(4)
thus…
xy=8
Changing the letters to numbers does not change the monomial identity of the term “xy,” which, in this case, equals 8, making the expression z÷xy translate into…
8÷8=1
…in accordance with Liebniz’s Law/Substitution Property of Equality.
— Dee
Comment by Dee R. — December 11, 2025 @ 2:44 pm
Hi Scott —
I had a chat with AI on the subject of Liebniz’s Law/Substitution Property of Equality, and the upshot of AI’s conclusion is that because according to Liebniz’s Law/Substitution Property of Equality, an expression such as 48÷2(12) can ALWAYS be substituted for with letter-variables representing the numbers, as in 48÷2(12)=z÷xy when z=48, x=2 & y=12, which removes any ambiguity of the obelus & the use of implied multiplication by juxtaposition. That’s because “xy” is a monomial — one term holding a single value which is the PRODUCT of its factors — 2(12) representing “xy” MUST maintain the same value as each other; in this case, xy=2(12)=24, making the expression 48÷2(12)=48÷24=2.
Here is the original question I asked:
https://www.google.com/search?q=Applying+Liebniz%27s+Law%2FSubstitution+Property+of+Equality%2C+calculate+z%C3%B7xy+when+z%3D8%2C+x%3D2+%26+y%3D4%2C+taking+into+account+that+z%3Dxy%2C+usaing+the+same+division+notation+of+the+obelus+%26+the+same+implied+multiplication+by+juxtaposition+of+%22xy%22+as+%222%284%29.%22&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCjExMjc4MGowajeoAgCwAgA&sourceid=chrome&ie=UTF-8&udm=50&fbs=AIIjpHxU7SXXniUZfeShr2fp4giZ1Y6MJ25_tmWITc7uy4KIeioyp3OhN11EY0n5qfq-zEMZldv_eRjZ2XLYc5GnVnMEZGNJVrGoAwGchCcSdnZLZy7WOGoGJoFvDj7uLOOMFKb6cnAvei6B3W0PAQ8NyPjGhRGWFiYyR0lZbHA9xesI0pbZYCfZs1JxgzZOyTlvOwd9piJXaHeaTSP1ngntRmVCyZaXWA&ved=2ahUKEwjikMTW66SRAxWXGVkFHZfMBzYQ0NsOegQIAxAB&aep=10&ntc=1&mstk=AUtExfDOB0N2GBJv7JgHNRFRawSkLWnRAfNCjS1-zOOZWNjS26Z946tex5oKRDt6GpryaltKRDHOgVKUgOmmYXeWVtIu1yX_Ya72Z8cn730sq14F6XHxYL7t6NAOLESXW_q1gSfsHzRbYha8-jGluF19sAlFsmLlafdk-rI-OhP7i9QPZRwlQVshBV85h3AGVgK1dBGxPwiVxp6GrtgLF7X0h8P4O0kOkORfkzHrY9wqBBbMHleRaqaxpFSoNg&csuir=1
When a convention like PEMDAS is applied in such a way as to produce a different answer for the same expression substituted for with letter-variables, that method is being incorrectly applied because it violates Liebniz’s Law/Substitution Property of Equality. Due to the fact that a monomial (formed via implied multiplication by juxtaposition) is treated as a single quantity in algebra (without needing to encase it inside parentheses), simply substituting numbers, which the letter-variables represent, CANNOT change the value of the monomial term!
When x=2 & y=4, the monomial term “xy” ALWAYS equals 8 when substituted for as 2(4). And no parentheses around xy are needed to be understood as a single term (monomial), so no parentheses surrounding the entire monomial are necessary when simply substituting in the corresponding numbers. See AI answer to z÷xy when z=8, x=2 & y=4, substituted in as z÷xy=8÷2(4)=8÷8=1.
According to the concepts specified in Liebniz’s Law/Substitution Property of Equality, 8÷2(4)=8÷8=1, whether written with letters representing numbers as z÷xy when z=8, x=2 & y=4, or written with the numbers themselves — because they are IDENTICAL values. No doubt about it!
— Dee
Comment by Dee R. — December 4, 2025 @ 5:19 pm |
That’s an important bit of data to have. Not sure how convincing it will be to the hard-line Pemdasians, but it’s another validation of juxtapositional binding.
Comment by Scott Stocking — December 4, 2025 @ 5:22 pm |
Yes, it is a validation of juxtapositional binding.
Comment by Dee R. — December 4, 2025 @ 5:48 pm
An important fact to note in Liebniz’s Law/Substitution Property of Equality is that the equals sign (=) is bidirectional: When z=8, then 8=z; when x=2, then 2=x; when y=4, then 4=y. That substitution can go back & forth without ever changing a monomial term’s value — because they are IDENTICAL.
Comment by Dee R. — December 5, 2025 @ 9:26 am