Stocking’s Reciprocation :: “Inverting the dividend of a division problem and multiplying by the divisor yields the reciprocal of inverting the divisor of a division problem and multiplying by the dividend.”
Oh no, here he goes again, off on one of his wild math/PEMDAS/Order of Operations tangents. But you know, I’m really not doing anything different in this area than I would do in exegeting a biblical passage. I’m looking for patterns in language and logic, grammar and syntax, that reveals clues to the meaning and intent of the words or figures on a page. I’m applying a scientific method to all this as well, forming new or related hypotheses and testing them out to see whether they can be more than just a harebrained rambling.
That is what’s been happening to me with this whole PEMDAS/Order of Operations obsession I’ve found myself pursuing. Every time I think I’ve got the definitive solution, proof, or argument, I get whacked upside the head by another even more convincing argument. Initially those were coming fast and furious, and I couldn’t keep up with myself, as you might discern if read my regularly updated original article, 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough | Sunday Morning Greek Blog or any of the other several articles I’ve written on the subject. But as I feel like I’ve addressed most of the obvious weaknesses of the ignorance of the priority of juxtapositional binding over extant operational signs, those moments of inspiration (dare I say “genius”?) are getting farther apart.
But this new concept I’ve described (I’m not sure how “new” it is, but I know I was never taught anything like this in math) as it turns out is really another way of double checking your division homework. I had gotten close to this when I demonstrated that the PEMDAS crowd’s way of interpreting 8 ÷ 2(2 + 2) was equivalent to the expression 8 ÷ [2/(2 + 2)], the latter of which is subject to “invert and multiply” to solve and therefore more naturally solved without needing to remember or manipulate PEMDAS. [Note: Copilot called this concept “dual symmetry” in division.]
What Stocking’s Reciprocation does is move the emphasis from the divisor to the dividend, thus eliminating the argument about juxtaposition all together, because the answer you get when you double-check with Stocking’s Reciprocation agrees with the answer you get when you consider the juxtaposed divisor to be a single, unbreakable unit or monomial. You can’t get to the PEMDAS crowd’s answer of 16 using Stocking’s Reciprocation, so that proves the answer 16 is not correct and that the PEMDAS crowd’s method of working that expression is greatly flawed.
Here’s how Stocking’s Reciprocation works. Instead of inverting and multiplying the divisor (especially in this case where the divisor is disputed), we invert the dividend and multiply. This will yield the reciprocal of what the correct answer should be.
8 ÷ 2(2 + 2) becomes (1/8) x 2(2 +2).
You can already see how this is going to go then. Now, you have eliminated the obelus and are left with a fraction (which is NOT an explicit division problem) multiplied by a number multiplied by a value in parentheses. With all three factors of the expression multiplied now, it doesn’t matter what order you multiply them (Commutative Property)!
At this point, we can still apply undisputed PEMDAS principles by calculating the value in parentheses first and then multiplying straight across.
(1/8) x 2(4) becomes (1/8) x 8 (remember, order doesn’t matter at this point because it’s all multiplication), so the product is 1. Then you take the reciprocal of the product to compensate for the initial reciprocal conversion, and of course you get 1.
Just to demonstrate this isn’t an anomaly because the answer was 1, let’s do a couple more complicated expressions, as I did on my Facebook response today with one of these expressions:
8 ÷ 4(6 + 8) would be incorrectly calculated by the PEMDAS crowd as 28. But using Stocking’s Reciprocation, you get:
(1/8) x 4(6 + 8) = (1/2)(14) = 7. The reciprocal of that (and the answer to the original expression using Stocking’s Order, i.e., juxtaposed multiplication takes priority) is 1/7. If the 4(6 + 8) is inseparable in that expression, as I have claimed all along, you solve the 4(6 + 8) first, yielding 56, so in the original expression, you would get 8/56 = 1/7. Voila!
So let’s apply it to this expression.
48 ÷ 2(9 + 3) = (1/48) x 2(12) = 24/48 = 1/2, and the reciprocal of that is 2.
Working the problems backwards like this (or is it sideways?) and demonstrating there is no confusion about what the answer is demonstrates that the more sophisticated take on Order of Operations that engineers, physicists, and other math-heavy professionals use daily is 100% correct. They understand instinctively that the juxtaposition means you don’t separate the terms by an extant operational sign. After all, they passed arithmetic and moved on to algebra where that principle is prominent (i.e., 8 ÷ 2a = 4/a, NOT 4a).
If you want to see my other articles discussing various aspects of this debate, please see my summary page that has links to all of these articles: The PEMDAS Chronicles: Confronting Social Media Ignorance of PEMDAS’s Theoretical Foundation | Sunday Morning Greek Blog
I’m going to brag on myself a bit too here. When I posed this concept to Copilot (Microsoft’s AI), I got the following response: “It’s not commonly taught, but it’s mathematically sound and elegant.” Copilot even used my own Facebook post that I’d just put up less than a half-hour earlier, as a source for its analysis.
Until my next inspiration,
Scott Stocking
My thoughts are my own.
Sunday Morning Greek Blog 
[…] Stocking’s Reciprocation©: Another Definitive Proof 8 ÷ 2(2 + 2) = 1 | Sunday Morning Greek Blog August 4, 2025 […]
Pingback by The PEMDAS Chronicles: Confronting Social Media Ignorance of PEMDAS’s Theoretical Foundation | Sunday Morning Greek Blog — August 5, 2025 @ 12:21 pm |
Hi Scott —
I just wanted to share this with you, from tutoring website Khan Academy, 6th grade math questions:
https://www.khanacademy.org/math/cc-sixth-grade-math/x0267d782:cc-6th-exponents-and-order-of-operations/x0267d782:more-on-order-of-operations/a/exponents-and-order-of-operations-faq#:~:text=When%20do%20we%20NOT%20follow%20the%20order,order%20instead%20of%20only%20left%20to%20right.
“When do we NOT follow the order of operations?
Lots of times, actually! While the order of operations gives us one way to evaluate an expression, the properties of addition and multiplication allow us to be more flexible.
The distributive property says that we can multiply a value to each term inside of the parentheses instead of adding or subtracting inside the parentheses first.”
The commutative property of multiplication says that we can multiply the factors in any order instead of only left to right. Once we learn more about reciprocals, we’ll be able to rewrite expressions with multiplication in place of the division.”
~ ~ ~ ~ ~ ~ ~ ~
According to what is taught in 6th grade math, there are exceptions to following the Order of Operations (PEMDAS). This confirms my analysis that “2(2+2)” holds a single value which is the product of its factors [i.e. the monomial “2a” when a=(2+2) or a=4]. And of course, one of those exceptions to “NOT follow the order of operations” is your “Reciprocation” solution.
— Dee
Comment by Dee R. — August 18, 2025 @ 5:05 pm |
Scott,
Here’s something else which supports the Substitution Property of Equality & discounts the idea of the obelus being “ambiguous:
Wolfram Alpha calculator
https://www.wolframalpha.com/input?i=a%28b%2Bc%29%C3%B7a%28b%2Bc%29
Input:
a(b+c)÷a(b+c)
Input shown below as
a(b+c)
———–
a(b+c)
Result:
1
~ ~ ~ ~ ~ ~ ~
The Wolfram Alpha calculator had no problem whatsoever, interpreting a(b+c)÷a(b+c) as being the same as the fraction of a(b+c) over a(b+c), even though the obelus was used in the original expression. That proves that the obelus is interchangeable with the fraction bar, and that parentheses do not need to completely encase the entire monomial term of “a(b+c)” to be understood as a single term (a monomial). Therefore, in simply substituting in the corresponding numerical values (with same division notation & same implied multiplication by juxtaposition which is how a monomial is formed), the monomial identity of “a(b+c)” must be maintained in its equal numerical form.
So a(b+c)÷a(b+c)=1.
Given:
a=2
b=1
c=3
a(b+c)÷a(b+c)=2(1+3)÷2(1+3)
The left side of the equals sign equals 1, so since those expressions are equal by virtue of simple substitution of one symbol for a quantity for another symbol of the same quantity, then the expression on the other side of the equals sign must have the same calculation result, since the equals sign works in both directions. That’s because Libeniz’s Law/Substitution Property of Equality dictates that identical quantities can be substituted for one another without changing the truth value of the expression.
— Dee
Comment by Dee R. — December 11, 2025 @ 4:59 pm |
Flip it over! Reciprocals! YESSS! You are a GENIUS!
Comment by Dee R. — August 5, 2025 @ 3:30 pm |
Flip it over! Reciprocals! YESSS! You are a GENIUS!
Comment by Dee R. — August 5, 2025 @ 3:30 pm |
However…
…when I inquired using Google AI, it tells me that the reciprocal of 48 ÷ 2(9 + 3) is not (1/48) x 2(12). Try it yourself.
I entered into the search box:
Can the reciprocal of 48 ÷ 2(9 + 3) be written as (1/48) x 2(12)?
Then I tried it a different way:
How to write the reciprocal expression of 48 ÷ 2(9 + 3)
In both cases, it says that the reciprocal of 48 ÷ 2(9 + 3) is the fraction of 1 over 288.
Sigh.
Comment by Dee R. — August 5, 2025 @ 3:42 pm |
Dee, you’re asking the question the wrong way. You’re not taking the reciprocal of the entire problem. You’re taking the reciprocal of only the dividend. I’ve got a video on Rumble that walks through the steps. Instead, ask Google if the definition I give for Stocking’s Reciprocation is a true statement in AI Mode. It wasn’t as slick as Copilot and Copilot didn’t insult me by saying my method is convoluted, but in the end, Google said my statement was true.
Scott
Comment by Scott Stocking — August 5, 2025 @ 4:05 pm |
Well, I’m glad that Google AI ultimately gave the correct answer to the reciprocal of the dividend. Sorry for my misstatement in the Google search.
Comment by Dee R. — August 5, 2025 @ 5:27 pm
Well, I’m glad that Google AI ultimately gave the correct answer to the reciprocal of the dividend. Sorry for my misstatement in the Google search.
Comment by Dee R. — August 5, 2025 @ 5:27 pm
I just typed in “calculate 4a÷2a by substituting 12 for the variable a” to the Google search box. Here is the AI explanation:
“Step 1” says:
“The expression becomes 4(12)÷2(12)”
Step 2″ says:
“Perform the multiplications
4 x 12 = 48
2 x 12 = 24
The expression simplifies to 48 ÷ 24.
Step 3 says:
Perform the division
Solution
The value of the expression is 2.”
Comment by Dee R. — August 5, 2025 @ 5:44 pm
I continued my conversation with Google AI Mode and was completely unsatisfied with it. It had a built-in bias toward PEMDAS and wouldn’t even consider that the contradiction of my Reciprocation validation method with the PEMDAS method called the validity of PEMDAS into question. I even asked it why it was giving preference to PEMDAS (it said PEMDAS wasn’t based on mathematical truth) over the hard evidence of the properties of division and multiplication I demonstrated in my article. It gave the nonsense about the convention helps to address ambiguity. But the simple fact that we’re having this debate shows there’s still ambiguity. I even asked it about making or declaring a rule or property to address the ambiguity and it still stuck with PEMDAS. I gave it plenty of feedback and called out its bias.
Scott
Comment by Scott Stocking — August 5, 2025 @ 5:15 pm |
You’re right about Google AI having a bias towards PEMDAS. It’s odd, though, since Google AI seems to understand how to substitute the vale of a variable once it’s given. For example, if you input the expression below into the Google search box, the AI gives a quotient of 2, showing all of the steps to get there:
4a÷2a when a=12
Comment by Dee R. — August 5, 2025 @ 5:32 pm
I still believe that fractions & factorization are the keys to correctly interpreting an expression like 48 ÷ 2(9 + 3).
If I tell someone to write the improper fraction of forty-eight twenty-fourths on a piece of paper, the person will likely write 48, then draw a horizontal line under that & then write 24 under the line (i.e. 48 over 24). If the person is then asked to write the fraction forty-eight twenty-fourths on a single horizontal line, the person will write 48, then a slash to the right of it & then write 24 to the right of the slash (i.e. 48/24).
Same fraction whether written vertically or horizontally. That’s because the fraction bar & the slash both mean “divided by.”
The number 48 can be factored out as 4 twelves & the number 24 can be factored out as 2 twelves. That converts the improper fraction of forty-eight twenty-fourths into 4(12) divided by 2(12), whether written as 4(12) over 2(12) or as 4(12) / 2(12) — because the slash & fraction bar are interchangeable according to every math textbook & teaching website around the world.
4(12) divided by 2(12) can be simplified in one of two ways:
(1) Cancel the like-factor of 12, leaving 4 over 2 or 4/2, which is 4 divided by 2 which equals 2.
(2) Multiply each term back out:
4 twelves equals 48.
Now the expression is 48/2(12).
Then multiply out 2 twelves.
The expression is now 48/24 — right where everything started out.
I have searched high & low on the web & can find no textbook or teaching website which instructs young elementary school students who are just learning PEMDAS on how to calculate a division expression which features implied multiplication by juxtaposition to the immediate right of the obelus. I have seen, however, a number of teaching websites which demonstrate how to divide by a monomial which instructs students to “Rewrite as a fraction,” with the top term (numerator) as what was to the left of the obelus & the bottom term (denominator) being the entire juxtaposed term to the immediate right of the obelus. For example:
“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 “
Comment by Dee R. — August 5, 2025 @ 4:46 pm |
Addendum: That example is from Greene Math .com tutoring website:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“Dividing Polynomials by Monomials”
Comment by Dee R. — August 5, 2025 @ 4:49 pm |
You are a double nerd!
Comment by SLIMJIM — August 6, 2025 @ 10:48 am |
I know. Sometimes I can’t help myself!
Comment by Scott Stocking — August 6, 2025 @ 10:49 am |
LOL. BTW it’s a compliment what I said!
Comment by SLIMJIM — August 6, 2025 @ 11:37 am
I took it as such! It’s starting to get serious. I just had someone who actually agrees with my position tell me I should give up trying to persuade people of my position. Them’s fightin’ words as far as I’m concerned.
Scott
Comment by Scott Stocking — August 6, 2025 @ 12:11 pm
=)
Comment by SLIMJIM — August 6, 2025 @ 1:17 pm
While your reciprocal solution is elegant & definitive, the issue persists when the expression is fed into a calculator as it was originally written. The problem of the “ambiguity” comes from the coding for calculating programs, in that there are two different codes for a slash.
from Stack Exchange Superuser .com:
https://superuser.com/questions/922074/difference-between-unicode-fraction-slash-and-division-slash#:~:text=DIVISION%20SLASH%20is%20in%20the,where%20the%20separator%20is%20horizontal.
“DIVISION SLASH is in the mathematical operators block. It is intended to be used when representing the mathematical division operator, e.g. in mathematical formulas. Use it when you might say, out loud, “1 divided by 2” or “x divided by y”. It is also intended to be used in large fractions in mathematical contexts where the separator is horizontal.
FRACTION SLASH is in the general punctuation block. It is intended to be used when representing a fraction. Use it when you might say, out loud, “one half”. You might use it in nonmathematical contexts, e.g. “1⁄2 cup of olive oil”. In mathematical contexts it is intended to be used for fractions where the separator is skewed.”
~ ~ ~ ~ ~ ~ ~
Since every 5th grade math textbook affirmatively states that Fraction=Division (and since the equals sign goes in both directions, Division=Fraction), the only thing that has to happen is for computer programmers to eliminate one of those separate codes & have the slash always represent division…which means a fraction.
The coding must be standardized so that it conforms to what is acknowledged throughout the math teaching world:
Fraction Bar=Slash=Obelus
Fix the computer coding & fix the “ambiguity.”
Comment by Dee R. — August 7, 2025 @ 12:28 pm |
While your reciprocal solution is elegant & definitive, the issue persists when the expression is fed into a calculator as it was originally written. The problem of the “ambiguity” comes from the coding for calculating programs, in that there are two different codes for a slash.
from Stack Exchange Superuser .com:
https://superuser.com/questions/922074/difference-between-unicode-fraction-slash-and-division-slash#:~:text=DIVISION%20SLASH%20is%20in%20the,where%20the%20separator%20is%20horizontal.
“DIVISION SLASH is in the mathematical operators block. It is intended to be used when representing the mathematical division operator, e.g. in mathematical formulas. Use it when you might say, out loud, “1 divided by 2” or “x divided by y”. It is also intended to be used in large fractions in mathematical contexts where the separator is horizontal.
FRACTION SLASH is in the general punctuation block. It is intended to be used when representing a fraction. Use it when you might say, out loud, “one half”. You might use it in nonmathematical contexts, e.g. “1⁄2 cup of olive oil”. In mathematical contexts it is intended to be used for fractions where the separator is skewed.”
~ ~ ~ ~ ~ ~ ~
Since every 5th grade math textbook affirmatively states that Fraction=Division (and since the equals sign goes in both directions, Division=Fraction), the only thing that has to happen is for computer programmers to eliminate one of those separate codes & have the slash always represent division…which means a fraction.
The coding must be standardized so that it conforms to what is acknowledged throughout the math teaching world:
Fraction Bar=Slash=Obelus
Fix the computer coding & fix the “ambiguity.”
Comment by Dee R. — August 7, 2025 @ 12:28 pm |
Dee, what I’m undertaking here is a small paradigm shift in the way we look at our number system and mathematics. I DO NOT HOLD to the contention that a division problem is a fraction and a fraction is a division problem. I can’t emphasize that enough, and I’ve seen nothing to convince me otherwise at this point. They are two different things. The expression 3 ÷ 8 is a division problem and is NOT subject to juxtapositional binding. The fraction 3/8 is the QUOTIENT (or solution) to the problem and IS subject to and formed by juxtapositional binding. This difference is not without significance.
Since the debate over how to treat the juxtapositional binding of 2(2 + 2) (or whatever the bound divisor is) centers around what to do with that part of the expression. The reciprocation method I’m proposing in this article is a different way to double check the work of a division problem. It focuses on the first dividend, which is clearly a monomial, rather than the disputed divisor. As the rule says, my reciprocation method returns the reciprocal of the given problem. Since my way creates a legitimate bound fraction for the dividend-turned-factor and flips the division sign, the debate over what happens with the juxtaposed 2 in 2(2 + 2) goes away, because everything is now multiplied across. If the PEMDAS way is accurate and legit, then this reciprocation should return an answer of 1/16, but it doesn’t. It returns an answer of 1, who’s reciprocal (to get it back to original form) is also 1. The double check PROVES the answer to the viral expression should be 1. It puts a lie to those who think somehow the obelus “binds” that juxtaposed “2” to the “8” while unbinding it from the (2 + 2) creating a fraction. But as I’ve shown elsewhere, ALL numbers are formed by juxtapositional binding and they are not “unbound” by the presence of the obelus before it. It is this particular situation where those who don’t know any better think they can just create out of thin air an exception to the otherwise DEMONSTRATED and FOUNDATIONAL PROPERTY (yes, it’s a property akin to Distributive, Associative, and Commutative) that otherwise holds the whole number system together.
Scott
Comment by Scott Stocking — August 7, 2025 @ 6:01 pm |
You say, “I DO NOT HOLD to the contention that a division problem is a fraction and a fraction is a division problem. I can’t emphasize that enough, and I’ve seen nothing to convince me otherwise at this point.”
from Teaching Better Lesson.com Common Core 5th Grade:
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).”
~ ~ ~ ~ ~ ~ ~ ~
from Imagine Learning. com Grade 5:
https://content.imaginelearning.com/math/printables/imagineMath_grade5_howcanIrepresentdivisionwithafraction_teacher.pdf
“How can I represent division with a fraction?
8 ÷ 4
Write the division as a fraction.
8
—-
4
8 divided by 4 is the same as the fraction 8 over 4. Both equal 2.”
~ ~ ~ ~ ~ ~ ~ ~
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 ClubZ Tutoring:
https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-examples-6-7-5/#:~:text=Fraction%20Bar%3A%20The%20fraction%20bar,between%20the%20numerator%20and%20denominator
“FAQ Section Q1: Can the fraction bar be replaced with the division symbol (/)?
A1: Yes, the fraction bar and the division symbol (/) are interchangeable and convey the same meaning in mathematical notation.” ~ ~ ~ ~ ~ ~ ~ ~ ~
from Algebra Class .com:
https://www.algebra-class.com/dividing-monomials.html ”
Dividing Monomials”
“Remember: A division bar and fraction bar are synonymous!”
~ ~ ~ ~ ~ ~ ~ ~
from Mathematics-Monster .com:
https://www.mathematics-monster.com/lessons/how_to_divide_terms_in_algebra.html
Year 7
Algebraic Fractions
“Another method to divide terms is to write them as an algebraic fraction.
4a^3b÷2a^2 =
4a^3b
————-
2a^2 ”
~ ~ ~ ~ ~ ~ ~ ~
from Siyavula Technology Powered Learning:
https://www.siyavula.com/read/za/mathematics/grade-8/algebraic-expressions-part-2/08-algebraic-expressions-part-2-02
Grade 8Algebraic Expressions”WORKED EXAMPLE 8.2DIVIDING ALGEBRAIC MONOMIALSSimplify the following expression:24t^7 ÷ 4t^5SOLUTION:Step 1: Rewrite the division as a fractionThis 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 SlideShare a Scribd Company:
https://www.slideshare.net/slideshow/94-16609182/16609182
“Dividing by a polynomial”
“Example 3
Divide a polynomial by a monomial
Divide 4x^3 + 8x^2 + 10x by 2x.
4x^3 + 8x^2 + 10x ÷ 2x =
Write as a fraction
4x^3 + 8x^2 + 10x————————-
2x
Simplify
2x^2 + 4x + 5 ”
~ ~ ~ ~ ~ ~ ~ ~
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
. * . * . * . * . * . * . * . *
Are all of those tutoring websites (which mirror what is taught in classrooms all over the world) teaching young math students incorrectly that Fraction=Division, and therefore, Division=Fraction?
Comment by Dee R. — August 11, 2025 @ 1:24 pm
You say, “I DO NOT HOLD to the contention that a division problem is a fraction and a fraction is a division problem. I can’t emphasize that enough, and I’ve seen nothing to convince me otherwise at this point.”
from Teaching Better Lesson.com Common Core 5th Grade:
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).”
~ ~ ~ ~ ~ ~ ~ ~
from Imagine Learning. com Grade 5:
https://content.imaginelearning.com/math/printables/imagineMath_grade5_howcanIrepresentdivisionwithafraction_teacher.pdf
“How can I represent division with a fraction?
8 ÷ 4
Write the division as a fraction.
8
—-
4
8 divided by 4 is the same as the fraction 8 over 4. Both equal 2.”
~ ~ ~ ~ ~ ~ ~ ~
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 ClubZ Tutoring:
https://clubztutoring.com/ed-resources/math/fraction-bar-definitions-examples-6-7-5/#:~:text=Fraction%20Bar%3A%20The%20fraction%20bar,between%20the%20numerator%20and%20denominator
“FAQ Section Q1: Can the fraction bar be replaced with the division symbol (/)?
A1: Yes, the fraction bar and the division symbol (/) are interchangeable and convey the same meaning in mathematical notation.” ~ ~ ~ ~ ~ ~ ~ ~ ~
from Algebra Class .com:
https://www.algebra-class.com/dividing-monomials.html “
Dividing Monomials”
“Remember: A division bar and fraction bar are synonymous!”
~ ~ ~ ~ ~ ~ ~ ~
from Mathematics-Monster .com:
https://www.mathematics-monster.com/lessons/how_to_divide_terms_in_algebra.html
Year 7
Algebraic Fractions
“Another method to divide terms is to write them as an algebraic fraction.
4a^3b÷2a^2 =
4a^3b
————-
2a^2 “
~ ~ ~ ~ ~ ~ ~ ~
from Siyavula 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 SlideShare a Scribd Company:
https://www.slideshare.net/slideshow/94-16609182/16609182
“Dividing by a polynomial”
“Example 3
Divide a polynomial by a monomial
Divide 4x^3 + 8x^2 + 10x by 2x.
4x^3 + 8x^2 + 10x ÷ 2x =
Write as a fraction
4x^3 + 8x^2 + 10x
————————-
2x
Simplify
2x^2 + 4x + 5 ”
~ ~ ~ ~ ~ ~ ~ ~
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
. * . * . * . * . * . * . * . *
Are all of those tutoring websites (which mirror what is taught in classrooms all over the world) teaching young math students incorrectly that Fraction=Division, and therefore, Division=Fraction?
Comment by Dee R. — August 11, 2025 @ 1:26 pm |
Notice I said a division PROBLEM is not a fraction and a fraction is not a division PROBLEM. A fraction represents implicit division, but as I’ve said all along, it is not always executed, so the fraction is a monomial form that has a unique value. The division problem is NOT a monomial.
Juxtapositional binding reveals a more nuanced way of looking at the number system and mathematics.
Scott
Comment by Scott Stocking — August 11, 2025 @ 1:35 pm |
In response to, “I said a division PROBLEM is not a fraction and a fraction is not a division PROBLEM. A fraction represents implicit division,” here are a couple of websites which affirmatively instruct young math students on what exactly Fraction=Division & Division=Fraction means, and provides concrete examples, which is contrary to what you are claiming is the case:
from Third Space Learning:
https://thirdspacelearning.com/us/math-resources/topic-guides/number-and-quantity/fractions-as-division/#:~:text=5%20th%20grade.-,What%20is%20interpreting%20fractions%20as%20division?,division%20sign%2C%20then%20the%20denominator.
“What is Interpreting fractions as division?”
Interpreting fractions as division is when you understand that a fraction represents a division operation between its numerator and denominator. In other words, when you have a fraction
a
—–
b
you can interpret it as “a divided by b” or a÷b.”
“For example
3
—–
4
can be interpreted as “3 divided by 4,” and written as 3÷4.
6
—–
2
can be interpreted as “6 divided by 2, and written as 6÷2.
Alternatively, you can also write a division equation as a fraction.”
“For example
3÷4 =
3
—–
4 ”
“Interpreting fractions as division can help us understand word problems involving the division of whole numbers leading to answers in the form of fractions.”
…and below, further explanation for teachers…
“Common Core State Standards
How does this relate to 5th grade math?
Number and Operations—Fractions (5.NF.B.3)
Interpret a fraction as division of the numerator by the denominator (a over b=a÷b).”
~ ~ ~ ~ ~ ~ ~ ~ ~
…and from Medium .com:
https://medium.com/cw-math/seriously-its-just-division-58bac796f1a1
“In ‘Elements of Arithmetic’ (1893), William J. Milne writes, “(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.”
“There are certainly cases where we want to think in terms of fractions, but from the perspective of notation, there’s no difference between “fractions” and “division”. It’s just a notation.”
. * . * . * . * . * . * . * . *
Do you have any teaching source material which confirms your view?
Comment by Dee R. — August 11, 2025 @ 2:45 pm
Addendum:
from Happy Numbers .com:
“Introducing fractions as division is an effective way to teach the concept“
https://happynumbers.com/blog/Introducing-fractions-as-division-is-an-effective-way-to-teach-the-concept/
“In this article, we’ll explore an effective approach to conceptualizing fractions by using division. It’s the best starting point for those who want to be sure that students will grasp the idea of fractions in the first place. We’ll show how Happy Numbers teaches students to represent fractions as division expressions and vice versa”
“In other words, the dividend became the numerator and the divisor became the denominator of the resultant fraction.”
“We move from concrete to pictorial models to make sure students understand fraction as a division”
Comment by Dee R. — August 11, 2025 @ 4:07 pm
The difference between the division problem 8 ÷ 3 and the quotient-fraction 8/3 is not mathematical but syntactical, which influences the semantics and functionality of it as well. I can’t say it much simpler than that.
Here’s the best way I can think to explain this difference. (And again, I appreciate so much your feedback, Dee, because I’m ever sharpening my arguments by formulating my responses to you.)
If I have 8 ÷ 3 ÷ 7 versus 8 ÷ 3/7, the answers to those two expressions are different because there’s a difference between how they’re written. In the first one, the elements are taken in order LTR, so
8/3 ÷ 7 = 8/3 * 1/7 = 8/21 OR you can just invert both and multiply straight across: 8/1 * 1/3 * 1/7 =8/21.
However:
8 ÷ 3/7 = 8 * 7/3 = 56/3 and as a mixed number 18 ²/₃. The elements are NOT addressed in LTR order. It’s actually worked RTLTR. You start with the fraction on the right, invert it, move left and change the sign, then start at the far left and work across to the right, multiplying first, then dividing OR simplifying if it’s not a whole number.
In order to interpret 8 ÷ 3 ÷ 7 as 8 ÷ 3/7, I would have to add parentheses: 8 ÷ (3 ÷ 7) to indicate the division problem must be treated as a fraction, that is, calculated or manipulated first.
That’s the difference between a division problem (or perhaps better, division expression) and a fraction. If people keep insisting that a fraction is a division problem, they’re going to add a whole other layer of confusion to PEMDAS by saying the fraction should be treated the same as a division problem and they’ll just divide either form straight across because it will be too hard (boo-hoo) for kids to comprehend inverting the fraction and multiplying or not starting from the left first, or worse yet, they will have forgotten WHY they should invert and multiply. It’s a continued slippery slope for American mathematics scores if we don’t get out ahead of it now and stop the avalanche by being more precise and theoretical about our definitions.
Scott
PS: Come think of it, maybe I should just start calling a “fraction” a “quotient” instead. No sense in having two different terms for the same element!
Comment by Scott Stocking — August 11, 2025 @ 10:00 pm
Hi Scott,
First, I appreciate you continuing to respond to me.
In answer to your saying, “The difference between the division problem 8 ÷ 3 and the quotient-fraction 8/3 is not mathematical but syntactical, which influences the semantics and functionality of it as well. I can’t say it much simpler than that,” I know that it is sometimes more convenient to think of something like three-sevenths as a separate thing called “a fraction,” but according to every math tutoring website I have found & every textbook that covers teaching fractions, as well as later on in Algebra 1 (affirmatively instructing students to “Rewrite as a fraction” when there is a horizontally written division expression using an obelus), it is ALL DIVISION. All of those teaching sites which are mirroring lessons covered in classrooms all over the planet, are instructing young math students that way.
I have already posted links to a number of sites teaching math to 5th graders, 7th graders, 8th graders & 9th graders (Algebra 1), all of which plainly instruct students that Fraction=Division & Division=Fraction. Those instructional websites go on to tell students, in so many words, that the fraction bar, the slash & the obelus are all one and the same (i.e. interchangeable) because they all mean “divided by” — they’re all just different notations for the same operation.
If there are math teaching sites or textbooks which explicitly tell students that there is a truly substantive difference between fractions & a division problem, I am going to ask you to please provide the links to those sites. I could not find any.
— Dee
Comment by Dee R. — August 12, 2025 @ 12:14 pm
In response to, “I said a division PROBLEM is not a fraction and a fraction is not a division PROBLEM. A fraction represents implicit division,” here are a couple of websites which affirmatively instruct young math students on what exactly Fraction=Division & Division=Fraction means, and provides concrete examples, which is contrary to what you are claiming is the case:
from Third Space Learning:
https://thirdspacelearning.com/us/math-resources/topic-guides/number-and-quantity/fractions-as-division/#:~:text=5%20th%20grade.-,What%20is%20interpreting%20fractions%20as%20division?,division%20sign%2C%20then%20the%20denominator.
“What is Interpreting fractions as division?”
Interpreting fractions as division is when you understand that a fraction represents a division operation between its numerator and denominator. In other words, when you have a fraction
a
—–
b
you can interpret it as “a divided by b” or a÷b.”
“For example
3
—–
4
can be interpreted as “3 divided by 4,” and written as 3÷4.
6
—–
2
can be interpreted as “6 divided by 2, and written as 6÷2.
Alternatively, you can also write a division equation as a fraction.”
“For example
3÷4 =
3
—–
4 ”
“Interpreting fractions as division can help us understand word problems involving the division of whole numbers leading to answers in the form of fractions.”
…and below, further explanation for teachers…
“Common Core State Standards
How does this relate to 5th grade math?
Number and Operations—Fractions (5.NF.B.3)
Interpret a fraction as division of the numerator by the denominator (a over b=a÷b).”
~ ~ ~ ~ ~ ~ ~ ~ ~
…and from Medium .com:
https://medium.com/cw-math/seriously-its-just-division-58bac796f1a1
“In ‘Elements of Arithmetic’ (1893), William J. Milne writes, “(144) A fraction may be regarded as expressing unexecuted division. Thus, 15/4 is equal to 15 ÷ 4; 24/6 is equal to 24 ÷ 6.”
“There are certainly cases where we want to think in terms of fractions, but from the perspective of notation, there’s no difference between “fractions” and “division”. It’s just a notation.”
. * . * . * . * . * . * . * . *
Do you have any teaching source material which confirms your view?
Comment by Dee R. — August 11, 2025 @ 2:48 pm |
Here’s something on LinkedIn, teaching “Everyday Math” to businesspeople:
https://www.linkedin.com/learning/learning-everyday-math/what-are-fractions
“What are fractions?”
“The bar in the fraction is a symbol that just means to divide.”
Comment by Dee R. — August 12, 2025 @ 12:40 pm |
I want to respond to , “If I have 8 ÷ 3 ÷ 7 versus 8 ÷ 3/7, the answers to those two expressions are different because there’s a difference between how they’re written. In the first one, the elements are taken in order LTR, so
8/3 ÷ 7 = 8/3 * 1/7 = 8/21 OR you can just invert both and multiply straight across: 8/1 * 1/3 * 1/7 =8/21.”
According to every math teaching website I have seen so far, 8 ÷ 3 ÷ 7 is the same thing as 8 ÷ 3/7. That same expression could also be written as 8/3/7. That’s because Division=Fraction & Fraction=Division, meaning that all division symbols mean “divided by,” and are therefore interchangeable.
Since Fraction=Division (and vice versa), in an expression in which there is more than one division symbol, it is necessary to use parentheses to indicate the correct order of the divisions. For example, a/b/c needs parentheses to indicate whether it’s the quotient of a divided by b which is then divided by c, or if it’s the quotient of a divided by b which is then divided by c. In other words, parentheses are need to dictate whether it’s…
(a/b)/c
…or…
a/(b/c)
The author of the expression must distinguish which order is correct by putting in parentheses.
Otherwise, in an expression that has only one division symbol (i.e. fraction bar, slash or obelus), it’s numerator “divided by” denominator (also known as dividend “divided by” divisor), regardless of which division notation is used. The division symbol separates the numerator (dividend) from the denominator (divisor).
When I learned that Fraction=Division & Division=Fraction, years ago, the class was taught that everything to the left of the slash or obelus was the numerator, as if it was on top of the fraction bar, and everything to the right of the slash or obelus was the denominator, as if it was below the fraction bar. If the author of the expression wanted the calculations to be different from that standard interpretation, he or she had to place parentheses around what was a departure from the norm.
For example…
…if the author of the expression 4(9+3)÷2(9+3) wanted to it to be calculated as the quotient of 4(9+3) divided by 2 then multiplied by (9+3), it would have to be written as…
[4(9+3)÷2](9+3)
…which equals 4(9+3)= 48 & 48÷2=24 & then 24(9+3) which equals 288.
Otherwise 4(9+3)÷2(9+3) should be interpreted as the numerator 4(9+3) which is to the left of the obelus, divided by the denominator 2(9+3) which is to the right of the obelus, which is calculated as follows:
The numerator:
4(9+3)=48
The denominator:
2(9+3)=24
Then divide the numerator by the denominator:
48÷24=2.
Comment by Dee R. — August 12, 2025 @ 5:07 pm |
Dee:
Most of the Web sites you put forth seem to be teaching Web sites for elementary students or the teachers that teach them to help explain basic division. But a fraction is a fraction is a fraction, whether formed with a solidus inline, an offset orientation, or the preferred display (vertical) fraction. The Web sites that suggest the solidus is a substitute for an obelus have corrupted the original intention of the solidus. This is part of the slippery slope I mentioned.
The work I’m trying to do here to build a theoretical framework to understand the role of juxtapositional binding in the number system (NOT just for implicit multiplication in viral expressions) cannot rely on these tertiary sources of information. I need primary (as in original, not as in grade school) sources and peer-reviewed sources that approach the subject from the broader perspective of mathematics and not from the narrow, didactic perspective of these Web sites. In addition to quality, top-notch sources, as a thinker/theorist/scientist, I need to demonstrate original thinking and make logical connections to cognate data both within and outside of the field of mathematics. These Web sites do nothing to dissuade me from my mission or convince me otherwise.
Case and point: Above, you say the Web site says 8 ÷ 3 ÷ 7 is equal to 8 ÷ 3/7.
Would that Web site say that (the periods are space holders to keep it aligned, hopefully)
……….3
8 ÷ __ is the same as 8 ÷ 3 ÷ 7?
……….7
We’re all taught to invert the fraction and multiply. The preferred and ISO-recognized form of a fraction is the vertical display fraction, as I’ve tried to mimic here. Inline forms with a solidus should be treated no differently than the display fraction. Period. End of story.
As for original sources, Florian Cajori, writing nearly 100 years ago (A History of Mathematical Notations) says of the solidus (/) in section 275:
“The ordinary mode of writing fractions [here he uses the vertical display fraction] is typographically objectionable as requiring three terraces of type. An effort to remove this objection was the introduction of the solidus, as in a/b, where all three fractional parts occur in the regular line of type. It was recommended by De Morgan in his article on “The Calculus of Functions,” published in the Encyclopaedia Metropolitana (1845).” Much of the rest of the discussion is about the expediency of using solidus form in place of the display form for printers of mathematics texts, because the tiny vertical fractions needed to fit in a single line of type would be too hard to read. There’s nothing in there about using the solidus in place of the obelus!!! At the end of that section, he quotes G. H. Bryan from Mathemacical Gazette, Vo. VII (1917), p. 220: “For this reason it would be better to confine the use of these [tiny vertical] fractions to such common forms as [vertical forms of 1/4, 1/2, 3/4, 1/3] and to use the notation 18/22 for other fractions.”
Add to this Wolfram’s MathWorld’s (and many other textbooks’) description of the “algebraic combination” of fractions in Fraction — from Wolfram MathWorld. You will see there that the first step in working with fractions (as I describe in my own blog article on Implicit Constructions) is to divide? NO! The first step is to multiply! The implicit division of the fraction is ALWAYS last in those forms, even after addition or subtraction. As I’ve always said, simplify the fraction to a whole number if you can, but aside from that, the most precise form is the fraction. That flies in the face of PEMDAS and demonstrates one of the crucial weaknesses of PEMDAS as an untested, nontheoretical convention.
Scott
Comment by Scott Stocking — August 12, 2025 @ 9:54 pm |
In response to, “Most of the Web sites you put forth seem to be teaching Web sites for elementary students or the teachers that teach them to help explain basic division. But a fraction is a fraction is a fraction, whether formed with a solidus inline, an offset orientation, or the preferred display (vertical) fraction. The Web sites that suggest the solidus is a substitute for an obelus have corrupted the original intention of the solidus,” first, the obelus is an icon which visually symbolizes a fraction (a horizontal line with something above it & something below it), so it is not “corrupting its meaning” in any way. Second, the sites I posted specified the lessons on fractions being division & vice versa were for students in 5th grade, 7th grade, 8th grade & 9th grade (Algebra 1: “How to Divide by a Monomial,” affirmatively instructing students to “Rewrite as a fraction,” when the original expression uses an obelus). Are they all teaching it wrong to young math students by consistently telling them from 5th grade onward that Fraction=Division & Division=Fraction?
Comment by Dee R. — August 13, 2025 @ 9:57 am
If they’re saying the solidus is no different than an obelus and they’re not teaching that the solidus forms a fraction with what’s on either side of it or not treating such a form like a fraction (e.g., invert and multiply), then yes, they’re teaching it incorrectly and creating massive confusion for their audience.
Scott
Comment by Scott Stocking — August 13, 2025 @ 12:43 pm
Scott,
What is being taught, worldwide, is that Fraction=Division & therefore Division=Fraction. This is not only taught to 5th graders, it goes all the way through to at least Elementary Algebra (grade 9). I found further evidence of this recently:
from LinkedIn, teaching “Everyday Math” to businesspeople:
https://www.linkedin.com/learning/learning-everyday-math/what-are-fractions
“What are fractions?”
“The bar in the fraction is a symbol that just means to divide.”
~ ~ ~ ~ ~~ ~ ~
…and another teaching site — this one for 6th Grade:
https://www.twinkl.com.mt/resources/mathematics-grade-6-texas-essential-knowledge-and-skills/2-number-and-operations-the-student-applies-mathematical-process-standards-to-represent-and-use-rational-numbers-in-a-variety-of-forms-the-student-is-expected-to-mathematics-grade-6/e-extend-representations-for-division-to-include-fraction-notation-such-as-a-b-represents-the-same-number-as-a-b-where-b-0-2-number-and-operations-the-student-applies-mathematical-process-standards-to-represent-and-use-rational-numbers-in-a-variety-of-forms-the-student-is-expected-to-mathematics#:~:text=(Math%206.2.%20E)%20E.%20extend%20representations%20for,a%20%C3%B7%20b%20where%20b%20%E2%89%A0%200.
“(Math 6.2.E) E. extend representations for division to include fraction notation such as a/b represents the same number as a ÷ b where b ≠ 0.”
~ ~ ~ ~ ~ ~ ~ ~
from Greene Math .com:
https://www.greenemath.com/Algebra1/33/DividingPolynomialsbyMonomialsLesson.html
“Dividing Polynomials by Monomials”
“In this lesson, we will learn how to divide a polynomial by a monomial (polynomial with one term). In order to perform this action, we will think back on operations with fractions. Let’s think about the following problem: 12 ÷ 3 = 4 We can rewrite this problem using a fraction bar. A fraction bar represents the division of the numerator by the denominator. In our example, 12 is being divided by 3, this means in fractional form 12 is our numerator and 3 is our denominator:
12
—-
3
=4
“Dividing a Polynomial by a Monomial”
” * Set up the division problem as a fraction.”
“Example 1
Find each quotient (4x^4 + 2x^3 + 32x^3) ÷ 8x^2
Step 1) Let’s set up the division problem using a fraction:
4x^4 + 2x^3 + 32x^3—————————–
8x^2 ”
Note where the explanation says, “We can rewrite this problem using a fraction bar.”
~ ~ ~ ~ ~ ~ ~ ~ ~
from Enchanted Learning .com:
“Mathematical Symbols”
https://www.enchantedlearning.com/math/symbols/
See mathematical symbols chart:
“Math symbols are shorthand marks that represent mathematical concepts.”
“Symbol
÷ or /
Name
division sign
~ ~ ~ ~ ~ ~ ~ ~
and from Third Space Learning:
https://thirdspacelearning.com/us/blog/what-is-division/
“Division is a mathematical operation which involves the sharing of an amount into equal-sized groups. For example, “12 divided by 4” means “12 shared into 4 equal groups”, which would be 3.
In mathematics, this is written with the division symbol, called an obelus → ÷. Division can also be presented as a fraction: in this case, 12/4 (the division sign is represented as a forwards slash) or 12 [over] 4 (the division sign here is represented as a horizontal line). These are all equal to 3.”
~ ~ ~ ~ ~ ~ ~
So it appears that the concept of Fraction Bar=Solidus=Obelus is not being taught wrong, since they all mean “divided by.” What is being taught wrong is the Order of Operations as PEMDAS because Division=Fraction (whether written vertically or horizontally), and to calculate the value of a fraction (i.e. a division problem), DIVISION MUST GO LAST (after calculating the numerator & denominator)!
— Dee
Comment by Dee R. — August 14, 2025 @ 2:01 pm
Scott —
I understand that you are aiming to create a new paradigm, and your “Reciprocation” solution does that in many respects, but the existing mathematical construct is that expressions such as 8÷2(2+2) & 6÷2(1+2) are fractions which are dividing one monomial by another monomial.
Let’s remember that included in the definition of a monomial (a single term) & examples thereof is that a constant is a monomial, as is a variable by itself. In both those cases, it is understood that they each have a numerical coefficient of “1.” In other words, x÷x=1x÷1x & 8÷4=1(8)÷1(4). No parentheses are necessary for a monomial to be understood as one term having a single combined value. For example, When a=4, the monomial 2a ALWAYS equals 8, regardless of what operational sign precedes it (going left-to-right).
With that being the case, the most concise way to convey this construct is that monomials feature implied multiplication (sometimes simply multiplied by 1), which means that there is a single value to all monomials, whether they are a constant, a variable by itself, or consist of multiple components without an explicit operational sign separating elements of the monomial (e.g. 4xyz or 3ab^2). That monomial single value does not change because there is a division sign to the immediate left of it in a horizontally written expression. In 4a÷2a, “2a” still equals 24, whether written vertically or horizontally (with an obelus or slash).
Those who believe otherwise need to refresh their memories on “How to Divide by a Monomial,” and how to substitute a value for a variable, once it is known. Googling the expression below will explain the process, step-by-step.
Copy & paste into the search box:
4a÷2a substituting 12 for a
The AI says 2 is the answer, solving it in two different ways:
(Method 1)
“Substitute, then divide”
“Substitute 12 for a: 4(12)÷2(12)
Perform the multiplications: 48÷24
Perform the division: 2”
(Method 2)
“Simplify first, then substitute:
4a÷2a=
4a
——
2a
“Cancel out the common factor a:
4
——
2
Perform the division: 2
In this case, both methods yield the same result. The value of 4a÷2a when a=12 is 2.”
Note that parentheses were not used to surround the monomial in either case, yet each monomial was still understood to have a single value. Also note that the expression is shown as a fraction in the second method.
— Dee
Comment by Dee R. — August 14, 2025 @ 4:05 pm
Scott —
The bottom line is that until someone can show math textbooks and/or lessons on bona fide math tutoring websites (which mirror lessons taught in the classroom), which include solved examples for students to learn the proper procedure for applying PEMDAS (or BODMAS) to an expression that features implied multiplication to the right of an obelus (or slash) where the method is shown as inserting an explicit multiplication sign and treating the number to the left of the open-parentheses as a completely separate entity, then that idea is unproven. Calculators may be programmed incompletely or incorrectly , so showing calculator results is not proof of anything.
What is proven on a number of math tutoring websites from all over the world, is that “dividing by a monomial” is described & demonstrated for students as a top-and-bottom fraction, because students have been taught from 5th grade onward that Fractions=Division & therefore Division=Fraction. Elementary Algebra students are affirmatively instructed to “Rewrite as a fraction,” when there is a horizontally written division expression which uses an obelus — and shows the denominator as the whole monomial term, not just the numerical coefficient (i.e. the number to the left of the open-parentheses, with no operational sign in between).
Let’s see solved examples which show young math students how to apply PEMDAS in a division expression which features implied multiplication to the immediate right of the obelus. It’s time to show proof of concept!
— Dee
Comment by Dee R. — August 15, 2025 @ 1:30 pm
Numbers like 2(9 + 3) and 3/7 ARE monomials by virtue of juxtapositional binding. This is my whole argument. Yet in your comment on 8/12, you said, “According to every math teaching website I have seen so far, 8 ÷ 3 ÷ 7 is the same thing as 8 ÷ 3/7.” That is NOT treating the fraction like the monomial it should be. Every math text I know of teaches when you divide by a fraction, you invert the fraction and multiply first. So the Web site is, as I said before, DEAD WRONG to teach otherwise. This is the inherent weakness in the whole PEMDAS/OOO paradigm, that it does NOT account for the manipulations that take place with a fraction PRIOR TO standard OOO.
Our whole number system depends on place value. The place value system depends on implied operations between the juxtaposed digits to give the numbers their full meaning. We don’t use parentheses around the numbers because that would just be too bulky of a system. We have juxtaposed numbers using implicit addition, implicit division, AND implicit multiplication. The Roman numeral system even has implicit subtraction built into their juxtapositional system with place “ordering” rather than place value (i.e., the biggest values come first unless you’re subtracting).
For the Arabic system:
Here’s yet another way to look at it from a place-value perspective.
And there you go again, Dee! :-) In responding to your valuable feedback, I’ve just come up with yet ANOTHER way of explaining how this juxtapositional binding operates in our number system (or at least how it should), and it doesn’t contradict anything else I’ve said about it.
Thank you for contributing.
Scott
Comment by Scott Stocking — August 18, 2025 @ 6:20 pm
Scott —
In response to your saying, “Numbers like 2(9 + 3) and 3/7 ARE monomials by virtue of juxtapositional binding. This is my whole argument. Yet in your comment on 8/12, you said, ‘According to every math teaching website I have seen so far, 8 ÷ 3 ÷ 7 is the same thing as 8 ÷ 3/7.’ That is NOT treating the fraction like the monomial it should be,” there is an explicit operational symbol in every fraction, and what is taught in classrooms all over the world is that the fraction bar is a division symbol (i.e. meaning ‘divided by’), just as the slash & obelus are. As a consequence, all division symbols are interchangeable. Conversely, juxtapositional binding via implied multiplication (i.e. with no explicit operational symbol) is taught as being a monomial — one term with a single value which is the product of its factors. Therefore, 2(9+3)=2a when a=(9+3) 0r a=12 –and that expression equals 24.
A fraction such as 3/7 is simply unexecuted division. That division problem of “3 divided by 7” can actually be divided further, but would have to be expressed as a decimal.
Teaching “Dividing Fractions” is showing the methods & techniques for executing successive divisions (including unexecuted divisions), which may use reciprocals to calculate the total value of the expression.
— Dee
Comment by Dee R. — August 19, 2025 @ 1:33 pm
Hi Scott,
On a site called The Math Doctors (“Implied Multiplication 3: You Can’t Prove It”), Dr. Peterson argues that it can’t be proven that a grammatical rule is correct or incorrect. Previously, in Dr. Peterson’s blog piece “Implied Multiplication 2: Is There a Standard?,” he asked for proof of what is taught, saying, “I would welcome examples of books, from anywhere and any time, that do explicitly teach about order of operations with implied multiplication, regardless of their view.”
I sent Dr. Peterson the following from teaching websites, but it does not appear in the comments — I think he has blocked me, perhaps for presenting information that disproves his stance that such expressions are “ambiguous.” I thought you might be interested in what I tried to send, as it contains links to different sites to what I have referred to before:
[to Dr. Peterson]
I get that you tepidly endorse IMF but you argue that it can’t be proven that a grammatical rule is correct or incorrect. In this case, the mathematical grammar to work out division with implied multiplication is actually taught in school lessons.After looking high, low and in between on the web, there does not seem to be a school lesson that tells elementary school students first learning PEMDAS how to calculate an expression using a division sign with an implied multiplication right after it.However, there are algebra lessons on sites geared for kids being homeschooled and for kids to get extra help to keep up with what their teacher has gone over, that directly address expressions with that exact construction or “grammar,” if you want to call it that.Some examples of those lessons:Prodigy website:https://www.prodigygame.com/main-en/blog/distributive-property#:~:text=What%20is%20the%20distributive%20property,and%20then%20adding%20them%20together.As one of the most commonly used properties, it’s important to learn how to perform and apply the distributive property. Without it, clearing the parentheses wouldn’t be possible.another site:Story of Mathematics:https://www.storyofmathematics.com/use-the-distributive-property-to-remove-the-parentheses/We can use the distributive property to expand and solve complex expressions. It tells us how to remove parentheses in an equation.another site:Study:https://study.com/skill/learn/how-to-solve-an-equation-with-parentheses-explanation.html#:~:text=Combine%20any%20like%20terms%20within,variable%20to%20solve%20the%20equation.The order in which we perform mathematical steps is known as the order of operations and is a critical part of solving equations properly. One general rule is to resolve any parenthetical statements before performing operations outside of the parentheses. Here are some steps to solving equations containing parentheses for an unknown variable.1. Combine any like terms within the parentheses.2. Distribute any coefficients to the parenthetical expression to remove the parentheses^^^^^^^^^So based on mathematical “grammar” lessons taught in school, in 6 ÷ 2(1+2) if you substitute (a+b) for (1+2) and follow the explicit instructions of those lessons, the distributive property can be used to expand and clear the parentheses as follows:6 ÷ 2(a+b)using the distributive property to expand the parentheses by multiplying out the coefficient before going any further:2(a+b)=(2a+2b)which makes the expression
6÷(2a+2b)
If a=1 and b=2, then replacing the variables with their values…6÷(2(1)+2(2))followed by doing the addition inside the expanded parentheses(2+4)=6Now the expression is 6÷6 according to how algebra students are told how to calculate an expression with implied multiplication to the right of the division sign.Is that the proof you asked for?
~ ~ ~ ~ ~ ~ ~
I think this proves the case as to what is taught in school, regarding how to solve an expression with a “grammatical” construction which contains an implied multiplication to the right of an obelus.
— Dee
Comment by Dee R. — August 30, 2025 @ 2:30 pm
I’m not on Facebook, but I think you are — have you seen this latest one from May of this year?
https://www.facebook.com/story.php?story_fbid=1143230534494204&id=100064218036538
48÷12(2+2)
You may want to weigh in on that.
If I may put in my two cents here, it is the same thing as 12a÷12a when a=4 or a=(2+2), because 48 can be factored out as 12(4) or as 12(2+2). And according to every math tutoring website I have seen which is teaching “How to Divide by a Monomial,” students are instructed to “Rewrite as a fraction,” and demonstrated in solved examples as the entire monomial term to the right of the obelus as the whole denominator (not just the numerical coefficient), making this proposition…
12a
——–
12a
…when a=4, it becomes…
12(4)
———–
12(4)
…which equals 1.
Comment by Dee R. — August 13, 2025 @ 10:12 am |
Hi Scott —
I just wanted to share this with you, from tutoring website Khan Academy, 6th grade math questions:
https://www.khanacademy.org/math/cc-sixth-grade-math/x0267d782:cc-6th-exponents-and-order-of-operations/x0267d782:more-on-order-of-operations/a/exponents-and-order-of-operations-faq#:~:text=When%20do%20we%20NOT%20follow%20the%20order,order%20instead%20of%20only%20left%20to%20right.
“When do we NOT follow the order of operations?
Lots of times, actually! While the order of operations gives us one way to evaluate an expression, the properties of addition and multiplication allow us to be more flexible.
The distributive property says that we can multiply a value to each term inside of the parentheses instead of adding or subtracting inside the parentheses first.”
The commutative property of multiplication says that we can multiply the factors in any order instead of only left to right. Once we learn more about reciprocals, we’ll be able to rewrite expressions with multiplication in place of the division.”
~ ~ ~ ~ ~ ~ ~ ~
According to what is taught in 6th grade math, there are exceptions to following the Order of Operations (PEMDAS). This confirms my analysis that “2(2+2)” holds a single value which is the product of its factors [i.e. the monomial “2a” when a=(2+2) or a=4]. And of course, one of those exceptions to “NOT follow the order of operations” is your “Reciprocation” solution.
— Dee
Comment by Dee R. — August 18, 2025 @ 5:07 pm |
Hi Scott,
On a site called The Math Doctors (“Implied Multiplication 3: You Can’t Prove It”), Dr. Peterson argues that it can’t be proven that a grammatical rule is correct or incorrect. Previously, in Dr. Peterson’s blog piece “Implied Multiplication 2: Is There a Standard?,” he asked for proof of what is taught, saying, “I would welcome examples of books, from anywhere and any time, that do explicitly teach about order of operations with implied multiplication, regardless of their view.”
I sent Dr. Peterson the following from teaching websites, but it does not appear in the comments — I think he has blocked me, perhaps for presenting information that disproves his stance that such expressions are “ambiguous.” I thought you might be interested in what I tried to send, as it contains links to different sites to what I have referred to before:
[to Dr. Peterson]
I get that you tepidly endorse IMF but you argue that it can’t be proven that a grammatical rule is correct or incorrect. In this case, the mathematical grammar to work out division with implied multiplication is actually taught in school lessons.
After looking high, low and in between on the web, there does not seem to be a school lesson that tells elementary school students first learning PEMDAS how to calculate an expression using a division sign with an implied multiplication right after it.
However, there are algebra lessons on sites geared for kids being homeschooled and for kids to get extra help to keep up with what their teacher has gone over, that directly address expressions with that exact construction or “grammar,” if you want to call it that.
Some examples of those lessons:
Prodigy website:
https://www.prodigygame.com/main-en/blog/distributive-property#:~:text=What%20is%20the%20distributive%20property,and%20then%20adding%20them%20together.
As one of the most commonly used properties, it’s important to learn how to perform and apply the distributive property. Without it, clearing the parentheses wouldn’t be possible.
another site:
Story of Mathematics:
https://www.storyofmathematics.com/use-the-distributive-property-to-remove-the-parentheses/
We can use the distributive property to expand and solve complex expressions. It tells us how to remove parentheses in an equation.
another site:
Study:
https://study.com/skill/learn/how-to-solve-an-equation-with-parentheses-explanation.html#:~:text=Combine%20any%20like%20terms%20within,variable%20to%20solve%20the%20equation.
The order in which we perform mathematical steps is known as the order of operations and is a critical part of solving equations properly. One general rule is to resolve any parenthetical statements before performing operations outside of the parentheses. Here are some steps to solving equations containing parentheses for an unknown variable.
1. Combine any like terms within the parentheses.
2. Distribute any coefficients to the parenthetical expression to remove the parentheses
^^^^^^^^^
So based on mathematical “grammar” lessons taught in school, in 6 ÷ 2(1+2) if you substitute (a+b) for (1+2) and follow the explicit instructions of those lessons, the distributive property can be used to expand and clear the parentheses as follows:
6 ÷ 2(a+b)
using the distributive property to expand the parentheses by multiplying out the coefficient before going any further:
2(a+b)=(2a+2b)
which makes the expression
6÷(2a+2b)
If a=1 and b=2, then replacing the variables with their values…
6÷(2(1)+2(2))
followed by doing the addition inside the expanded parentheses
(2+4)=6
Now the expression is 6÷6 according to how algebra students are told how to calculate an expression with implied multiplication to the right of the division sign.
Is that the proof you asked for?
~ ~ ~ ~ ~ ~ ~
I think this proves the case as to what is taught in school, regarding how to solve an expression with a “grammatical” construction which contains an implied multiplication to the right of an obelus.
— Dee
Comment by Dee R. — August 30, 2025 @ 2:32 pm |