(NOTE: This article was modified September 1, 2024, to include two new terms I recently coined: “functional monomial” and “operational monomial.” New material is in italics. When I say “PEMDAS” I’m implying Order of Operations (OOO) as well.
My original PEMDAS article has received quite a bit of traffic in the 11 months since I posted it. I’m well over 10,000 views at this point, which for the narrow appeal of my usual topics on this blog, is significant for me. It represented about one-third of the total views I had last year overall. The article has generated enough discussion that has allowed me to continue to fine-tune my arguments and further solidify my position that, bottom line, juxtaposed multiplication takes priority over any signed operations. I want to present a couple different views here, neither of which I feel has been adequately disputed by those who think showing me a page from a grade school math textbook is enough to convince me my arguments are faulty. I’ll start with an even stricter interpretation of PEMDAS then what the online PEMDASians apply, and then I’ll discuss the linguistic and syntactical arguments that has yet to be refuted by the paltry evidence the PEMDASians try to put forth.
A Stricter(!) Application of PEMDAS/OOO
(NOTE: This section is modified from post I made on Facebook to a different expression with similar issues.)
To refresh your memory, here is the expression in question:
8 ÷ 2(2 + 2)
Let’s start at the parentheses level of PEMDAS, shall we? The first step is simple enough. Add 2 + 2 to get 4, leaving us with:
8 ÷ 2(4)
Let’s set aside for a moment that this violates the distributive property to work what is inside the parentheses first, which would imply the expression becomes 8 ÷ (4 + 4} before solving what the parenthetical expression.
You’ll notice that we have parentheses around the 4, so we need to perform the function syntactically suggested by the parentheses, multiplication, to remove them. Notice I say “syntactically,” because math isn’t just about numbers, signs, and symbols; it’s about how those are arranged in an expression. This is the fatal error that the PEMDASians make: They fail to acknowledge the implicit relationships suggested by the presence of the parentheses.
The presence of the parentheses represents two issues: First, the juxtaposition itself implies parentheses around BOTH juxtaposed numbers as a single unit of value. We see this with mixed numbers, where the horizontal juxtaposition of a whole number and a fraction IMPLY addition, yet we treat the mixed number as one value. You’d be hard pressed to find a textbook that always puts parentheses around a mixed number in any expression. We also see this with fractions. Keep in mind that a fraction is not always intended to be a division problem. The figure 3/4 (diagonally juxtaposed with a solidus; or its display equivalent with vertical juxtaposition separated by a fraction bar or vinculum) would typically be pronounced “three-fourths.” The expression “3 ÷ 4” would be pronounced “3 divided by 4.” There is a qualitative difference between the two expressions, and we cannot assume that one substitutes for the other. It would be rare, but not unheard of, to see such a fraction with parentheses around it in an expression, because the fraction, like the mixed number, represents a single value.
The logical conclusion from this line of reasoning is this: If a juxtaposed mixed number is considered a single value and a juxtaposed fraction is a single value, then juxtaposed cofactors that use parentheses around one or more of the numbers to distinguish the values of the cofactors (i.e., “2(4),” “(2)(4),” and “(2)4”) all represent a single value of “twice four” as opposed to “24,” which represents “twenty-four” with no intervening parentheses. The lack of an operational sign (remember, the vinculum and solidus in in-line texts are grouping symbols first and foremost, NOT operational signs; only the obelus is an actual operational sign) in these three formats (implied multiplication, mixed number, display fraction) suggests that these forms do NOT fall with the last four steps of the “order of operations”/PEMDAS and instead should be given a higher priority. I would consider these “functional monomials,” because the function they perform is not explicitly stated by the use of operational signs but by the syntax of the format. This priority is heightened by the fact that when you have to divide by a mixed number or a fraction, you have to manipulate the fraction and the extant operational sign to properly work the expression.
Because “functional monomials” do not use explicit operational signs outside of their use in parentheses, they belong in the first two steps of PEMDAS/OOO where we find other functional monomials that reflect implied operations: powers (e.g., 33) and roots, factorials (e.g., 5!), and trig and log functions (e.g., 2 cos2 x; log10 423), among others. They are calculated before any other operational signs acting upon them and are not disturbed or separated by preceding operational signs. (See below for the discussion of “operational monomials” in contrast to “functional monomials.”
If you accept my first point above, then, this second point is moot, but I’ll address it anyway. We can’t get rid of the parentheses (remember, we’re still in the parentheses step) until we perform the function inherent in the parentheses. The PEMDAS/OOO charts get it wrong when they interpret “inside the parentheses” as only what is in-between the parentheses. “Inside” also means “inherent in the nature of,” so the function of the parentheses must be performed as well in the absence of any extant sign. The parentheses are still present, so we’re still in the parentheses step of PEMDAS. It’s at this point that the PEMDASians want to just simply replace the parentheses with a multiplication sign. But where in the parentheses step does it allow that kind of substitution? You have a syntactical relationship between the 2 and the (4) that simply disappears if one makes such a substitution. That substitution is not a valid or necessary math function when evaluating the written expression! Yes, you must use the multiplication key on a calculator if you really need to use one for this type of problem, but that is a matter of technology and not of math theory. The only way to address the parentheses at this point in the process and finish the parentheses step in PEMDAS is to perform the implicit multiplication first, because the juxtaposition creates an implied set of parentheses around the 2(4). Only then are we done with the parentheses step and are left with the simplified expression:
8 ÷ 8
Which of course equals 1.
Still not convinced that substituting the multiplication sign for the parentheses isn’t valid (except when you’re entering it into a calculator, but we’re not using a calculator here), then consider this. The relationship between the two 8’s is that of a dividend to a divisor only. We wouldn’t look at the way that is written and say 8 is the numerator and the other 8 is the denominator. It’s not written that way. We could only do that if we used a vinculum or fraction bar. As such, then, the vinculum, which is a juxtapositional symbol implying division, creates a unique relationship between the two numbers not implied by the obelus. As such, it’s not a valid substitution! With the vinculum, the numbers represent a part of a whole or the whole divided into parts. (I work in a field that requires a significant amount of government reporting on data, and they are always speaking of the populations in terms of which set is the numerator and which set is the denominator.) If the vinculum is such a juxtapositional tool that it creates or represents a unique relationship between the two numbers, then the parentheses serves the same purpose for multiplication. The 2(4) can’t be reduced to a simple multiplication. Depending on the context in which such an expression might arise, it may refer to a single quantity, like a bundle of 2 packages of golf balls with 4 balls in each package (8 golf balls).
Therefore, the only way one can claim that 8 ÷ 2 is somehow the term multiplied by what is in parentheses is to unequivocally declare it so by putting it in parentheses or constructing it as a block fraction using a vinculum vertically centered on the (2 + 2). As I said above, an expression using the obelus is NOT syntactically or linguistically equivalent to a fraction using the same numbers. Since the PEMDASians have failed to clarify the function of 8 ÷ 2 by enclosing it in parentheses, they do not have any solid ground to stand on to insist the answer is 16. This is where my other new term applies, “operational monomial.” If the problem had been written 8 ÷ 2 x (2 + 2), then those who believe the answer is 16 would have a point, because PEMDAS/OOO rules tell us division comes first, so they would then be correct to say the 8 ÷ 2 is the “term” as they define it that serves as the coefficient to the parenthetical expression. Since the expression uses an extant operational sign, it falls in the last four steps of PEMDAS/OOO, after the first two steps that represent “functional monomials.”
Explaining the Linguistic and Syntactical Arguments
(NOTE: This is copied from a response I made on the original article, with a few minor edits.) When I say there is a “linguistic” or “syntactical” component to the given expression, what I’m talking about is how Merriam-Webster defines the term: “The study of human speech including the units, nature, structure, and modification of language.” I take “speech” to mean the written word as well as the spoken word, especially since as a preacher I’ve gotten into the habit of writing out my sermons so I can make more intentional use of my language as opposed to speaking extemporaneously. And in the context of this article, I don’t just mean words alone, but any symbols or figures that we use to communicate, calculate, or cantillate (how’s THAT for an alliteration!): numbers, punctuation, “character” words (e.g., ampersand, &), mathematical and scientific symbols, proofreading symbols, and even music notation.
All of these elements of language, and linguistics more broadly, have their place in their appropriate contexts, and they are subject to their own respective set of rules for putting them together in a coherent form that communicates the message and meaning we intend subject to the rules and conventions of their respective contexts. When someone composes a musical score, the main melody or tune is subject to certain patterns that follow the chords that underlie the melody. If the tune doesn’t match the chords, it sounds, well, discordant. The notes of the melody, harmony, or even a descant are not strictly random. They typically have some relationship with the chord, and often playing a note that doesn’t exactly fit the chord prefigures a change in the chord or even a change in the key signature. Intentional discordancy is not without significance either, as it can communicate chaos or irrationality.
When we write a sentence, we generally expect a subject and verb to be close together and to arrange direct and indirect objects appropriately with any modifiers or prepositions, and so forth. For example, consider the difference between the three sentences, which have the exact same words.
- I eat fish only on Friday.
- I eat only fish on Friday.
- I only eat fish on Friday.
Sentence #1 is truly ambiguous, because the placement of “only” can be taken either way. Is it “Fish is the only thing I eat on Friday” (akin to Sentence #2) or “Friday is the only day I eat fish” (akin to Sentence #3)? Does that sound familiar in the context of this post? More on that in a bit.
In the original article, I make reference to the relationship between the definite article, noun, and adjective in a Greek adjectival phrase. The position of (or absence of) the definite article impacts how the phrase can be interpreted. I’ll use transliterated words to demonstrate.
- kalos logos [beautiful word]
- ho kalos logos OR logos ho kalos [the beautiful word]
- ho logos kalos OR kalos ho logos [the word is beautiful]
In Greek, Phrase #1, which has no definite article (the indefinite article “a” can fairly be implied absent other contextual clues), would be considered ambiguous by itself. We would need contextual clues to know whether it means “a beautiful word” or “a word is beautiful.” (Greeks do not have to use a form of the copulative verb “to be” if that is the only verb in the sentence.) In Phrase #2, the definite article precedes the adjective, which means the adjective is attributive, that is, it directly modifies the noun (“The beautiful word”). It doesn’t matter if the noun is first or last; it’s attributive either way. Phrase #3 has a predicate construction. This means that the noun is the subject of a sentence, and the adjective would come after the verb in that sentence. In this case, it doesn’t matter where the adjective is, although there may be a nuanced implication one way or the other. Either way, the translation is still “The word is beautiful,” so no difference there.
Given those three examples (music, English adverb placement, and Greek definite article placement), I think anyone who’s reading this is starting to see the bigger picture of how linguistics (in this case, specifically syntax) influences mathematics as well, especially in the context of the expression at hand. So let me use the expression in the same way I used the sample phrases above:
- 8 ÷ 2(2 + 2) = 1 (in my worldview and the worldview of those who are of the same mind) OR 16 (in the competing worldview)
- (8 ÷ 2)(2 + 2) = 16 (in both worldviews; NOTE: if the expression had been written with (8 ÷ 2) as a block fraction with a vinculum centered vertically on the (2 + 2), there would be no argument that it equals 16; see text for my critique of that, however.)
- 8 ÷ (2(2 + 2)) = 1 (in both worldviews)
Expression A seems unambiguous from the perspective of one’s worldview then. But are both worldviews equally valid? We can make arguments from our respective worldviews to try to convince the other side, but it is very difficult to convince one to change their worldview without a powerful defining event that shakes their worldview to the core. Otherwise, we’re comfortable with our ways. I happen to think that several of the arguments I’ve made to support my worldview are quite devastating to the competing worldview, but alas! there has been very little evidence of any change of heart among their hardliners.
Just like the position of adverbs and definite articles, so then is the generous use of parentheses needed to clearly avoid the ambiguity of the given expression. But let me make yet another appeal here for the case that the given expression, in light of my demonstration here, is not really ambiguous at all. The juxtaposition of the 2 to (2 + 2) is akin to Phrase #2 in my Greek examples above. The attachment between the two places them in an attributive relationship (the 2 is the definite article; the (2 + 2) is the adjective). The 2 directly modifies the (2 + 2) by telling us how many of that quantity we need to divide by and keeps the monomial on one side of obelus without an extant multiplication sign. In other words, it isn’t separated from its cofactor by the “action” of the obelus. There is no need for the extant multiplication sign because the relationship is clearly defined. If one were to place a multiplication sign between the 2 and (2 + 2), that would emphasize that the 2 and (2 + 2) are not cofactors and sever the relationship between them. This would make the expression like Greek Phrase #3 above, where the modifier is divorced from what it modifies modified and dragged kicking and screaming all alone into the action of the obelus. That which appeared to modify the (2 + 2) now modifies the 8. The implications of the expression change by substituting the multiplication sign. Additionally, in the case of Greek Phrase #3, if we would add the implied copulative verb where it is not technically needed, that would also place emphasis on the verb and suggest a more nuanced meaning.
Greek verbs demonstrate a similar phenomenon; most Greek verb forms have an ending that tells you what “person” [1st, 2nd, 3rd, or I/we; you/you; he, she, it/they] is the subject of the verb. If there is no explicit subject accompanying the verb, the corresponding pronoun is implied [“He eats”]. If a Greek pronoun is used as the subject, that implies emphasis [“He himself eats”]; so an extant multiplication sign emphasizes the function of the sign over the relationship between the two cofactors when the multiplication is implied by parentheses. The bottom line for the Greek phrases, then, is when you add a word that isn’t necessary for the base form of what you’re communicating, you alter the meaning of what you’re communicating. You also alter the meaning when you add a multiplication sign that isn’t necessary for the basic calculation of the given expression.
This may seem kind of heady to some, but I hope I’ve made my position a little easier to understand. My worldview and what I consider the strength of my arguments here and elsewhere, along with a ton of historical evidence, do convince me that the given expression is unambiguous and has no need for extra parentheses to understand the answer to be 1. For those who think writing ambiguous expressions is somehow educational and instructive when you know there are those who think otherwise, I declare that you have met your match in me. Game over. Checkmate!
Scott Stocking
My opinions are my own.
Sunday Morning Greek Blog 
[…] If you like this article, you may also like https://sundaymorninggreekblog.com/2023/05/29/toward-an-active-pemdas-strengthening-its-theoretical-foundation/ or my latest article, https://sundaymorninggreekblog.com/2024/03/16/8-%c3%b7-22-2-1-part-2-a-defense-of-the-linguistic-arg… […]
Pingback by 8 ÷ 2(2 + 2) = 1: Why PEMDAS Alone Is Not Enough | Sunday Morning Greek Blog — March 16, 2024 @ 12:36 pm |
Just read it now. Some thoughts…
You say:
“…the fatal error that the PEMDASians make: They fail to acknowledge the implicit relationships suggested by the presence of the parentheses. We can’t get rid of the parentheses (remember, we’re still in the parentheses step) until we perform the function suggested by the parentheses. …As such, we’re still in the parentheses step of PEMDAS.”
100% CORRECT!
You also say:
“The relationship between the two 8’s is that of a dividend to a divisor only. We wouldn’t look at the way that is written and say 8 is the numerator and the other 8 is the denominator. It’s not written that way. We could only do that if we used a vinculum or fraction bar. As such, then, the vinculum, which is a juxtapositional symbol implying division, creates a unique relationship between the two numbers not implied by the obelus. As such, it’s not a valid substitution! With the vinculum, the numbers represent a part of a whole or the whole divided into parts.“
My comment on that…
I have seen several math teaching websites (and Wikipedia) which are teaching young math students that the division sign (obelus), the slash (solidus) & the fraction bar (vinculum) are synonymous — they all mean “divided by.” Here’s an example, from the Basic Algebra teaching site called “Algebra for Dummies”:
https://www.dummies.com/article/academics-the-arts/math/algebra/recognizing-operational-symbols-in-algebra-194538/
“Division (÷, −, /): The division, fraction line, and slash symbols all mean divide. The number to the left of the ÷ or / sign or the number on top of the fraction is the dividend (in this example, 6). The number to the right of the ÷ or / sign or the number on the bottom of the fraction is the divisor (in this example, 2). The result is the quotient (in this example, 3).
6 ÷ 2 = 3
6
__ = 3
2
6 / 2 = 3 “
————–
It is possible to have an “improper fraction,” which is not “a part of a whole or the whole divided into parts.“
Because all division symbols indicate the same operation (i.e. something divided by something), “numerator” and “dividend” are synonymous with one another & “denominator” and “divisor” are synonymous with one another.
————–
If I were to tell you that I have 4 dozen eggs, you would understand that I have a total of 48 eggs, as “4 dozen” is a single term which holds a value of a dozen taken 4 times, which can be written as…
4 dozen = 1 dozen + 1 dozen + 1 dozen + 1 dozen
…or…
4 dozen = 12 + 12 + 12 + 12
So because the term “4 dozen” has a single value of 48, the “4” in “4 dozen” cannot be separated from its factor (“dozen”) & used in some other operation in the statement before calculating the value of the single term (monomial) itself.
Comment by Dee — March 18, 2024 @ 11:32 am |
For a simple division problem of two numbers or monomials, I would agree with the definition. However, if you have 6 ÷ 2 ÷ 3 and take that left to right, you get 1. But if all three symbols are identical in function, the you should be able to get the same answer. But that’s not the case. If you sub in the vinculum between the 2 and 3, you get 6 divided by two-thirds, which, after you invert the fraction after the obelus and change the obelus to multiplication, you get the answer 9. So the symbols are not a symbolic “identity.”
Comment by Scott Stocking — March 18, 2024 @ 12:02 pm |
With different division symbols being used, many people might “see” it the way you’re saying. In actuality, the obelus, solidus & vinculum all mean “divided by,” so there is no substantive difference in the meaning of one division symbol vs. another.
With that being the case, in a horizontally written statement containing multiple division operations, I believe parentheses would be necessary to delineate where the numerator (dividend) is & where the denominator (divisor) is. In a horizontally written statement containing only one division symbol, however, the numerator (dividend) is everything to the left of the obelus (division sign) or solidus (slash) & and the denominator (divisor) is everything to the right of the obelus or solidus, just as in a vertically written fraction (something-over-something) the numerator (dividend) is everything above the fraction bar (vinculum) & the denominator (divisor) is everything below the fraction bar (vinculum).
Comment by Dee — March 18, 2024 @ 12:41 pm
Vertical juxtaposition by vinculum is the complement to horizontal juxtaposition by parentheses. Or put another way, the vinculum is to implied division what the parentheses are to implied multiplication.
Comment by Scott Stocking — March 18, 2024 @ 2:05 pm
Here’s Symbolab’s online teaching site’s calculator:
https://www.symbolab.com/solver/trigonometric-simplification-calculator/4x%20%5Cdiv%202x?or=input
I typed in 4x ÷ 2x, and in the steps showing how the quotient was derived, it turned it into the top-and-bottom fraction of…
4x
___
2x
In so doing, the teaching website is instructing young math students that the statement “4x divided by 2x” can use an obelus (or solidus) when written horizontally, or can be written vertically as a top-and-bottom fraction, with the vinculum to be read as “divided by.”
Comment by Dee — March 18, 2024 @ 4:21 pm
I appreciate the feedback, and you’re sharpening my thinking even more. I’m not denying that in a simple division problem, what you describe is 4x divided by 2x. I’m saying that when you start to put such fractions (a fraction is different from a linear expression with an extant obelus) in the context of a larger expression, there are other issues that arise. Wolfram describes the vinculum as “a horizontal line placed above multiple quantities to indicate that they form a unit.” https://www.wolframalpha.com/input?i=vinculum You’ll notice in the link that one of the examples is the horizontal line extended from the root symbol to indicate the radicand. He also has a cross-reference to the “long division symbol,” which uses the vinculum extended from the vertical bar to indicate the dividend or numerator in such an expression. Merriam-Webster defines it as “a straight horizontal mark placed over two or more members of a compound mathematical expression and equivalent to parentheses or brackets around them.” It even has an illustration with the expression a – b – c, with a vinculum over the b – c, and equates that to a – [b – c]. I’ve seen older math texts describe vinculum in similar terms, as a form of grouping.
So if the vinculum is a form of grouping, then any part of any expression that uses it must be addressed in the P step of PEMDAS! Here’s the proof, as I’ve already spelled out in previous responses: If we have an expression 6 ÷ ⅔ (where the solidus is actually a vinculum; have to pretend in this limited formatting environment), then all of us have been taught to address the fraction first, right? How? We invert the fraction first, then multiply across! That expression should NOT be read “six divided by two divided by three,” but “six divided by two-thirds.” The use of the ordinal (or would it be partitive???) word “thirds” is the division equivalent to “thrice” in multiplication. Do you see what I’m getting at there? The vinculum should be considered a bracket (a la BODMAS) or parentheses. That’s why in the broader usage of the vinculum, it is NOT an identity with the obelus. For the same reason, then, in 8 ÷ 2(2 + 2) you cannot ungroup the 2 by imposing a vinculum ONLY on the 2. You’d be disrupting an existing juxtaposition by creating a different juxtaposition. The 2(2 + 2) is already grouped, so if you add another grouping symbol in front of that, it must group that which is already grouped.
Again, thank you for the feedback. This is yet another discovery for us that further solidifies our position that 8 ÷ 2(2 + 2) = 1.
Scott
Comment by Scott Stocking — March 18, 2024 @ 9:45 pm
You make some very good points about the vinculum being a grouping symbol. At the same time, the obelus & solidus are actually the same as the vinculum, in terms of where the “divider” is between two groups — the obelus & solidus also separate the numerator (everything to the left of the obelus or slash) from the denominator (everything to the right of the obelus or slash).
from Algebra Class. com:
https://www.algebra-class.com/dividing-monomials.html
Dividing monomials”
“Remember: A division bar and fraction bar are synonymous!”
…and from Algebra Practice Problems .com:
https://www.algebrapracticeproblems.com/dividing-monomials/“
“Any division of monomials can also be expressed as a fraction”
————-
Division is a fraction. A fraction is division.
————-
With regard to a multi-division statement such as 6/2/3, it could mean (6/2) divided by 3, or it could mean 6 divided by (2/3) — parentheses would be necessary to delineate which it is. In a vertical fraction, it could be grouped as (6/2) over 3, or as 6 over (2/3). Using the smaller solidus-vinculum form (i.e. ⅔) would be one way to visually group a fraction within a larger fraction — but that would be akin to using parentheses to differentiate the divisions within the statement, as far as where the numerator is & where the denominator is in the “main” fraction. Another way is to use gigantic parentheses around the numerator or denominator in a top-and-bottom fraction, which I haven’t figured out how to manage on my computer. In any case, parentheses are necessary to delineate the order of divisions in a multi-division statement.
Comment by Dee — March 19, 2024 @ 10:28 am
I thought I had replied to this already, but I don’t see it posted. Oh, well.
I see what you mean in defining the vinculum as a grouping symbol, separating the numerator from the denominator (indicating precisely what is being divided by what). In the context of division, though, the obelus or slash is also the “divider” between the numerator (everything to the left of the obelus or slash) & the denominator (everything to the right of the obelus or slash). As such, the obelus & slash are also grouping symbols in a horizontally written division statement. According to every math teaching website I have seen, in a division statement, the obelus or slash are the exact equivalent of the vinculum in the sense that they all separate the numerator from the denominator & they all mean “divided by.”
from Algebra Class. com:
https://www.algebra-class.com/dividing-monomials.html
“Remember: A division bar and fraction bar are synonymous!”
…and from Algebra Practice Problems .com:
https://www.algebrapracticeproblems.com/dividing-monomials/
“Any division of monomials can also be expressed as a fraction”
—————-
Division is a fraction. A fraction is division.
—————-
In a horizontally written multi-division statement such as 6/2/3, that could mean (6/2) divided by 3 or it could mean 6 divided by (2/3), which yields different quotients, depending on the order in which the divisions are done. The same statement written as a vertical fraction could mean six-halves over 3, or it could mean 6 over two-thirds. In a vertically written fraction, using gigantic parentheses to encompass the numerator or denominator dictates the order of divisions in that multi-division statement. Using the small solidus-vinculum (e.g. ⅔) would also be a way to indicate grouping of the numerator or denominator. The upshot is that in a multi-division statement, an extra grouping symbol is required to define the numerator vs. the denominator, whether the statement is written horizontally or vertically.
Comment by Dee — March 19, 2024 @ 1:03 pm
Addendum:
If the vinculum, via its being a grouping symbol, should be considered as part of the “Parentheses” step in PEMDAS (i.e. as if the numerator & denominator were each already encased in parentheses), then it should be considered the same way in a horizontally written division statement which uses an obelus or solidus — because those division symbols are all synonymous with one another.
Comment by Dee — March 19, 2024 @ 1:12 pm
This is where I would disagree with you, as I’ve already explained. If the obelus is present, then that is the D step in PEMDAS. If the obelus is NOT present and a grouping vinculum or solidus is used, any such term is grouped and should be considered in the P step of PEMDAS prior to any extant operational signs (+-x÷). Notice I said “considered” and not “calculated.” That is an intentional distinction. The obelus may represent the same *mathematical* function as the vinculum by itself apart from any other terms or functions, but the obelus has a different *linguistic* or *semantic* function than the vinculum, especially in the larger context of an expression.
Comment by Scott Stocking — March 19, 2024 @ 9:29 pm
Here’s an example from a math teaching website, instructing young math students on how to do monomial division…
from YouTube: “Working with Monomials”
https://www.youtube.com/watch?v=i3TcoTkztqU
At 5:03 into the video, this example is shown, using an obelus:
a^8 ÷ a^5
…which is then rewritten as the top-and-bottom fraction:
a^8 over a^5.
~ ~ ~ ~ ~ ~
and another math teaching website…
Algebra Practice Problems,com:
https://www.algebrapracticeproblems.com/dividing-monomials/
“How to divide a monomial by a monomial”
“Any division of monomials can also be expressed as a fraction:
8x^3 y^2 z ÷ 2x^2y = “
…which is then shown as a top-and-bottom fraction with 8x^3 y^2 z as the numerator & 2x^2y as the denominator — without any parentheses in the original horizontally written division statement, giving the quotient as: 4xyz.
~ ~ ~ ~ ~ ~
and from Algebra Class. com:
https://www.algebra-class.com/dividing-monomials.html
Dividing monomials“
“Quotient
A quotient is an answer to a division problem.
Let’s take a look at what happens when you raise a fraction (or a division problem) to a power. Remember: A division bar and fraction bar are synonymous!”
~ ~ ~ ~ ~ ~
from Cue Math .com:
“Dividing Monomials“
https://www.cuemath.com/algebra/dividing-monomials/
Example:
15mn ÷ 5m
See site for the steps to finding the quotient to this horizontally written division statement that uses an obelus as the division symbol, but does not use parentheses to group the numerator or denominator.
~ ~ ~ ~ ~ ~
another video from YouTube:
“Dividing Monomials“
The example is first written horizontally, without parentheses anywhere in the statement:
12a^5 ÷ 4a^3
…which is then promptly rewritten as the top-and-bottom fraction of 12a^5 over 4a^3.
~ ~ ~ ~ ~ ~
This all proves that young math students are currently being taught that when dividing monomials, a monomial needs no parentheses around it to be understood to be one term with a single value of the PRODUCT of the coefficient multiplied by the variable or variables (factor or factors).
Comment by Dee — March 21, 2024 @ 1:58 pm
If the statement 8 ÷ 2(2 + 2) was written as a top-and-bottom fraction as…
8
__________
2(2+2)
…then I think everyone in the online community would agree that the quotient is 1. That’s because everyone understands that the order of operations in a fraction is as follows: Do all of the operations indicated in the numerator, then do all of the operations indicated in the denominator, and then finally, divide the numerator by the denominator.
In this case, the numerator “8” is divided by the denominator 2(2+2). The denominator breaks down as follows:
(2=2) = 4
2(4) = 8
…which makes the fraction…
8
__
8
…which equals 1.
Done another way, the numerator of “8” can be factored out as…
2(2+2)
…which would then make the fraction…
2(2+2)
________
2(2+2)
…which still yields a quotient of 1, if the steps listed above are followed.
In the case of the fraction
2(2+2)
________
2(2+2)
…we can replace what’s inside the parentheses with the variable “x,” making the fraction…
2x
___
2x
To calculate this monomial division, the like-variable (the “x’s”) cancels out, leaving the fraction
2
__
2
…which, of course, equals 1.
The division sign (obelus) & slash (solidus) are the exact equivalent of the fraction bar (vinculum), in that all of those division symbols separate the numerator from the denominator & they all mean “divided by.” Therefore, the vertical fraction of 2x over 2x is the precise equivalent of 2x ÷ 2x (or 2x / 2x).
Conclusion: The Order of Operations as PEMDAS does not apply to fractions — in fractions, division must go LAST! And according to every math teaching website I have seen, young students are instructed that all division statements can be written as a fraction. That’s because in a horizontally written division statement, everything to the left of the division symbol is the numerator & everything to the right of the division symbol is the denominator. In a case in which there are multiple divisions in the same statement, then parentheses would be needed to differentiate the numerator from the denominator.
————
What do you think?
Comment by Dee — March 19, 2024 @ 2:57 pm
We may have to agree to disagree on this, but let me try to explain from a different direction. The word “cleave” has two distinct and opposite meanings, “to cling to” and “to separate, esp. with a knife.” The word by itself can have either one of those meanings but when used in a sentence, the context will generally indicate which meaning is intended. An obelus, a vinculum, and a solidus can carry with it division, but in context, what is the intent of each. With a fraction, we may not want to do any division at all, that is, we may not want the final result to be a decimal, especially if it’s not a finite decimal value. The obelus retains the two separate values of the dividend and divisor, but the vinculum and the solidus create a single value treated as a unit. When you put those in the context of an expression, only then can we know if there’s a fully executed division there. In my 6 ÷ 2/3 example, the 2/3 is never divided by itself. I think most of the math sites you’re referencing aren’t looking beyond the basic use of the signs in a simple expression or fraction. The 2/3 is part of the parentheses step precisely because *in context* it has to be dealt with first. Inverting undoes the grouping, then it can be calculated linearly 6 x 3 ÷ 2.
Comment by Scott Stocking — March 19, 2024 @ 7:06 pm
All I can tell you is that young math students are being instructed, right now, that the obelus, solidus & vinculum are synonymous (as evidenced by the links I posted, to Basic Algebra teaching websites). In other words, all division statements can be written horizontally with an obelus or solidus, or can be written as a vertical fraction. In fact, every math teaching website I have seen, tells students that in order to find the quotient of a monomial division, they should rewrite the division statement as a top-and-bottom fraction, with everything to the left of the obelus or solidus as the numerator & everything to the right of the obelus or solidus as the denominator — sans parentheses in the horizontally written version of the division statement.
Comment by Dee — March 21, 2024 @ 1:04 pm
I don’t doubt that, but that’s just the basics of division. What I’m talking about is the underlying and often unspoken and untaught principle of the linguistics and syntax of math. This failure to teach critical thinking when it comes to math is why we have these debates about PEMDAS, because as you’ve probably noticed, there’s a wooden allegiance to it, but nowhere have I ever seen any kind of theoretical defense for PEMDAS. I’m trying to develop the theoretical underpinnings for PEMDAS that don’t seem to exist yet.
Comment by Scott Stocking — March 21, 2024 @ 1:19 pm
The failure appears to be classroom teachers apparently continuing to misidentify the “Parentheses” step of PEMDAS as ONLY what is inside of the parentheses, rather than instructing students that the parentheses must be COMPLETELY resolved, first, before moving onto the next step:
Implied multiplication by juxtaposition to a quantity inside parentheses means you still have to resolve the remaining parentheses by multiplying those quantities.
Young math students also need to be helped to understand that implied multiplication by juxtaposition indicates the presence of a monomial, which holds a single value (the PRODUCT of the coefficient multiplied by the variable, which is a factor). To see the existence of the monomial more clearly, simply substitute a variable such as “x,” for what is inside the parentheses.
In the case of the statement 8 ÷ 2(2 + 2), the “8” can be factored out as 2(2+2), making the statement 2(2+2) ÷ 2(2 + 2). Replacing what’s inside the parentheses with the variable “x,” makes the same division statement…
2x / 2x
…or the top-and-bottom fraction of 2x over 2x, since division is a fraction.
Comment by Dee — March 21, 2024 @ 2:29 pm
I think you & I are in agreement that he use of spoken language can be very helpful in understanding the meaning of a mathematical statement. For example, I don’t believe anyone would read “2x ÷ 2x” out loud as, “Two times x, divided by two, times x,” which is how the strict PEMDAS crowd is conceptualizing the statement. Hearing the statement read aloud as, “Two x divided by two x,” helps clarify that “2x” is actually a single quantity.
The concept of multiplying the coefficient by the factor to get the product (i.e. the overall value of the monomial), before doing anything else in the mathematical statement, is made clearer when hearing spoken phrases such as, “two dozen,” or saying “two pairs”– everybody easily comprehends that two dozen eggs is a total of 24 eggs (2 times 12) & that 2 pairs of socks is a total of 4 socks (2 times 2).
Comment by Dee — March 21, 2024 @ 3:13 pm
Yes, we are definitely in agreement about the importance of the spoken word and the nonverbal clues that accompany the verbal output. Emphasis, pregnant pauses, voice pitch, etc. all factor into the communication process as do the visual cues of the letters, numbers, and symbols that reflect the written word.
Comment by Scott Stocking — March 22, 2024 @ 9:21 pm
It is my understanding that the vinculum is read as “divided by,” the same way that the obelus & slash are read as “divided by.” Can you find a link to a reputable source which says that the vinculum is “division by juxtaposition,” as opposed to meaning “divided by”?
Comment by Dee — March 18, 2024 @ 2:23 pm
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I didn’t read all of this post but I read the part where you explained that whatever is immediately outside parentheses is part of it. Thank you!
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