Sunday Morning Greek Blog

(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.

1. I eat fish only on Friday.
2. I eat only fish on Friday.
3. 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.

1. kalos logos [beautiful word]
2. ho kalos logos OR logos ho kalos [the beautiful word]
3. 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:

1. 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)
2. (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.)
3. 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.