This post is a rebuttal to a Twitter debate with Scott Leishman. My response is too long to post on Twitter, so here it is. This is where the thread left off:

We're almost to the crux. I'm going to stop posting now, go make dinner, and come back and post to this thread and only this thread. If you have not muted me, I won't further waste your time on other side issues.

— Derek Ramsey (@ThyRamMan) August 13, 2019

This is in response to the Twitter troll post:

A simple math problem has divided the internet.

It's either 1 or 16, depending on which mathematician you believe. | @CBCKids pic.twitter.com/hACth9O4KP

— CBC (@CBC) August 2, 2019

Primary Argument

By the polynomial definition and Scott’s argument, “8”, “2”, and “(2+2)” must all be separate coefficients, not variables, that is, a coefficient (“2”) of a coefficient (“2+2”). A coefficient of a coefficient is odd phrasing.^[1] Regardless, this leads to an apparent contradiction: if “8” is a coefficient, then the expression “8/y” (i.e. “8y^-1”) is not a polynomial. If it is not a polynomial expression, then Scott’s argument fails.

You cannot evaluate “8/2(2+2)” without using division or negative exponents, but it’s okay to use division or negative exponents in the coefficient of a monomial. Thus, the whole expression must be a single monomial with coefficient “8/2(2+2)”, that is, “(8/2*(2+2))x^0”. When evaluating the coefficient from left to right, you get 16.

Consider the claim that “2(2+2)” can be represented as “2y”:

I’m not.

Take the form 2y.

2 is the coefficient of y.

After substitution, y is implied to have parentheses around it.

If y = 2+2, then 2y is 2(2+2).

2 is the coefficient of (2+2).

This isn’t difficult.

— Scott Leishman (@scottleish) August 5, 2019

This treats “(2+2)” as if it were a variable (with a coefficient), but it is not indeterminate. You can’t “invent” an indeterminate “y” to replace a determinate.^[2] It is invalid to imply a coefficient to a variable.^[3] Instead of saying “2 is the coefficient of the coefficient (2+2)”, simply say “2 multiplied by the sum of 2 and 2”, that is, 2*(2+2). That is what the notation means. It’s basic arithmetic, not polynomial algebra. 8/2*(2+2) is 16.

Or, resolve the parentheses first. Rewrite it as “8/2(4)”. If this is a polynomial expression, then the coefficient of “(4)” must be “8/2”, otherwise it isn’t polynomial form (as stated above).^[4] Thus, 8/2 is 4, so 4(4) is 16.

Secondary Arguments

The primary argument presumes that parentheses imply a “coefficient in a polynomial”. But this is inconsistent. Given “c/ab”, where c=8, a=2, and b=4, there are no parentheses, but you’d still do the same thing. Indeed, the parentheses don’t matter for Scott’s argument at all: you’d do the same thing with or without them. If the parentheses don’t matter, then how are we so certain that the original expression is a polynomial expression and not just plain arithmetic? Obviously we can’t be certain: that’s the whole point of the original Twitter post. The default, however, should be to treat basic arithmetic as basic arithmetic.

A polynomial expression consists of variables and coefficients. Examples of coefficient ‘a’ include ax², ax¹, and ax⁰. Neither 2 nor (2+2) are variables, so 2 is a coefficient of the coefficient (2+2). You typically see polynomials in the form “ax² + bx + c”. It is likely that most mathematicians have rarely, if ever, used polynomials in the form of  “ax² + bx + c(d)” where c and d are known integer constants. Moreover, if we are going to write our expressions that way, why not write them like this: “a(d)x² + b(e)x + c(f)”. Everyone would recognize that this is equivalent to “(a*d)x² + (b*e)x + (c*f)” because the parentheses don’t mean anything other than shorthand multiplication. They certainly don’t mean that, for example, a is the coefficient of d, yet Scott’s argument contradicts this.

Scott has made the claim that nearly all mathematicians would get the answer “1” to the original expression because they all recognize that it’s a “2” is polynomial coefficient. Scott posted this YouTube video. It contains the evidence that  “PVMg/RT” means “(PVMg)/(RT)”. But contained in this are two entirely different arguments, because there are no parentheses. First, the reason why implicit multiplication is done the way it is, is because of assumptions regarding the obelus. It is assumed that the obelus has lower precedence all on its own. If all that is required is an obelus that means “everything multiplied on the left over everything multiplied on the right”, then there is no reason to worry about any other rationale. Second, is this a polynomial? We can’t tell. If all it takes for implicit multiplication to be higher precedence than explicit multiplication, then the whole argument about polynomials and coefficients is a complete red-herring. All you need to know is that there is an expression “2(2+2)”. It doesn’t matter if it is a coefficient or not. Indeed, half a Twitter thread about polynomials, monomials, binomials, a 4-page paper proof, and coefficients is completely irrelevant. That’s Twitter for you, I guess.

But it gets worse. Consider the expressions “x+1  /  x-1” and “x  +  1/x  – 1”. Are they the same or different? If you said different, it is because you consciously or unconsciously applied spatial grouping on terms based off the distance between them, rather than strictly apply order of operations. Many of these Twitter math trap problems are formulated with spatial traps for exactly this reason. The people who calculate “1” based off spatial rules are not following (1) the rules of polynomial coefficients, (2) implicit multiplication, or (3) the placement of the obelus. They are deriving their answer from a fourth standard. It’s a complete mess.

There are four different ways to arrive at the answer “1”, but there is only one way to arrive at the answer “16”: PEMDAS. It’s not required to use this standard, but the method is completely deterministic and applies equally in all situations. There is no ambiguity if you use this method. This is why all the major programming languages and North American teachers and textbooks (up to and including college) prefer PEMDAS. 80% of the market share of graphing calculators, as well as Google, Wolfram, and the calculator on my Android phone, all use PEMDAS. When polled 40% of people choose PEMDAS. 60% choose the answer derived from one or more of the other four methods: it is not clear which methods are most popular and many cannot explicitly articulate which method they use. By contrast, people who use PEMDAS are able to explain their answer with ease.

Built into Scott’s argument is the notion that a(b) cannot ever mean just a×b, even though it obviously means multiplication.^[5] In other words, implicit multiplication is a different kind of multiplication from explicit multiplication. But that’s absurd. Multiplication is multiplication, whether you notate it explicitly or implicitly.

This notation originated with polynomial mathematics in algebra. But it is such a useful shorthand that it was adopted in other contexts, including pre-algebra levels. Allowing shorthand notations to be incorporated into different contexts is useful.^[5] There is no reason why pre-algebra cannot use this notation. Mathematicians have long used implicit multiplication outside the context of polynomials anyway, so why not pre-algebra? A turf war of authorities is unnecessary.

^[1] This terminology is not universal. It is unusual to find examples of mathematicians using a constant coefficient of a constant coefficient. Some definitions, like this one, explicitly exclude it. Other definitions, like this one, explicitly exclude this claim by Scott.

^[2] If you understand “y” as a shorthand for symbolic replacement, then it’s really saying nothing. “a+b” where a=1 and b=1 is still “1+1”. They are equivalent and no additional semantic meaning can be inferred from the act of replacing constants with arbitrary symbols.

^[3] The purpose of this implication is to support the claim that “2” is the coefficient of “(2+2)”. By writing the latter in variable form, it gives the false appearance that “2” is a coefficient of a variable rather than another coefficient. It is not.

^[4] It isn’t that you can’t evaluate implicit multiplication before explicit division and multiplication, but that it makes no sense to do so. It leads to the absurdities stated. Of course you can do what you want with conventions, but that doesn’t mean it’s precise or clear.

^[5] It originated because mathematicians wanted to save print. Less to print saved money on printing. It also meant less physical writing.