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2, 4, 6, 8, What Does Not Associate?

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Last month, I wrote about group theory via monkeys, and it got me thinking about the associative property. A mathematical group consists of a collection of stuff: integers, or rational numbers, or even something more abstract; and an operation that combines any two elements of your stuff into another element of stuff.

One of the rules the stuff and operation have to obey in order to be a group is associativity: it can’t matter what order you combine your stuff in. The prototypical example of a group is the integers with the operation of addition. In that case, the associative law says that if you are adding 1+2+3, it doesn’t matter whether you add 1 and 2 first, or 2 and 3 first.

Addition and subtraction are similar operations. In school, we generally learn addition first and then subtraction. Later, when we learn about negative numbers, we learn that subtraction is just addition of a negative number. So if addition follows the associative law, then it seems like subtraction should as well.

But it doesn’t! In the expression 5-4-3, we get different answers depending on how we group it. (5-4)-3=-2, and 5-(4-3)=4. Likewise, subtraction doesn’t associate with addition. (5-2)+4=7, while 5-(2+4)=-1.

The problem goes away if we replace subtraction signs with addition of negative numbers. In other words, if we see the expression 5-4-3, we convert it to 5+(-4)+(-3). Then the answer is unambiguously -2 because by replacing the subtraction signs by addition of negative numbers, we’ve made a choice about what order the subtraction comes in.

We can peel another layer off this onion: what is a negative number? That’s a philosophical question, but one way of thinking about the minus sign at the beginning of a negative number is as the symbol for unary subtraction. A unary operator is an operator that takes only one input. We’re introduced to subtraction as a binary operation, for example, 5-3, but when we see a negative number, we’re omitting and understood 0 from the beginning of the expression. The number -2 is shorthand for 0-2.

I started thinking about this when my Internet pal and one of the editors of the excellent math(s) blog The Aperiodical tweeted this:

Multiplication and division share a similar relationship to addition and subtraction. (In the form of an SAT analogy, may they rest in peace, addition:subtraction::multiplication:division.) Multiplication is the “default” operation, and division is an inverse operation. And once again, multiplication is associative, but division is not. (12÷6)÷2=1, while 12÷(6÷2)=4. Division doesn’t associate with multiplication either, and this is a source of confusion in those awful math problems that sometimes make the rounds on Facebook. 6÷2(1+2) is ambiguous because of the non-associativity of division with multiplication and inconsistency in how we interpret the order of operations, sometimes called PEMDAS, BEMDAS, BODMAS, or BIDMAS.

Unary subtraction is one reason similar problems that only use addition and subtraction are not ambiguous. Because we’re so used to using unary subtraction, we automatically do the problem 5-4-3 as 5+(-4)+(-3), which is the same as doing it strictly from left to right. (Not being an expert at the history of mathematical notation, I do note that this might be a chicken-and-egg situation: did unary subtraction develop because we are so committed to doing addition/subtraction problems from left to right? Or does unary subtraction, and our comfort with negative numbers, make that the natural way to handle addition/subtraction problems?)

Would we have Facebook flame wars over the order of operations if we had unary division? As Perfect tweeted, it wouldn’t be hard to notate it: unary subtraction uses the additive identity, 0, as the understood first term. Analogously, unary division would use the multiplicative identity, 1, as the understood first term. ÷2 would mean 1÷2, or ½, just like -2 means 0-2.

If we used unary division the same way we automatically use unary subtraction in an expression like 5-4-3, then 6÷2(1+2) could mean 6×½×(1+2), and 9 would be the unambiguous answer. Unary division would make the choice for us of what order to do the division and multiplication in. It would reinforce the idea that multiplication and division are of equal precedence and should be performed from left to right.

But another source of ambiguity in this problem is the missing times sign: 2(1+2) means 2×(1+2). When written without the times sign, it’s sometimes called “multiplication by juxtaposition.” Some people interpret multiplication by juxtaposition as taking precedence over multiplication that uses the times sign. Because there’s no such thing as addition by juxtaposition, it’s not clear how we would treat multiplication by juxtaposition in a world that also had unary division. (Cue dramatic movie trailer music, with voiceover: “In a world with unary division…”)

The solution to the infamous Facebook problem is not 1 or 9, it’s to put more parentheses into the expression to keep it from being ambiguous in the first place! Mathematical notation is not a God-given sacred scroll that we must interpret correctly, it’s something we made up to make communication easier. If we’re in danger of being misunderstood, we should just add a couple of symbols to clear it up.

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.

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  1. 1. M Tucker 6:45 pm 06/23/2014

    “If we’re in danger of being misunderstood, we should just add a couple of symbols to clear it up.”


    I do like the idea of unary division because it seems to lend itself to defining division as multiplying by the reciprocal. To me it seems to go naturally along with defining subtraction as addition of negative numbers. BUT, our multiplicity of symbols does seem to lead to confusion unless we have a universally accepted order of operation. In that wiki link we find 2x/2x = x^2. That kind of stuff can influence a kid who feels unsure at math from attempting higher math classes.

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  2. 2. SJCrum 7:36 pm 06/23/2014

    As for the equation of 6/2(1+2), what they are misunderstanding is SIMPLY that it is a rock-solid, dead-on-right rule of real mathematics that anyone doing the math from LEFT-to-right, you have to, DUH-UH, start at the left. The first part then is to ALWAYS divide the 6 by 2, which equals 3, and THEN you take the very evil 3 and them multiply that by the, already fully known concept of adding 1 + 2 inside the brackets to get 3. The real math answer is 1.

    By the way, a reallllly cool math thing is for anyone in the entire world to describe how imaginary numbers actually work? The best math experts in the world know they work, but before just a few days ago, not a one knew why.

    An example of imaginary numbers is we all know that -1 x -1 = +1. But, in imaginary numbers, -1x-1=-1. Cool, huh? And the totally silly imaginary varmint thinking actually works. Hee, hee, huh?

    The silly real math is simply that it still uses the idea of two negatives multiplied by each other ends up having a positive result. The issue then is where are the two negatives. In -1x-1 the first negative is that left-most -1. But, the second -1 isn’t the second negative. Instead, the entire negatively ridiculous thinking that it could actually have a positive result RIGHT THERE is the other negative. Cool, huh? It’s as naughty as duck spit, but the furry tushed varmint thinking only works because there are FACTUALLY still two negatives multiplied by each other.

    The point is that an equation of (-1 squared)x6/2+(36/3)=15 can be true by actually changing the sign in front of the 15 to a minus sign. By doing that at the end, it all gets changed back into the real world again.

    There are actual mathematics calculations that have great results by using imaginary numbers, and where they cannot get the answers without doing so.

    Anyway, that is the real math involved.

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  3. 3. Evelyn Lamb in reply to Evelyn Lamb 3:26 am 06/24/2014

    I’m not quite sure what you’re getting at with the imaginary numbers here. We never say -1x-1=-1, we use a different symbol, i, to define the square root of -1. The number i is defined as “that which when squared is -1.”
    I wrote a little bit about complex numbers last year:

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  4. 4. Zeitgeist64 3:28 am 06/24/2014

    ” The solution to the infamous Facebook problem is not 1 or 9, it’s to put more parentheses into the expression to keep it from being ambiguous in the first place! ”

    Ok , so here we go !

    6/ 2(1+ 2)= 6/[{ 2(1)} ] +[ 2(2)] = 7
    Can it be right answer then ?

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  5. 5. roguewhistler 3:26 pm 06/25/2014

    I’m not sure of the utility of unary division since we already have the notation ^-1 for reciprocals. If you want a new notation that is consistent with both this and the negative notation, then consider the following: -5 is (-1)5, so we place the minus sign where me might see multiplication by juxtaposition. We could likewise use a minus sign as a superscript to mimic the exponent of -1: one-fifth, or 1/5 could be written as 5^-1 or as simply 5^- (try it using a superscript instead of ^).

    Regarding “what is a negative number”, I find it worthwhile to first distinguish cardinal numbers and signed numbers, aka scalars and vectors. The + or – is an indication of direction. Confusion stems not from negative numbers, but from the mistaken assumption that all numbers are quantities. In both physics and finance we need both direction and magnitude.

    Regarding associativity, it would have been more interesting for you to quickly write off subtraction and division as nonissues due to negatives and reciprocals, and instead push further to exponentiation. With respect to the set of natural numbers, we can regard addition as fast counting, multiplication as fast addition, and exponentiation as fast multiplication. One might ask why we do not teach a new symbol for fast exponentiation. We do, in fact have one, namely an upward arrow, but exponentiation is the first binary operation in the above sequence that is not associative: 3^(3^3) is not equal to (3^3)^3. This implies that the next level requires greater precision of undestanding than the previous operations.

    Finally, on the topic of order of operations, I explain to students that it is merely a convention, not a “fact”, that is meant to reduce the parentheses we require. Further objections are met with an exercise requiring 12 sets of nested parentheses to pursuade them, along with a discussion of the associative and distributive properties to emphasize that we can rid ourselves of all of the parentheses used to group within a sequence of like operations, but only half of the parentheses in a sequence of mixed operations. Half is better than none, so we take it, and the “order” must be agreed upon, hence the convention we call “order of operations”. No silly acronyms or mnemonics are necessary; just do more complicated levels of operations before simpler ones (^ before * before +).

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