The Math Meme That Would Not Die

Some version of this question keeps getting asked on the internet. What is \(8\div2(2+2)\)?

Some background

The strength of mathematical notation is at the intersection of clarity and simplicity. We could be completely clear, which would leave us writing details we don’t need and make it hard to read. We could oversimplify, and lose clarity. In general, we should begin with full clarity and then simplify only to the point that clarity is not lost.

For instance, let us say we want to write the formula for a cubic function that has no squared term. Here’s one example, with full clarity: \[(3\cdot (x^3)) + (0\cdot (x^2)) – (4\cdot (x^1)) – (5\cdot (x^0))\]

There is no mistaking what order operations occur in here, nor the structure of the polynomial as having one term per exponent, starting from 3 and going down to 0. This reinforces that all constants are polynomials of degree 0. But it contains a lot of information that isn’t needed and can be implied if we come to some basic agreements.

First of all, it’s typical to agree that exponents should be evaluated before multiplication (and division), and that multiplication (and division) should be evaluated before addition (and subtraction). Once we agree to that, the so-called “Order of Operations,” we can get rid of all those parentheses:  \[3\cdot x^3 + 0\cdot x^2 – 4\cdot x^1 – 5\cdot x^0\]

This admittedly lacks some amount of clarity, in that it requires us to learn an arbitrary rule about what order to evaluate things in, but it makes things much faster to write, and it’s not that unclear.

We can also use a few rules to simplify individual terms here:

  1. Anything times 0 is 0.
  2. Anything times 1 is itself.
  3. The first power of any number is itself.
  4. The zeroth power of any number is 1.

The last rule glosses over a debate about the value of \(0^0\), but we’ll just assume that \(0^0=1\) for the purposes at hand and move on.

These four rules let us get rid of the second term entirely and to simplify the third and fourth terms, leaving us with:  \[3\cdot x^3 – 4\cdot x – 5\]

Finally, at some point mathematicians got tired of writing multiplication symbols and decided they weren’t needed: When two distinct things appear next to each other with nothing between them, that implies multiplication.

This is one of the more confusing rules in mathematical notation. It doesn’t apply when:

  1. Two integers are next to each other. \(53\) is not a simplification of \(5\cdot 3\) but rather of the much more complicated \(5\cdot 10^1 + 3\cdot 10^0\).
  2. An integer is next to a fraction. \(1 \frac{1}{2}\) is \(1 + \frac{1}{2}\), not \(1\cdot \frac{1}{2}\).
  3. A letter represents a function name instead of a variable. \(f(x)\) is generally “the function f on the variable x,” not \(f\cdot x\), although there’s nothing preventing us from using f as a variable.

In summary, two things next to each other should be multiplied except when they shouldn’t. That’s a lousy rule, but it’s a widespread one.

Applying it to our examples gives us \[3x^3 – 4x – 5,\] which is full of assumptions that mathematicians take for granted but students struggle to learn. To a mathematician, though, it’s perfectly clear.

Which brings us back to …

So what’s the problem with \(8\div2(2+2)\)?

This version has a few problems.

The first is with the implicit multiplication (that’s what it’s called when we don’t put the multiplication sign in). Some people just get confused with “PEMDAS” and think that we have to do everything to get rid of the parentheses first. This leads to \(8\div 2(2+2) = 8\div 2(4) = 8\div 8 = 1\).

A related problem is with implicit multiplication. What is \(8/2x\)? Is it \((8/2)x = 4x\) or is it \(8/(2x)=4/x\)? Among mathematicians, there’s no universally accepted answer to this. \(8/2x\) is genuinely unclear. We can apply the same reasoning to \(8/2(4)\).

Finally, there’s the obelus sign (\(\div\)). I’ve seen some people insist that it means that you evaluate everything to its left and everything to its right before you evaluate the division. I think the confusion here is that the obelus is rarely seen beyond middle school: It’s simply not a sign that serious mathematicians typically use. There’s nothing inherently wrong with it, but it comes from an era when proper typesetting was more difficult.

The ideal ways of writing the expression in question leave no question of clarity: \(\frac{8}{2}(2+2) = 16\) and \(\frac{8}{2(2+2)} = 1\). No questions there.

Technically, by the most standard conventions of mathematics, implicit multiplication involving constants should be treated the same as explicit multiplication, and the obelus is just plain old division, so \(8\div 2(2 + 2) = 8 / 2 \cdot (2 + 2) = 8 / 2 \cdot 4 = 4 \cdot 4 = 16\).

But that’s “technically.” In reality, it’s just plain sloppy. It involves a simplification that gets in the way of clarity, and therefore should be avoided.

In practice, in a mathematician’s handwritten notes, were they to write something like \(8\div 2(2 +2)\), you would need the surrounding text or thoughts to determine what they meant. It’s possible that it might mean 16, and it’s possible that it might mean 1. Just as, for instance, \(-1^2 = -1\), but a mathematician might write \(-1^2\) in their personal notes to mean \((-1)^2\), which is equal to 1.

So what’s the answer?

The answer is fairly simple: Don’t write it that way.

 

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