Division by zero

A regular topic that crops up in every maths class I’ve taken, ever, is division by zero, and why it does/doesn’t work. This pattern is repeated everywhere: Twitter, forums, graffiti in toilets, you name it.

Here, therefore, is a (hopefully) succinct and understandable explanation of why division by zero is impossible, and what would happen if it wasn’t, as collated from explanations by maths teachers and (hopefully) sound reasoning.

What is division anyway? Proof by $0 \times k = 0$ , and why the cake is a lie

The cake is a lie

A common problem with maths teaching in schools is that it’s taught as “sharing”—for example, dividing cake amongst people at a party. For example, if you divide two cakes by six guests, because $2 \div 6 = \frac{1}{3}$ each guest gets one third of a cake.

The trouble is that this analogy quickly breaks down when you divide by a fraction. How do you share one cake between half a person? It’s impossible.

A far better way of thinking of division is as the inverse function of multiplication. When we write $7 \times 8$ , we mean “eight lots of seven.” Division is the opposite: if we say $56 \div 8$ , we mean “how many lots of eight do we need to get 56?”

We can generalise this (write it down as rules for all numbers) like this: if $x \times y = z$ , $z \div y = x$ .

Now, returning to our original problem, dividing by zero, let’s assume for a moment that we can divide by zero. $x$ and $y$ can be any numbers, so let’s let $z = 1$ and $y = 0$ . If we divide one by the other, we get $\frac{z}{y} = \frac{1}{0} = a$ where $a$ is our answer.

The trouble is, this makes no sense at all if you follow on in a logical progression:

$1 \div 0 = a$
$1 = a \times 0$
$0 \times k = 0$
$\therefore 1 = 0$

What we’ve just proved is that $1=0$ if you allow division by zero. The same applies for any number: you could prove that $4642 = 0$ , $34 + \frac{\pi}{2^e} = 0$ , anything.

The infinity fallacy

A common, but completely incorrect, statement that you’ll find on the Internet is this: $k \div 0 = \infty$ . This is quite easy to disprove, and there are multiple ways to do it.

Using $k \times 0 = 0$

$1 \times 0 = 0$
$2 \times 0 = 0$
$1,000,000 \times 0 = 0$
$\infty \times 0 = 0$
$\therefore k \div 0 \neq \infty$

Using the graph of $\frac{1}{x}$

Here is a graph.

This is a graph of the function:

$x \in \mathbb{R}$

$f(x) = \left\{ \begin{array}{ll} \frac{1}{2}x + 1 & \mbox{if } x < 6 \\ -2x + 21 & \mbox{if } x > 6 \end{array} \right.$

Unlike a continuous function such as $f(x) = x^2$ , this function is discontinuous. If we follow the line, and approach $x=6$ from the left, $y \rightarrow 4$ . However, if we approach it from the right, $y \rightarrow 9$ .

This means that $f(x)$ is undefined when $x=6$ . $f(6) \neq 4$ , and $f(6) \neq 9$ . $f(6)$ has no value whatsoever, not even zero.

Now, let’s take a look at the graph of $f(x) = \frac{1}{x}$ .

This function, like the one above, has a region for which it is undefined. If we approach the y-axis ( $x=0$ ) from the right, $\lim_{x \to 0^+} {1 \over x} = +\infty$ . However, if we approach it from the left, $\lim_{x \to 0^-} {1 \over x} = -\infty$ . Therefore, the function is undefined for $x=0$ .

Zero and nothing: $k \div 0 \neq 0$

Upon being told that $1 \div 0 = (nothing)$ , a common assumption is that $1 \div 0 = 0$ . The flawed cake analogy presented above supports this: if I share six cakes between no people, no-one gets any cake.

However, this is also wrong. You can show this with the graph for $y = \frac{1}{x}$ : as $x$ approaches 0, $y$ asymptotically approaches $\infty$ or $-\infty$ . It never touches the axis, so $1 \div 0 \neq 0$ .

We can also prove this by distinguishing between the number 0 and the abstract concept of “nothing.” When we use the number 0, we really mean “no units.” For example, no litres of water, no nuts, no seconds. $5 \times 0$ means “no lots of five.”

“Nothing,” on the other hand, has no units: it means precisely that, nothing. Dividing by zero gives you nothing, because it is undefined; it literally doesn’t mean anything in mathematical terms. It’s gibberish. If I were to make up a mathematical operation, called the blankwart and represented by the $(\ddot\smile)$ symbol, that would also be gibberish since I haven’t defined it as anything.

In short, writing $1 \div 0$ is about as useful as writing $\pi (\ddot\smile) (phartiphukborlz)^\phi$ . It literally means absolutely nothing.

It’s undefined for a reason

I’ve already covered one reason why division by zero is undefined, but here’s another old chestnut in proof that $2 = 1$ .

Let $x=1$ , $y=1$

$x=y$ (multiply by $x$ )
$x^2=xy$ (subtract $y^2$ )
$x^2 - y^2 = xy - y^2$ (factorise)
$(x+y) (x-y) = y (x-y)$ (divide through by $(x-y)$ )
$x+y = y$ (substitute in our values)
$2 = 1$ Q.E.D. (!)

What we’ve just proved is that two is equal to one, which is clearly wrong. The problem lies in step four, where we divide through by $(x-y)$ ; if you were paying attention, you’d realise that $x-y = 0$ , so we’ve screwed up by dividing everything by zero.