On $0 \div 0$

A question that frequently comes up in the insalubrious sort of place a mathematician might hang around is, what is that value of $0^0$. We generally sigh and answer that the same way every time.

It was nice, then, to see someone ask a more fundamental one: what is $0 \div 0$?

The short answer is, it's not defined, even though $0 \div a = 0$ pretty much everywhere. But why?

There are probably dozens of explanations for this. My favourite is to look at what division means.

$a \div b$ asks the question "what do you multiply by $b$ to get $a$?" So, $6 \div 2 = 3$ because $3 \times 2 = 6$.

In particular, $0 \div 0$ asks "what do you multiply by $0$ to get $0$?" The answer to that is "anything at all" - which is a Problem.

Basic arithmetic relies on operations giving unique answers to questions - if $2+2$ was both 4 and 5, we'd be in a terrible state. $0 \div 0$ can't possibly be 0 and 7 and $\pi$ and Graham's number and every other value you can multiply by 0 to get 0, for exactly the same reason.

In short, it's undefined because the answer could be anything, and that's not allowed.


Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.


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