There is a tear in the fabric of maths. I’m not talking about Gödel’s Incompleteness Theorem, however excellent a tear that is; I’m talking about something much more fundamental, some arithmetic that looks like it ought to be basic, but which completely breaks down when you look at it.

The question is, what is $0^0$? Zero raised to the zeroth power. Somewhat bizarrely, there are three distinct camps with different ideas of what it has to be. I’ll list the arguments below, so you can see what we’re dealing with.

Camp Zero

There’s a simple argument for zero: “if you multiply zero by itself any number of times, you get zero.”

Camp One

There are several arguments for one. There’s the dogmatic “anything to the power of zero is one, it’s an empty product”. There’s also the “well, $x^x$ approaches one as $x$ approaches zero” argument, and the quite reasonable “If you differentiate $x^1$, you get $x^0$, and the derivative of $x^1$ is one everywhere.

Camp Undefined

To quote the mathematician in the joke: “Come, come, gentlemen – it can’t be both.” The function $f(x,y) = x^y$ does not have a uniquely specified limit as $x$ and $y$ both approach 0 – if you come in from one angle ($x=0$), you get 0, and if you come in from another ($y=0$), you get 1.

You can also argue that since $0^{-1}$ is definitely undefined – that’s an “infinite” sort of undefined rather than “indeterminate” sort of undefined – and $0^0$ must equal $0^{-1} \times 0$, you get something that just doesn’t have a value. In a sea of hand-wavy arguments, this is the hand-waviest, but it’s worth mentioning.

My verdict

Context is significantly underrated in maths. While the answer to $6 \div 2(1+2)$ is always “write the bloody thing properly”, there are times when someone will write $1/2x$ and mean $\frac{1}{2x}$; other times when they will mean $\frac{1}{2}x$. The thing is, given a sense of what they’re working out, you can usually figure out which one they mean. (When someone write $6 \div 2(1+2)$, what they mean is “let’s have a fight for engagement purposes.”)

So: for the sake of rigour, we need $0^0$ to be undefined. It’s a discontinuity in the surface of the function $f(x,y) = x^y$, which gets even weirder when $x$ is negative. (For example, $(-1)^{\frac{1}{3}}$ has three reasonable values, two of which are perfectly justifiable in different contexts. Inverting $f(x) = x^3$ makes it clear that $(-1)^{\frac{1}{3}} = -1$, but the principle value of the cube root function gives $(-1)^{\frac{1}{3}} = \frac{-1 + i\sqrt{3}}{2}$. Don’t let’s get started on negative irrational powers.)

In short, the $y$-axis is a boundary of oddness.

Meanwhile, for the sake of convenience, in most of the mathematical applications you’re likely to come across, it’s a lot easier to say that $0^0=1$. I’ve been reading some information theory. Asserting that $0^0 = 1$ makes the whole field possible.

And there’s the rub: it’s absolutely ok to assert for your purposes that $0^0$ is one – or zero, or undefined, or a lemon-flavoured space-hopper if that’s what your application requires. You can do that if it helps you.

But you do have to state it. That’s because $0^0$, rigorously speaking, is undefined.