The Interesting Number Theorem is an interesting theorem from the field of number theory, which is the part of mathematics that deals with the theory of numbers. The Theorem works through a process of reductio ad absurdum. This is a Latin term for imagining that something is true in order to prove that it isn’t true. Mathematicians are fond of imagining things that are not true. For instance, when they needed to pretend that there is a square root of -1 they imagined the number called i. If you multiply i by itself the product is -1, even though i doesn’t actually exist. In order to make this clear, mathematicians refer to multiples of i as imaginary numbers. This is to distinguish them from non-imaginary numbers which also don’t exist in the physical world, but which can be represented in the physical world. For this representation, mathematicians often refer to something which is known as Zermelo–Fraenkel Set Theory. This is an interesting theory that uses sets as the basis of all mathematics. A set is a collection of things. What kind of things they are doesn’t really matter. They can be imaginary things or real things, but either way we are going to imagine that we have a set of them, and the number of things in the set is equivalent to the number we might use to represent them. So sets are a kind of a representation of numbers, which are a representational construct, used to count things in the real world, but which are also useful for counting imaginary things. If we want to represent zero, for instance, we imagine an empty set, or nothingness. It’s hard to imagine a set that contains i things, but that doesn’t matter because we can construct a set with one member, then imagine that set being subtracted from nothing, and then imagine finding the square root of something that is less than nothing.
So having established some interesting facts about real and imaginary numbers and ways of representing such numbers using Zermelo–Fraenkel Set Theory, we can move on to the next question: what is the Interesting Number Theorem? Essentially, the question here is whether some numbers are more interesting than other numbers. The answer to this is fairly obvious to a mathematician. It’s self-evident that 341 is more interesting than 66, or that 4095 is more interesting than 2491, for instance. And 253 is more interesting than 611. A quirky little number such as 42 might be seen as more interesting than 97, though that is obviously a slightly more controversial suggestion. 1722 is extremely interesting, but there isn’t very much that is interesting about 171 at all: that certainly seems like a pretty boring number and many mathematicians would concur.
So the next question that mathematicians like to imagine someone asking is whether there are any numbers that are simply uninteresting. If there are then we could call such numbers boring. There would then be a set of boring numbers and a finite or infinite number that could be associated with that set depending on whether the set of boring numbers was finite or infinite. To a mathematician, it’s obvious that the next question has to be, ‘What is the smallest boring number?’ At this point there is an interesting problem. Zero is a number. As we’ve seen, in order to represent zero we have to imagine nothingness. Imagine an empty space, in which there are no objects. Is that interesting? Imagine a snowy day but with no snowflakes. Imagine a desert without any sand. Imagine no raindrops, no sky, no sounds, nothing whatsoever. Imagine a long motorway stretching out into the distance, with no cars, no verge, no bridges, no horizon. Imagine a beach with no pebbles. Imagine the infinite vastness of space, with no stars, no planets, no comets, no particles of any sort. This is the empty set, the null set, the void, an imaginary representation of nothingness without end.