Dear Uncle Colin,

I’ve been struggling to get my head around what happens if you chop infinity in two? Is half of infinity still infinity? Help!

How Infinity Lies Beyond Every Reasonable Theory

Hi, HILBERT ((Didn’t we have a HILBERT before? I can’t remember, and can’t be bothered to check.))

The short answer is yes: halving infinity gives you infinity. (Once you get to infinity, you’re kind of stuck there: even multiplying by 0 or dividing by infinity give you undefined behaviour). However, this is fairly loose talk and I should tighten it up a wee bit.

Technically speaking, infinity isn’t part of the number system, and doing sums with it is strictly forbidden. If you try, you get answers that don’t make much sense (e.g.: $\infty + \infty = \infty$, which suggests $\infty = 0$ if you take away $\infty$ from both sides). That’s clearly not the case, so the normal rules can’t apply.

What we can do maths on is the size (cardinality) of infinite sets. The size of the set of whole numbers is infinite. Obviously. They go on forever, without an upper bound. The size of the set of even numbers is also infinite – obviously – but intuitively, there should only be half as many even numbers as whole numbers.

Your intuition is wrong.

There the same number of even numbers as whole numbers. One way to see this is to imagine a hotel with infinitely many rooms, numbered starting at 1. Suppose all of the rooms are full. Then move the occupants of room 1 into room 2, the occupants of room 2 into room 4, room 3 into room 6, and generally whoever’s in room $n$ into room $2n$.

Now only the even numbered rooms are occupied, but the hotel still has as many guests as before, since nobody’s left the hotel. This means there are as many even numbers as there are whole numbers!

* This idea of the Hilbert Hotel is one of many discussed in Cracking Maths…