A popular joke among mathematicians goes as follows: Infinitely many mathematicians walk into a bar, and the first one orders a pint of beer; the second orders half a pint, and the next orders a quarter of a pint, and the next one orders an eighth of a pint. The barkeeper says ‘I’ll stop you there’, pours two pints and puts them on the bar, saying ‘Sort yourselves out.’

Mathematically, this joke is very funny – but sadly, outside of mathematics, that doesn’t always correspond with jokes being actually funny. My suspicion would be that in order to get this joke, there’s a bit of mathematical information you’ll need first.

The series of drinks orders the mathematicians make might seem strange – certainly, in the UK, it’s normal to order a pint of beer, and a half-pint is also a pretty standard measure. Some places which serve particularly strong ales will also offer a ¼-pint option, but there aren’t many places you can get away with ordering an eighth of a pint, and assuming that the pattern was going to continue in the obvious way, none of the other drinks orders would be standard either.

The pattern goes:

1 1/2 1/4 1/8 1/16 1/32 1/64 1/128 1/256 …

Each time, the number on the bottom of the fraction doubles, meaning the amount of beer being ordered halves. We can define the sequence as 1/2ⁿ, for values of n ranging from 1 to infinity.

Let’s assume, for the sake of argument, that it’s possible to pour each of these measures, and that you can do so with a reasonable degree of accuracy – this might be difficult, since the tenth mathematician’s order is 1/512 of a pint, which is just under 1 millilitre. From experience, measuring quantities this small is quite difficult, especially with liquids like beer, where more than 1 millilitre of it will stick to the side of the glass while you’re pouring.

But this is maths, and we can conduct mathematical experiments (and jokes) in a perfect hypothetical world, where properties like surface tension and fluid adhesion don’t factor in. So, we have our measures of beer, which carry on forever in increasingly tiny glasses – each mathematician still orders a drink, even if it is a very small one for all but the first few – and the barkeeper seems to think they’ll be satisfied with just two pints of beer. Surely not?

The first mathematician’s drink accounts for one of the pints, so the second pint glass is where all the action takes place. Starting from an empty pint glass, we can pour in the half-pint, leaving exactly half a pint of space. Then the next drink is a quarter pint, filling half the remaining space and leaving a quarter pint gap – and so on.

As each drink is added to the glass, it takes up precisely half the space remaining in the glass – and this will always continue, no matter how many drinks we add. So the infinite series of drinks ordered, lined up along an infinite bar stretching along into the distance, can all fit into this one glass.

I wasn’t kidding when I said this was something I know from experience – I’ve performed this trick on stage, at comedy shows and on TV, in order to demonstrate this idea. Seeing it physically happen in front of you makes it much more comprehensible that you can have an infinite number of drinks which add up to a finite number. The trick is that the numbers are getting smaller, and doing so quickly enough that they get closer and closer to a finite number, without actually reaching it.

There are many examples of series like this, which are said to converge – that is, the sequence of sums of successive numbers of terms get closer and closer to a particular finite value. Convergent series include, for example, the series of sums of any sequence where the number on the bottom is an increasing power of the same fixed value. In general, the series of sums of numbers of the form 1/kⁿ, where k is a whole number, converges to k/(k-1).

If you take a sequence like the triangular numbers (numbers of dots that can be arranged in a triangle, like pins in a bowling alley, including 1, 3, 6, 10, 15, and so on) and consider the series of fractions 1, 1/3, 1/6, 1/10, 1/15 …, these sums also converge to 2. And the series of sums of fractions of 1 divided by a square number, 1/n²: 1, 1/4, 1/9, 1/16, 1/25 and so on, converges to π²/6 (of course). But even though these series keep getting closer and closer to this value, they’ll never actually reach it – there’s always a little gap left in the top of the pint glass.

In fact, the only way the second pint glass will actually be 100% full is if we do continue pouring drinks in to infinity – mathematically, we say that the limit of the series, adding the numbers together to infinity, is 2. This explains the additional punchline you can give to the original joke, in which that the barkeeper adds ‘The problem with you mathematicians is, you need to know your limits!’