Algebra is weird. This is a very short post in which I’ll show that algebra is a little odd.

Let’s start with a sum:

Where the dots mean that we keep on adding half the amount
we just added forever. We take an algebraic approach to this sum, set

So,

We solve this to see that x = 2.

This gave rise to a terrible joke. An infinite number of mathematicians
walk into a bar. The first orders a pint
of beer, the second a half pint, the third a quarter of a pint and so on. The barman sighs and just pours out two pints
telling them to share.

I don’t think x = 2 is a surprising result, its nice, but not surprising. Let’s try another:

where this time the dots
mean we keep doubling the last term and adding it on. Taking the same algebraic approach, we set