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
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
and double each side,
We solve this and find that .
Algebra is weird.