# Algebra is weird

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

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{2^4} + \dots$

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

$x = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{2^4} + \dots$

So,

$2x = 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{2^4} + \dots$

$2x= 2 + x$

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:

$1 + 2 + 4 + 8 + 2^4 + \dots$ where this time the dots mean we keep doubling the last term and adding it on.  Taking the same algebraic approach, we set

$y = 1 + 2 + 4 + 8 + 2^4 + \dots$ and double each side,

$2y = 2 + 4 + 8 + 2^4 + \dots$

$2y = y-1$

We solve this and find that $y = -1$

Algebra is weird.