Algebra is weird

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:

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.