Fun math facts

\((x+1)(x-1) = (x^2) − 1\)

And in general,

\((a+b)(a-b) = (a^2) − (b^2)\)

Polynomials are neat

Any polynomial of odd degree with real coefficients must have at least one real root and an even number of complex roots. (That number can be zero.)

A cool observation about powers of 2

\(2^0 + 2^1 = 2^2 -1\)

\(2^0 + 2^1 + 2^2 = 2^3 - 1\)

In general, \(2^0 + 2^1 + \dots + 2^{n-1} = 2^n -1\). (Since any \(2^n-1\) is written as \(111\dots111\) in binary, this is basically the lite version of how \(\dots111_{2}=-1\) in the 2-adic metric.)

But wait, there’s more!

It’s well known that \(2^{-1} + 2^{-2} + 2^{-3} + \dots = 1\) — in fact you can represent this in binary as \(0.\overline{111}\dots_{2} = 1\) (in much the same way that \(0.\overline{999}\dots_{10}=1\)).

But this means that:

\(2^{n-1} + 2^{n-2} + \dots + 2^0 + 2^{-1} + \dots = 2^n\)

Which is rather satisfying, is it not?