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?