## Fast Exponentiation, revisited

12/11/2019

Quite a while ago, I presented a fast exponentiation algorithm that uses the binary decomposition of the exponent $n$ to perform $O(\log_2 n)$ products to compute $x^n$.

While discussing this algorithm in class, a student asked a very interesting question: what’s special about base 2? Couldn’t we use another base? Well, yes, yes we can.

## Walk Count like an Egyptian (Part I)

11/03/2014

The strangest aspect of the Ancient Egyptian’s limited mathematics is how they wrote fractions. For example, they could not write $\frac{5}{7}$ outright, they wrote

$\displaystyle \frac{5}{7}=\frac{1}{2}+\frac{1}{5}+\frac{1}{70}$,

and this method of writing fractions lead to absurdly complicated solutions to trivial problems—at least, trivial when you can write a vulgar fraction such as $\frac{3}{5}$. But how much more complicated is it to write $\frac{1}{2}+\frac{1}{5}+\frac{1}{70}$ rather than $\frac{5}{7}$?