The decimal representations of oppositely signed powers of two and powers of five look alike, as seen in these examples: 2^{-3} = 0.125 and 5^{3} = 125; 5^{-5} = 0.00032 and 2^{5} = 32. The significant digits in each pair of powers is the same, even though one is a fraction and one is an integer. In other words, a negative power of one base looks like a positive power of the other.

This relationship is not coincidence; it’s a by-product of how fractions are represented as decimals. I’ll show you simple algebra that proves it, as well as algebra that proves similar properties — in products involving negative powers.

Continue reading “Seeing Powers of Five in Powers of Two and Vice Versa”