A Pattern in Powers of Ten and Their Binary Equivalents

In my article “One Hundred Cheerios in Binary”, I made a comment about the decimal number 100, and its binary equivalent, 1100100:

“And will they wonder if the two sub strings of ‘100′ in the binary number have any significance?”

What I meant is if a novice might wonder if a decimal string made up of 1s and 0s must appear in its binary equivalent. Of course that’s not true in general, but it is true for nonnegative powers of ten — the trailing digits of the binary number will match the power of ten!

You can see the pattern in these examples:

The pattern is easy to explain. A nonnegative power of ten is a multiple of a power of five and a power of two: 10ⁿ = 5ⁿ * 2ⁿ. A power of five always ends in ’5′, so it’s odd — its binary representation always end in ’1′. When you multiply by a power of two, you shift the power of five left by n bits, which adds n trailing 0s. So the binary representation ends with a ’1′ followed by n 0s, which looks like the power of ten!

Cool, huh?