A Pattern in Powers of Ten and Their Binary Equivalents
Copyright © 2008-2013 Exploring Binary
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:
|Power of Ten (in Decimal)||Power of Ten (in Binary)|
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!