A Pattern in Powers of Ten and Their Binary Equivalents
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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) |
|---|---|
| 1 | 1 |
| 10 | 1010 |
| 100 | 1100100 |
| 1000 | 1111101000 |
| 10000 | 10011100010000 |
| 100000 | 11000011010100000 |
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?
(No Ratings Yet)
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February 25th, 2012 at 5:51 pm
I know I’m 3 years late, but it is cool!
So, this should work for all bases which have exactly one “2″ in their prime factorisation. Indeed, I don’t see why not, but I ought to do a demo, and the simplest choice is base 6 – senary, as it’s called. It does actually live up to its punnable name in terms of fraction representation, but that’s beyond the point.
(senary = binary)
1 = 1
10 = 110
100 = 100100
1000 = 11011000
As expected, all good! Also, the power of 2 in the prime factorisation can be used to tell you how many trailing zeros to expect. In my familiar base 2^2 * 3, for example, you get:
(dozenal = binary (comment))
10 ^ 0 = 1 (2 * 0 trailing zeros)
10 ^ 1 = 1100 (2 * 1 trailing zeros)
10 ^ 2 = 10010000 (2 * 2 trailing zeros)
10 ^ n has 2n trailing zeros.
This of course holds for odd bases (where the power of 2 is 0, and all powers are odd). But it only looks interesting for the “exactly one 2″ bases.