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Powers of 2
 > Powers of 2 and 3
Powers of N
Fractions
Fractions in Base 2
Fractions in Base N

 Logarithmic Forms

Powers of 2 and 3

I don't really have anything to say about these numbers … I like them, they turn up over and over, and it will please me to have a table of them here on my site.

 1 3 9 27 81 243 729 2187 6561 19683 59049 2 6 18 54 4 12 36 108 324 8 24 72 216 16 48 144 432 1296 32 96 288 7776 64 192 576 1728 46656 128 384 1152 256 768 2304 512 1536 1024 3072 2048 6144 4096 12288 8192 24576 16384 49152 32768 65536

It's not a complete table because I've only included numbers that I recognize. The first three columns I know mainly from Apple II days; and of course the main diagonal I know from working out dice probabilities. I'd like to include 3456 (below 1728) and 3888 (above 7776), just because they're such nice numbers, but I have to admit I wouldn't have recognized them.

I'll also admit to having favorites. I like all the small numbers, but there familiarity breeds taking for granted. I like 243, partly because it reminds me of 73 = 343; I also like 59049 a lot. Then there's 216, which is very pleasing for some reason.

Among the larger powers of two, of course I'm very fond of 256, and also of 65536, which was the size of the Apple's address space. Speaking of which, some numbers are very familiar because they were nice round addresses in hexadecimal: 768 (300) and 24576 (6000). I don't think of 192 (C0) as an address, but it's also a nice round number, and very handy.

Now, here's an interesting thing I never noticed before. I was going to say that I liked how there happened to be some pairs of similar numbers, like 288 and 12288, or 576 and 24576. But, it turns out that's not just a coincidence, it actually follows from the useful equation

1024 = 1000 + 24.

If you multiply that by any entry N from the table, and rearrange a little, you get

210 N = 1000 N + 23 3 N,

which has a nice translation into English: the entry ten spaces down from N is equal to a thousand times N plus the entry three spaces down and one over.

What's more, you can do the same thing for any equation that contains only powers of 2, 3, and 5. For example, there's a rule based on the Pythagorean triangle.

32 + 42 = 52