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Powers of 2
Fractions in Base 2
Fractions in Base N
Powers of 2 and 3I 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.
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
Multiplication Table, The
Powers of N
Twelve-Note Scale, The
@ March (2004)