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Base 60

In Hexadecimal I mentioned the possibility of learning to think in base 60. I've never done that, and almost certainly never will, but it's fun to think about.

Part of the appeal is that 60 has so many divisors … that ought to make it a good base for computation. In particular, there are many more fractions that have finite expansions. If we use pairs of base-10 digits to represent base-60 digits, here are all the fractions 1/n that have single-digit expansions.

1/101.001/1000.06
1/200.301/1200.05
1/300.201/1500.04
1/400.151/2000.03
1/500.121/3000.02
1/600.101/6000.01

Of course any other fraction m/n with one of those denominators will also have a single-digit expansion.

You may have heard that the Babylonians (and Sumerians) used base 60. That's true (as far as I know), but … if you search around on the web to see how they actually wrote numbers, you'll notice that they didn't have sixty distinct symbols. In fact they had only two, a ones mark and a tens mark, and they just piled on as many as needed to make the desired number.

The idea of a ones mark should be familiar from Roman numerals. However, where the Romans compressed larger numbers like IIIIIIIII by using a fives mark and/or reversing the order to subtract (IX), the Babylonians compressed them by grouping them in rows of three.

The tens mark is a bit more subtle. The Roman system requires a new mark for every power of ten (and for five times every power of ten), but is fundamentally decimal. The Babylonian system is positional, no new marks needed, but it is not decimal. Tens are distinguished from ones, and the number of tens is allowed to range from zero to five.

The end result, of course, is that the successor to 59 is 01 00, and that's why the system is described as base 60. It seems to me, though, that it would be more accurate to describe it as alternating between base 6 and base 10. If you think of it that way, you can leave out the spaces between pairs of digits and drop the leading and trailing zeroes, so that the successor to 59 is 100, and the expansion of 1/2 is 0.3.

So, that brings me to another thing I like about base 60, which is that we still use it, in the exact same alternating form, to measure times and angles. I assume that can be traced back to the Babylonians, but I don't know for sure.

By the way, I imagine not everyone knows that degrees, like hours, are divided into 60 minutes (of arc), and the minutes into seconds. That is why I say we use base 60 to measure angles. (Although, I wouldn't be surprised if that died out within my lifetime, since it's just as easy to have a GPS with a decimal readout.) And, of course, there are also 360 degrees of arc in a circle, which we can think of as two base-6 digits plus a base-10 digit.

The last thing I wanted to mention about base 60 has to do with the imaginary language Speedtalk. You can tell from the paragraphs I quoted that the people who speak the language use base 60, with a full sixty symbols, but in the story (Gulf) there's a whole little paragraph about it. The key point: “three times four times five, a convenient, easily factored system”.

 

  See Also

  Duodecimal
  Footnote (Decimal Expansions)
  Four Digits
  In Mathematics
  In Other Bases
  Other Bases
  Usual Random Thoughts, The

@ April (2004)