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One of the publicity videos for this talks about the Babylonian base-60 system allowing them to write down fractions more accurately than we can. This is bunk.

One of the publicity videos for this talks about the Babylonian base-60 system allowing them to write down fractions more accurately than we can. This is bunk. You can't write down one-third exactly as a base 10 decimal, but you can work with it exactly by writing it down as 1/3. It's more laborious, and probably the Babylonians found base-60 very useful, but given we have obscenely powerful computers it's probably useless to us as a practical matter.

Koshkin 9y ago
... Except we use it all the time.
- Rhinobird 9y ago
  Just a minute! I'll second this.
jacobolus 9y ago
Notice that nothing like our modern notation for fractions existed at the time (the first such notation is from India in ~500 CE), and calculations throughout the ancient world were done using mental arithmetic, finger counting, or physical manipulation of tokens either in piles or on some kind of counting board, rather than using symbolic manipulation with columns of written digits the way we learn in schools today. In particular, long division is a real pain compared to multiplication, so being able to use a reciprocal table to turn division problems into multiplication problems would have been a big help. * * * Here’s my Reddit comment showing a bit about how sexagesimal can be nice for doing exact computations with numbers that can be exactly reciprocated: As a simple example, 0.4 (base 10) is a much more precise way of writing 2/5 than ~0.31 (base 8). In base sixty you get many more divisors which result in terminating positional fraction expansions: 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, etc. (any number which can be written as powers of 2, 3, or 5) Whereas in base ten, you only have 2, 4, 5, 8, 10, 16, 20, 25, etc. If you want to fill out a table of all the possible divisors between 1.0 and 2.0 which are “decimally smooth” and use up to 4 decimal digits (10000 possibilities) you get: 1.0, 1.024, 1.25, 1.28, 1.5625, 1.6, 1.6384, 2.0 If you fill out a table of all the divisors between 1:00 and 2:00 which are “sexagesimally smooth” and use up to 2 sexagesimal digits (3600 possibilities) you get: 1:00, 1:00:45, 1:04, 1:04:48, 1:06:40, 1:08:16, 1:12, 1:12:54, 1:15, 1:16:48, 1:20, 1:21, 1:23:20, 1:25:20, 1:26:24, 1:30, 1:33:45, 1:36, 1:37:12, 1:40, 1:41:15, 1:42:24, 1:44:10, 1:46:40, 1:48, 1:49:21, 1:55:12, 2:00 (I might have missed a couple here.) As you can see this is a much richer set of divisors, more usefully spaced throughout the interval.
- jonahx 9y ago
  thanks for posting this. what does "decimally smooth" mean in this context?
  - jacobolus 9y ago
    I just mean a number such that both itself and its reciprocal is a terminating decimal (rather than a repeating decimal like the reciprocal of 3; 1/3 = 1.33333...). That is, some rational number made up of 2 to some power times 5 to some power (either or both of the exponents could be negative). “Decimally regular” might be a more standard term.
kevin_thibedeau 9y ago
> but given we have obscenely powerful computers it's probably useless to us as a practical matter. Babylonian arithmetic methods have been used to give us our obscenely powerful computers.

Koshkin 9y ago

... Except we use it all the time.

Rhinobird 9y ago
Just a minute! I'll second this.

jacobolus 9y ago

Notice that nothing like our modern notation for fractions existed at the time (the first such notation is from India in ~500 CE), and calculations throughout the ancient world were done using mental arithmetic, finger counting, or physical manipulation of tokens either in piles or on some kind of counting board, rather than using symbolic manipulation with columns of written digits the way we learn in schools today. In particular, long division is a real pain compared to multiplication, so being able to use a reciprocal table to turn division problems into multiplication problems would have been a big help. * * * Here’s my Reddit comment showing a bit about how sexagesimal can be nice for doing exact computations with numbers that can be exactly reciprocated: As a simple example, 0.4 (base 10) is a much more precise way of writing 2/5 than ~0.31 (base 8). In base sixty you get many more divisors which result in terminating positional fraction expansions: 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 18, 20, 24, 25, 27, 30, etc. (any number which can be written as powers of 2, 3, or 5) Whereas in base ten, you only have 2, 4, 5, 8, 10, 16, 20, 25, etc. If you want to fill out a table of all the possible divisors between 1.0 and 2.0 which are “decimally smooth” and use up to 4 decimal digits (10000 possibilities) you get: 1.0, 1.024, 1.25, 1.28, 1.5625, 1.6, 1.6384, 2.0 If you fill out a table of all the divisors between 1:00 and 2:00 which are “sexagesimally smooth” and use up to 2 sexagesimal digits (3600 possibilities) you get: 1:00, 1:00:45, 1:04, 1:04:48, 1:06:40, 1:08:16, 1:12, 1:12:54, 1:15, 1:16:48, 1:20, 1:21, 1:23:20, 1:25:20, 1:26:24, 1:30, 1:33:45, 1:36, 1:37:12, 1:40, 1:41:15, 1:42:24, 1:44:10, 1:46:40, 1:48, 1:49:21, 1:55:12, 2:00 (I might have missed a couple here.) As you can see this is a much richer set of divisors, more usefully spaced throughout the interval.

jonahx 9y ago
thanks for posting this. what does "decimally smooth" mean in this context?
- jacobolus 9y ago
  I just mean a number such that both itself and its reciprocal is a terminating decimal (rather than a repeating decimal like the reciprocal of 3; 1/3 = 1.33333...). That is, some rational number made up of 2 to some power times 5 to some power (either or both of the exponents could be negative). “Decimally regular” might be a more standard term.

kevin_thibedeau 9y ago

> but given we have obscenely powerful computers it's probably useless to us as a practical matter. Babylonian arithmetic methods have been used to give us our obscenely powerful computers.