Wednesday, December 16, 2009

Wednesday Math, Vol. 101: The whispers of the ancients.

Last week, I wrote about the success of the Hindu-Arabic numeral system. The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are now effectively the only symbols used to do math by humans. The existing competitors, the Chinese and Roman numeral systems, aren't as easy to use when doing addition, subtraction, multiplication and division, and now are used sparingly and largely ornamentally.

What all three of these systems have in common is base ten. Because humans have ten fingers and ten toes, every written system decided that a group of ten would be a special bundle. For the sake of consistency, the Romans and the Chinese and the Hindus decided a bundle of ten bundles would be the next important number to get its own special name and symbol, and then ten bundles of ten times ten bundles would be the next important number, and so on. When numbers get big enough we use the words meaning hundreds and thousands and ten thousands and on and on.


A bundle of ten is always considered a natural grouping, but some civilizations decided the next useful grouping would be six bundles of ten. The idea of a base 60 system is credited to the Sumerians, and the Babylonians followed suit. Obviously, it isn't a consistent system, but 60 has a big advantage over 100, and that is divisibility.

The divisors of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100.

The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. Being able to divide a bundle into equal parts in several different ways was decided to be more useful than the consistency factor of always multiplying by ten to get the next biggest bundle. The main surviving places where we see the base 60 system is of course in the measurement of time, with 60 seconds in a minute and 60 minutes in an hour. The number of degrees in a circle, 360, is 6 bundles of 60, and 360 is the smallest number that is evenly divisible by ever number less than 10, except for 7.

Another ancient system that still has remnants today is base 12. Many civilizations decided time units should be split into twelve, from twelve hours of day and twelve hours of night, twelve months and twelve zodiac signs. The Chinese zodiac is based on a twelve year pattern. Again, 12 is a little more divisible than 10, since it can be divided evenly by 1, 2, 3, 4, 6 and 12, instead of just 1, 2, 5 and 10. The Romans used base 10 for regular numbers, but their fractional system always split the things they could break up into twelve equal parts called uncia, pronounced "OON-cha". Uncia is the root word for the English words inch and ounce. Ounce is a little confusing, because nowadays, it's 16 ounces in a pound and 16 fluid ounces in a pint. Technically, the Troy weight system (named for Troyes, France, not the ancient city of Troy) has a pound that is twelve ounces, but the only use for the Troy system today is the Troy ounce, slightly larger than a regular Avoirdupois ounce and used for measuring silver, gold and gemstones. No one bundle Troy ounces into Troy pounds anymore.

While the inch may someday fade from use by humans, the use of base 12 and base 60 in the measurement of time isn't going anywhere. The French had great success in exporting their metric system to the rest of continental Europe, and the measurement of distance and weight and volume became an internally consistent system using base 10. On the other hand, during the Revolution, the French decided their should be 10 hours in a day, 100 minutes in an hour and 100 seconds in a minute, and the next most important clump of days should be 10 days and not 7. These ideas went nowhere, not even popular in France. There would be 100,000 seconds in a French day, compared to the 86,400 seconds using the 60x60x24 second method, so the French second would be just a little faster than a regular second.

If you wander around the Internet into some of the weirder neighborhoods, there are people who advocate splitting the day into 100,000 units, but they think it makes sense to use different words to avoid confusion, like ticks, beats, grands and cycles. Maybe they should come up with words for these things in Esperanto, because both those systems have roughly equal chances of being adopted.

Americans are the last people on earth clinging to inches and ounces. Eventually, the English measurement system could get shoved down the memory hole. But this odd system of splitting time into ancient units based on 60 isn't going anywhere. If I were to guess at the reason why, people realize that time is a thing we can measure, but not bend to our will. The length of a day or the length of a year is a natural thing we cannot change, and it does not conform to our base ten system, so we feel no great need to overthrow the somewhat unwieldy ancient ways.

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