Tuesday, December 18, 2012

A Swarm of Bases

Looking at Mayan ways of herding numbers, where base-20 numeration (the vigesimal) is dominant, with one exception, I was reminded that ancient cultures, going far back into the BC regions, also liked working with numbers other than the decimal. The Mayans used positional notation for their numbers but, unlike us, who move from right to left as numbers increase, they moved from top to bottom. At each position they showed multiples of 20 except in the third place. There they used an 18-base structure (octodemical). Their numeration came from the construction of calendars. The first position (topmost) included units, the second weeks (20 days), and the third years. If they had been consistently vigesimal, they would have derived a year by multiplying 1 week of 20 days by 20. That would have yielded a 400-day year—a little too long for comfort. Therefore they used 18-base numeration in that position to get a year of 360 days. Thereafter, each successive period was 20 times the earlier. The Mayans also had a symbol for zero; it had the shape of an eye.

During Babylonian times (around 1800-1600 BC), base-60 numeration (sexagesimal) was common—whereas the Egyptians used the decimal system. Use of the sexagesimal seems to have been motivated by the problem of fractions, known to plague all elementary grade children in decimal eras. Sixty is evenly divisible into halves, thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, and thirtieths fractions, whereas 10 is divisible cleanly only into halves and fifths.

Now, of course, we also use a sexagesimal system daily—if not hourly—without having our eyes glaze up: the sixty seconds to the minute, 60 minutes to the hour, the 360 degrees of the circle, which is 6 x 60. And as we know instinctively that 15 minutes are a quarter hour, so we also know that 90° are a quarter of a circle. Our longer measurements of time are also derivable from 60—as in hour, and day—but when it comes to a year, Mother Nature shakes off our precision and insists on pesky fractions.

The Babylonians, unlike the Mayans, applied their numbers to geometry and mathematics, not to calendars. They stuck to a lunar calendar which is something of a nightmare to describe.

Modern people all over the world also daily use other bases without ever thinking about them. Two that are hidden—but the nerds know—are base-2 (binary), the core language of computers, and built on top of it the base-16 (hexadecimal), of which a half is the 8-bit byte. Some folks, stuck with very old computers, may also be served by 12-base (duodecimal) numbers.

The duodecimal is under our linear measurements (12 inches to the foot); hexadecimal hides in weights (a dram is 1/16 of an ounce, an ounce 1/16 of a pound), and in volumes (an ounce is 1/16 of a pint, the quart and gallon are multiples of pints). And so on. Note that bases other than 10 all have more clean fraction than the decimal. Sexagesimal wins, with ten such. The 12-, 18-, and 20-base have four, the 16-base has three.

So what is the point of all this? The point is that, before modern calculators, there was motivation (to use a word the nerds are fond of) for using all kinds of bases for ease of calculation—and habit has preserved those practices in ordinary life. And the final point is to praise the decimal point—which has freed us of calculating awkward fractions in the old way in an age that is becoming ever more fractionalized.
My source here is A History of Mathematics by Carl B. Boyer, revised by Uta C. Merzbach, John Wiley & Sons, on the Babylonians p. 25, on the Mayans p. 213.

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