114 5. COUNTINGNamed powers of 10. The powers of 10 that have a name, such as hundred, thou-
sand, and so on, vary from one society to another. Ancient Greek and modern
Japanese contain special words for 10 thousand: myrias in Greek, man in Japan-
ese. With this unit one million becomes "100 myriads" in Greek or hyakuman in
Japanese (hyaku means 100). In these systems it would make more sense to insert
commas every four places, rather than every three, to make reading easier. The
ancient Hindus gave special names to numbers that one would think go beyond
any practical use. One early poem, the Valmiki Ramayana, from about 500 BCE,
explains the numeration system in the course of recounting the size of an army.
The description uses special words for 10^7 , ÉÏ^12 , 10^17 , and other denominations, all
the way up to 10^55.2.2. Nondecimal systems. The systems still used in the United States—the last
bastion of resistance to the metric system—show abundant evidence that people
once counted by twos, threes, fours, sixes, and eights. In the United States, eggs
and pencils, for example, are sold by the dozen or the gross. In Europe, eggs are
packed in cartons of 10. Until recently, stock averages were quoted in eighths rather
than tenths. Measures of length, area, and weight show other groupings. Consider
the following words: fathom (6 feet), foot (12 inches), pound (16 ounces), yard (3
feet), league (3 miles), furlong (1/8 of a mile), dram (1/8 or 1/16 of an ounce,
depending on the context), karat (1/24, used as a pure number to indicate the
proportion of gold in an alloy),^2 peck (1/4 of a bushel), gallon (1/2 peck), pint
(1/8 of a gallon), and teaspoon (1/3 of a tablespoon).
Even in science there remain some vestiges of nondecimal systems of measure-
ment, inherited from the ancient Middle East. In the measurement of both angles
and time, minutes and seconds represent successive divisions by 60. A day is divided
into 24 hours, each of which is divided into 60 minutes, each of which is divided into
60 seconds. At that point, our division of time becomes decimal; we measure races
in tenths and hundredths of a second. A similar renunciation of consistency came
in the measurement of angles as soon as hand-held calculators became available.
Before these calculators came into use, students (including the present author) were
forced to learn how to interpolate trigonometric tables in minutes (one-sixtieth of
a degree) and seconds (one-sixtieth of a minute). In physical measurements, as
opposed to mathematical theory, we still divide circles into 360 equal degrees. But
our hand-held calculators have banished minutes and seconds. They divide degrees
decimally and of course make interpolation an obsolete skill. Since ð is irrational,
it seems foolish to adhere to any rational fraction of a circle as a standard unit;
hand-held calculators are perfectly content to use the natural (radian) measure,
and we could eliminate a useless button by abandoning the use of degrees entirely.
That reform, however, is likely to require even more time than the adoption of the
metric system.^3
(^2) The word is a variant of carat, which also means 200 milligrams when applied to the size of a
diamond.
(^3) By abandoning another now-obsolete decimal system—the Briggsian logarithms—we could elim-
inate two buttons on the calculators. The base 10 was useful in logarithms only because it allowed
the tables to omit the integer part of the logarithm. Since no one uses tables of logarithms any
more, and the calculators don't care how messy a computation is, there is really no reason to
do logarithms in any base except the natural one, the number e, or perhaps base 2 (in number
theory). Again, don't expect this reform to be achieved in the near future.