- BASES FOR COUNTING 113
of examples to show that in many languages the word for five also means hand,
and that words for eight, nine, and ten also designate specific fingers of the hand
in some languages. A survey of ways of counting around the world provides some
evidence for Gow's thesis. The Bororo of Mato Grosso, for example (Closs, 1986,
p. 23), use a phrase for the number five that translates literally "as many of them
as my hand complete." In that same language the number seven is "my hand and
another with a partner," 10 is "my fingers all together in front," and 15 is "now
my foot is finished."
2. Bases for counting
Children have to be taught to count before they can talk about groups of more
than four things. Beyond certain sizes, it becomes impossible for anyone to tell
at a glance how many objects are present. Most people, for example, can say
immediately how many letters are in a word of eight letters or fewer, but have to
count for longer words. When the limit of immediate perception is reached, human
ingenuity goes to work and always arrives at the idea of grouping the objects to
be counted into sets of some definite size, then counting the number of sets. Thus
arises the notion of a base for counting. It is well known and seems completely
natural that in most cases this base is five or ten, the normal number of fingers on
one or two hands.2.1. Decimal systems. Decimal systems arose spontaneously in ancient Egypt,
India, China, and elsewhere. The choice of 10 as a base is not itself a sign of superior
wisdom. Only when combined with an efficient notation does the usefulness of a
base make itself known. A place-value system greatly simplifies calculation, which
is just as difficult in base 10 as in any other base when done without a place-value
system.The modern decimal system. Modern American counting—and increasingly also,
British counting—has special words for thousand, million, billion,^1 then trillion
(a thousand billions), quadrillion, quintillion, and so forth. Because these names
change with every third decimal place, we are effectively using 1000 as a base for
counting large sets. That fact shows through in the use of a comma (or period, in
Europe) to separate each group of three digits from its predecessor. The largest of
these units that anyone is likely to encounter in newspapers is the trillion, since it
is the most convenient unit for discussing the national budget or the national debt
in dollars. The Greek prefixes kilo- (thousand), mega- (million), giga- (billion), and
tera- (trillion) are used to discuss the memory cells in computers, and the march
of technology has made the first of these essentially negligible. The prefixes milli-
(one-thousandth), micro- (one-millionth), and nano- (one-billionth) for reciprocals
are used in discussing computing time. These are the units needed nowadays, and
those that have names at present provide a comfortable margin around the objects
to which they will be applied, so that no new units will need to be invented in the
foreseeable future.(^1) A billion is a thousand millions in American and increasingly in British usage, where it originally
meant a million millions.