- COUNTING AROUND THE WORLD^121
0 12··3- · ·
4- · · ·
5 6
7··8- · ·
9- · · ·
10 1112
··13- · ·
14- · · ·
15 1617
··18- · ·
19- · · ·
20ü
2122··23
- · ·
24
- · · ·
25
2627- ··
28
- · ·
29
- · · ·
FIGURE 4. Maya numeralsthe immensity of infinity. He saw that once the process just described had been
completed, he could label the numbers that were named up to that point the first
period. The mind feels carried to dizzying heights by such a process. Archimedes
did not stop until he reached a number that we would represent as a 1 followed
by 80 quadrillion zeros! And of course, there is not the slightest reason to suspect
that Archimedes thought the creation of integers needed to stop there. It must stop
somewhere, of course.
By applying reasonable and generous estimates of the size of the universe as it
was known to him, Archimedes showed that the number of grains of sand needed
to fill it up could not go beyond the 1000 sixth-order units (1000 units into the
seventh order, or 10^51 in our terms). Allowing an assumption of an even larger
universe, as imagined by the astronomer Aristarchus, he showed that it could not
hold more than 10^63 grains of sand.
3.6. The Maya. Although geographically far removed from Egypt, the Maya cul-
ture that existed in what is now southern Mexico and Central America from 300
BCE to 1500 CE shows some intriguing resemblances to that of ancient Egypt,
especially in the building of pyramidal structures and in a hieroglyphic type of
writing. On the other hand, the Maya system of counting resembles more the
Mesopotamian sexagesimal system, except that it is vigesimal (base 20). As with
the Mesopotamian system, only two symbols are needed to write all the numbers
up to the base: a dot for ones and a horizontal line for fives. Thus the smaller base
on which the vigesimal system is built is five in the case of the Maya, whereas it
was 10 in the Mesopotamian system. The Maya numerals illustrate the principle
that higher-level groupings need not always have the same number of members as
the lower. As Fig. 4 shows, four groups of five are consolidated as a single unit of