120 5. COUNTING
1 2 3 4 5 6 7 8 9á' â' V <5' å' ò' C V' 0'
10 20 30 40 50 60 70 80 90é' X? Á' ì' í' î' ï' ôô' 9'
100 200 300 400 500 600 700 800 900Ñ' ô' í' ÷' ù'
FIGURE 3. The ancient Greek numbering systemthat the number 4 would be represented by the fourth letter of the Greek alphabet
(ä) and written ä'. When they reached 1000 (khilias), the Greeks continued with
numerical prefixes such as tetrakiskhilioi for 4000 or by prefixing a subscripted
prime to indicate that the letter stood for thousands. Thus, >ä' stood for 4000,
and the number 5327 would be written >å'ô'ê'æ'. The largest independently named
number in the ancient Greek language was 10,000, called myrias. This word is the
source of the English word myriad and is picturesquely derived from the word for
an ant (myrmex). Just how large 10,000 seemed to the ancient Greeks can be seen
from the related adjective myrios, meaning countless.
The Sand-reckoner of Archimedes. The Egyptian-Greek system has the disadvan-
tage that it requires nine new names and symbols each time a higher power of 10
is needed, and the Roman system is even worse in this regard. One would ex-
pect that mathematicians having the talent that the Greeks obviously had would
realize that a better system was needed. In fact, Archimedes produced a system
of numbering that was capable of expressing arbitrarily large numbers. He wrote
this method down in a work called the Psammites (Sand-reckoner, from psammos,
meaning sand).
The problem presented as the motivation for the Psammites was a childlike
question: How many grains of sand are there? Archimedes noted that some people
thought the number was infinite, while others thought it finite but did not believe
there was a number large enough to express it. That the Greeks had difficulty
imagining such a number is a reflection of the system of naming numbers that they
used. To put the matter succinctly, they did not yet have an awareness of the
immense potential that lies in the operation of exponentiation. The solution given
by Archimedes for the sand problem is one way of remedying this deficiency.
Archimedes saw that solution of the problem required a way of "naming" ar-
bitrarily large numbers. He naturally started with the largest available unit, the
myriad (10,000), and proceeded from there by multiplication and a sort of induc-
tion. He defined the first order of numbers to be all the numbers up to a myriad of
myriads (100,000,000), which was the largest number he could make by using the
available counting categories to count themselves. The second order would then
consist of the numbers from that point on up to a myriad of myriads of first-order
numbers, that is, all numbers up to what we would call (lO^8 ) — 10^16. The third
order would then consist of all numbers beyond the second order up to a myriad of
myriads of second-order numbers (10^24 ). He saw that this process could be contin-
ued up to an order equal to a myriad of myriads, that is, to the number (10^8 )^10.
This is a gargantuan number, a 1 followed by 800 million zeros, surely larger than
any number science has ever needed or will ever need. But Archimedes realized