118 5. COUNTINGf. I
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FIGURE 1. The cuneiform number 45.the second century CE, used the sexagesimal notation with a circle to denote an
empty place.3.3. India. The modern system of numeration, in which 10 symbols are used and
the value of a symbol depends only on its physical location relative to the other
symbols in the representation of a number, came to the modern world from India
by way of the medieval Muslim civilization. The changes that these symbols have
undergone in their migration from ancient India to the modern world are shown in
Fig. 2. The idea of using a symbol for an empty place was the final capstone on
the creation of a system of counting and calculation that is in all essential aspects
the one still in use. This step must have been taken well over 1500 years ago in
India. There is some evidence, not conclusive, that symbols for an empty place
were used earlier, but no such symbol occurs in the work of Arbyabhata. On the
other hand, such a symbol, called in Sanskrit sunya (empty), occurs in the work of
Brahmagupta a century after Arybhata.3.4. China. The idea of having nine digits combined with names for the powers
of 10 also occurred to the Chinese, who provided names for powers of 10 up to
100,000,000. The Chinese system of numbering is described in the Sun Zi Suan
Jing. A certain redundancy is built into the Chinese system. To understand this
redundancy, consider that in English we could write out the number 3875 in words
as three thousand eight hundred seventy-five. Since Chinese uses symbols rather
than letters for words, the distinction between a written number such as "seven"
and the corresponding numeral 7 does not exist in Chinese. But in writing their
numbers the Chinese did not use physical location as the only indication of the
value of a digit. Rather, that value was written out in full, just as here. To convey
the idea in English, we might write 3875 as 3 thousands 8 hundreds 7 tens and 5.
Because of this way of writing, there is no need for a zero symbol to hold an empty
place. For example, 1804 would simply be 1 thousand 8 hundreds and 4. Large
numbers were handled very efficiently, with a special name for 10,000 (wan). Its
square [wan wan, that is, 100,000,000 ( = 10^8 )] was called yi. Thereafter the
Chinese had special names for each power of 108. Thus, 1016 was zhao, 1024 was
jing, and so on, up to 10^80 (zai), which was surely large enough to meet any needs
of commerce or science until the twentieth century.^6 In that sense 108 amounted
to a second base for arithmetic in Chinese usage.
(^6) An estimate ascribed to Sir Arthur Eddington (1882-1944) of the number of protons in the
universe put the number at 136 · 2256 , which is approximately 1.575 • 1079.