116 5. COUNTINGRoman numerals (IV — 5 - 1). and we use subtraction to tell time in expressions
such as ten minutes to four and quarter to five?3. Counting around the world
We now examine the ways used to count in a selected set of cultures in which
mathematics eventually developed to the point of being written down.3 .1. Egypt. In Egypt the numbers appearing in hieroglyphics (the oldest writing)
are represented as vertical strokes (|) for each individual digit, up to 9; then 10 is
written as Ð, 20 as ÐÐ, and so on. To represent 100 the Egyptians used a symbol
resembling a coil of rope. Such a system requires new symbols to be invented for
higher and higher groupings, as larger and larger numbers become necessary. As
the accompanying photograph shows, the Egyptians had hieroglyphic symbols for
1000 (a lotus blossom), 10,000 (a crooked thumb), 100,000 (a turbot fish), and
1,000,000 (said to be the god of the air). With this system of recording numbers,
no symbol for zero was needed, nor was the order of digits of any importance,
since, for example, 111 Ð Ð and Ð Ð 11 j both mean 23. The disadvantage of the
notation is that the symbol for each power of 10 must be written a number of
times equal to the digit that we would put in its place. When hieroglyphics were
invented, the Egyptians had apparently not realized that it would be useful to have
names for the numbers 1 through 9, and then to name the powers of 10. Later on,
in the hieratic and demotic scripts that replaced hieroglyphics, they had special
symbols for 1 through 9, 10 through 90, 100 through 900, and so on, a system that
was reproduced in the Greek numeration with Greek letters replacing the hieratic
symbols.
3.2. Mesopotamia. As the examples of angle and time measurement show, the
successive divisions or regroupings need not have the same number of elements
at every stage. The sexagesimal system appears to have been superimposed on a
decimal system. In the cuneiform tablets in which these numbers are written the
numbers 1 through 9 are represented by a corresponding number of wedge-shaped
vertical strokes, and 10 is represented by a new symbol, a hook-shaped mark that
resembles a boomerang (Fig. 1). So far we seem to have a decimal system of
representation, like the Egyptian hieroglyphics. However, the next grouping is not
ten groups of 10, but rather six groups of 10. Even more strikingly, the symbol for
the next higher group is again a vertical stroke. Logically, this system is equivalent
to a base-60 place-value system with a floating "decimal" (sexagesimal) point that
the reader or writer had to keep track of mentally. Within each unit (sexagesimal
rank) of this system there is a truncated decimal system that is not place-value,
since the ones and tens arc distinguished by different symbols rather than physical
location. The number that we write as 85.25, for example, could be transcribed
into this notation as 1,25; 15, meaning 1 · 60 + 25 • 1 + 15 • ^j.
This place-value sexagesimal system goes back some 4000 years in the Middle
East. However, in its original form it lacked one feature that we regard as essential
today, a symbol for an empty place. The later Greek writers, such as Ptolemy in
(^5) Technology, however, is rapidly removing this last, vestige of the old way of counting from
everyday life. Circular clock faces have been largely replaced by linear digital displays, and ten
minutes to four has become 3:50. This process began long ago when railroads first imposed
standard time in place of mean solar time and brought about the first 24-hour clocks.