Sometime before 2000 B.C. the Babylonians developed a positional number system. It grouped numbers less than 60 using base-ten, and numbers of 60 or more using base-sixty.
Unit Symbol | Symbol for 10 |
OR |
Notice how the number 60 is written on the right using the unit symbol to indicate one power of 60, or 1 × 60, instead of 6 symbols for 10. From now on, to indicate the Babylonian sexagesimal place-value system, we will use our numbers separated by a semicolon to separate different orders of units. Thus, 69 would be written 1;9.
In the second millenium B.C. the Babylonian system had difficulties due to the lack of a zero (or special sign to mark any missing power of 60). For example, 3610 was written as 1;10, but this could mean 1 × 60^{2} + 10 = 3610 or 1 × 60 + 10 = 70.
To fix this problem Babylonian scribes sometimes left a blank space where a power of 60 was missing, but this could be confusing. Some scribes sometimes forgot to leave a space. If two or more consecutive powers were missing, indicating this by consecutive blank spaces left it up to the reader to determine how many orders were missing. Another method was needed.
During the fourth to first centuries B.C. (Seleucid era) Babylonian mathematicians and astronomers developed a true zero to indicate the absence of sexagesimal units of a certain order. Instead of the blank space they used either of the following two signs:
Babylonian mathematicians did not use a zero sign at the beginning or end of a written number, meaning 60 = 1;0 = 1 × 60 + 0 was not used. Babylonian astronomers did. They used the zero at the beginning, middle, and end of numbers.
1;0 = 1 × 60 +0 |
Babylonian astronomers used the zero at the beginning of a number (or initial position) to note sexagesimal fractions, those whose denominator is a power of 60. For example, 1/60 and 30/3600, written as the Babylonian astronomers did, are shown in the table below.
0;1 = 0 + 1/60 | 0;0;30 = 0 + 0/60 + 30/60^{2} |