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Zero as a Placeholder

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.

Babylonian unit symbol Babylonian symbol for 10
Unit Symbol Symbol for 10
A symbol was used for the unit. For numbers up to 9, this symbol was simply repeated. Another symbol was used for 10 and was repeated and combined with the unit symbol for numbers 11 to 59. Each of these are displayed in the table on the right. The numbers 60 and above were represented using the symbols for the numbers 1 to 59. To indicate multiples of 60 they used the position principle. Below is an example of the number 69.

Babylonian symbols for 69 OR Babylonian symbols for 69

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 × 602 + 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 Zeros
Babylonian zero Babylonian zero

Babylonian symbol for 3610
Babylonian symbol for 3610 without a 		zero or space
Babylonian symbol for 3610 with a 		space
Now 3610 could be written as in row 1 of the table to the right, 1;0;10 = 1 × 602 + 0 × 60 + 10 instead of the ways it is written in rows 2 and 3, which could easily be confused with 70 or left the interpretation of the width of a space to the reader.

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.

Babylonian symbol for 60
1;0 = 1 × 60 +0
In a Babylonian astronomical tablet, dated from the Seleucid era, the number 60 is written as in the table on the left: The zero sign here indicates the absence of units of the first order.

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.

Babylonian symbol for 1/60 Babylonian symbol for 30/3600
0;1 = 0 + 1/60 0;0;30 = 0 + 0/60 + 30/602