The Evolution of the Number Zero

Vindya Bhat

Term Paper, History of Mathematics, Rutgers University

Spring 1999

edited and reformatted by Professor Cherlin

The Oxford English Dictionary effortlessly defines zero as "Nought or nothing, reckoned as a number, denoted by the figure 0; the total absence of a quantity, considered as a quantity," (OED, p. 802) but it took ancient mathematicians m illenia to come to accept this "absence of a quantity" as a number. Although the question of the time and place of the first introduction of a true zero sign remains unsettled, the history of the development of the notion of zero proves to be quite fascinating. We will attempt to follow this process from the time of Old Babylonia to the introduction of the system of Hindu-Arabic numerals to the West.

The earliest positional numeral system known to historians of mathematics is the notation found in Old Babylonian texts. Numbers less than 60 were represented by a base ten scale, while numbers greater than 60 were built from these "digits" in a sexagesimal scale. This positional numeral system was developed by the ancient Babylonians between 1800 and 1600 B.C. (Neugebauer, p. 15, 29) and expressed numbers using three symbols in a style of writing called cuneiform from the Latin for "wedge-shaped" as shown below.

These wedge-shaped characters were inscribed with a short stylus onto wet clay tablets which were then baked in an oven, rendering them almost indestructible. Fragments of the semi-permanent records that resulted from this process still exist today and have been decoded by specialists. The symbols used include a subtractive symbol, a symbol for one, and a symbol for ten called the Winkelhaken (Held, p. 1), a German word meaning "corner hook". With the application of the multiplicative and additive principles, and less commonly a subtractive principle, the mathematicians of Old Babylonia could represent arbitrary integers efficiently. as well as many fractions, using these symbols. Some examples of written numbers employing these symbols follow:

However, it soon became apparent that in this notation the lack of any notation for absent positions, led to ambiguity and resulted in errors. For instance, in the middle of the second row from the bottom of Plimpton 322 (fig. 2), a Babylonian clay tablet with cuneiform numerals from about 1800 B.C., we can read a number which is decoded as 29; (Neugebauer and Sachs, Plate 7). Although this decoding is correct in that context, it is quite possible that these same symbols could be read as 1209 or 36,609, as follows:

Clearly, the cuneiform characters used to represent the numbers 29 , 1209 , and 36,609 are essentially identical.

Recognizing this problem, the mathematicians of Old Babylonia eventually introduced a separation sign (fig. 1) to indicate the absence of a sexagesimal order, which was less ambiguous than the blank space which may be seen for example in the celebrated "Pythagorean triple" text, Plimpton 322. This separation sign met the difficulties associated with the non-ciphered (lacking zero) positional numeral system of Old Babylonia, and was in use at least by the beginning of the Seleucid period, about 300 B.C. (Neugebauer, p. 29), if not earlier (textual evidence is incomplete). This symbol for zero was used merely to indicate the absence of a sexagesimal unit of a particular order, as illustrated by certain lines of a mathematical tablet from the late third century B.C. whose fragments were found in archeological digs at Uruk (Ifrah, p. 380, fig. 3). Lines 10, 14, and 24 of this tablet make use of the Babylonian zero, a cuneiform symbol of two adjacent slant wedges (fig. 1). The tenth line of this tablet presents the Babylonian zero in writing as shown below: