The Ancient Egyptian Concept of Zero and the Egyptian Symbol for Zero: A Note on a Little Known African Achievement

Beatrice Lumpkin

It is well known that a zero placeholder was not used or needed in Egyptian numerals, a system of numerals without place value. Values were expressed by grouping and addition of repeated ciphers. Still historians such as Boyer (1968) and Gillings (1965) have found examples of the use of the zero concept in ancient Egypt. But Gillings added, "Of course zero, which had not yet been invented, was not written down by the scribe or clerk; in the papyri, a blank space indicates zero." However, some Egyptologists did know that the ancient Egyptians used a zero symbol, but it may have been missed by historians of mathematics because the symbol did not appear in the surviving mathematical papyri.

The Egyptian zero symbol was a triliteral hieroglyph, with consonant sounds nfr (Gardner, 1978). This was the same hieroglyph used to represent beauty, goodness, or completion (Faulkner, 1976). There are two major sources of evidence for an Egyptian zero symbol:

1. Zero Reference Level for Construction Guidelines

Massive stone structures such as the ancient Egyptian pyramids required deep foundations and careful leveling of the courses of stone. Horizontal leveling lines were used to guide the construction. One of these lines, often at pavement level, was used as a reference and was labeled nfr, or zero. Other horizontal leveling lines wcre spaced 1 cubit apart and labeled as 1 cubit above nfr, 2 cubits above nfr, or 1 cubit, 2 cubits, 3 cubits, and so forth, below nfr (Arnold, 1991).

In 1931, George Reisner described zero leveling lines at the Mycerinus (Menkure) pyramid at Giza built c. 2600 BCE. He gave the following list collected earlier at Borchardt and Petrie from their study of Old Kingdom pyramids (Reisner, 1931).

nfrw	zero (Note the w suffix added to nfr for grammatical reasons.)
m tp n nfrw	zero line
hr nfrw	above zero
md hr n nfrw	below zero

2. Bookkeeping, Zero remainders

A bookkeeper's record from the 13th dynasty c 1700 BCE shows a monthly balance sheet for items received and disbursed by the royal court during its travels. On subtracting total disbursements from total income, a zero remainder was left in several columns. This zero remainder was represented with th same symbol, nfr, as used for the zero reference line in construction (Reisner, 1931).

These practical applications of a zero symbol in ancient Egypt, a society that conventional wisdom believed did not have a zero, may encourage historians to re-examine the everyday records of ancient cultures for mathematical ideas that have been overlooked.

References

Arnold, D. (1991). Building in Egypt. New York Oxford University Press.

Boyer, C. B. (1968). A history of mathematics. New York Wiley.

Faulkner, R. O. (1976). A concise dictionary of middle Egyptian. Oxford Griffith Institute.

Gardner, Sir A. A. (1978). Egyptian grammar. Oxford: Griffith Institute.

Gillings, R. J. (1965). Mathematics in the time of the pharaohs. Cam bridge, MA: MIT Press.

Reisner, G. A. (1931). Mycerinus: The temples of the third pyramid at Giza. Cambridge, MA: Harvard University Press.