A brief history of algebraic notation

School Science and Mathematics, May 2000 by Stallings, Lynn

In Greece, the wording was geometric. For example, xyz would be called the "solid whose sides are the first, second, and third unknown quantities" (Nelson, 1993, p. 33). A possible reason for their use of geometric explanations were the Greeks' conceptual difficulties with fractions and irrational numbers, such as the Pythagoreans' shock at the discovery of the seemingly counterintuitive irrational numbers. Although the Greeks found it hard to accept (square root of) 2, they could accept that (square root of) 2 could be the length of the diagonal of a square with sides of length 2 (Baumgart, 1969). Another possible explanation suggested by Gullberg ( 1997) was the Greek "system of numeric notation, using letters of their alphabet sometimes built several stories high, and littered with accents, apostrophes, and other diacritic signs" (p. 298).

Syncopated Algebra

Rhetorical algebra was the only algebra for several centuries until Diophantus of Alexander (c. AD 250) pioneered syncopated algebra (Nelson, 1993). This next stage in algebra's development, which used some shorthand or abbreviations, was useful with powers up to the sixth and their reciprocals. In most parts of the world other than Greece and India, rhetorical algebra persisted for a longer period; for example, Western Europe used it until the 15th century (Eves, 1983).

Diophantus wrote Arithmetica, a book that tremendously influenced algebra and number theory (Struik, 1967). The word arithmetic comes from the Greek word arithmetike, which is composed of the Greek words for number (arithmos) and for science (techne). The notation in Arithmetica helped move algebraic notation from rhetorical to syncopated, as well as influencing the evolution of some individual symbols (Eves, 1983).

Table I shows some examples of Diophantus' syncopated algebra using the Greek ciphered system for numerals. Historians have found explanations for many of Diophantus' selections of symbols. His symbol for the unknown may have been a combination of the first two Greek letters, alpha and beta, although over time it came to resemble the Greek letter sigma. The symbol for an unknown squared, (delta)^sup Y^, comes from the first two letters of the Greek word for power (dunamis or (Delta)YNAM(Sigma)). Similarly, the symbol for an unknown cubed, K^sup gamma^, comes from letters of cube (kubos or KYBOZ) (Nelson, 1993). Expressions such as (delta)^sup gamma^(delta) (square square), (delta)K^sup gamma^ (square cube), and K^sup gamma^K (cube cube) are derived from the previous examples. The sign used for minus is a combination of the first two letters of the word lacking (leipis or (Lambda)EI(psi)I(Sigma)). The sign for constant term (see Table 1) was an abbreviation of the Greek word for units (monades or MONA(delta)E(sigma)) (Eves, 1983). Some of the other basic conventions he used follow:

Negative terms are grouped together after the minus symbol:

Juxtaposed terms are to be added.

When an alphabetic Greek numeral follows the power symbol, it is the power to which the unknown is to be raised. (Nelson, 1993)