A brief history of algebraic notation

School Science and Mathematics, May 2000 by Stallings, Lynn

Roots

Roots in the form of an inverted L appear in early egyptian papyri found at Kahun. Other symbols used included R, l, O, or a fractional exponent. The l represented the Latin word latus, which means "side of a square" (Cajori, 1993). The need for l to represent logarithms is one reason the use of l for root did not persist. The radical sign was introduced by German Christoff Rudolff (c. 1500 - c. 1545) in 1525 in his book on algebra Die Coss. It is hypothesized that the symbol was chosen because it resembles a small r from the, word radix, which means base (Eves, 1983). Some historians think that the radical descended, not from the letter r, but from the use of dots in manuscripts from the late 1400s: One dot meant square root and two dots meant square root of square root (Cajori, 1993).

Other Symbols

Leibniz, who was a diplomat, librarian, linguist, and philosopher, invented calculus concurrently with but independently of Englishman Sir Isaac Newton (1642-1727). He first used the integral symbol, 1, an elongation of the first letter of the Latin word summa (sum) (Eves, 1983). Leibniz also introduced the - for similarity and -for congruence (Cajori, 1993). Leibniz is one of the few mathematicians who systematically experimented with different notations. For that reason, he devised a better symbol system for calculus than Newton, but the dispute over the discovery kept many English mathematicians from adopting Leibniz's symbolism for some time (Eves, 1983).

Some symbols were adopted for practical reasons. Factorials have been notated in various ways, including M by Euler in 1751, n* by Basedow in 1774, [pn] by Vandermonde in 1772, 1^sup m|l^ by Kramp in 1808, II(n) by Weber in 1893, and In by Jarret in 1827 (Cajori, 1993). In 1808, Christian Kramp (1760-1826) first introduced the n! (along with I^sup m|l^) (Eves, 1983). It was commonly adopted for factorial n around the early 1900s because it was easier to print than the previous symbol most commonly used, In (Cajori, 1993).

Oughtred's student John Wallis (1616-1703), who was known as one of the most creative mathematicians of his time, is credited with introducing the symbol for infinity,oo (Eves, 1983). It is hypothesized that this symbol came from a late Roman symbol for 1,000 (Cajori, 1993).

Leonard Euler (1707-1783), the most prolific writer ever to have written about mathematics may have introduced more of the symbols used today than any other mathematician. He is credited with flx) for functional notation; Sigma for the base of natural logarithms; S for summation; and i for the imaginary unit, square root of -1 (Eves, 1983).

This brief summary of the history of algebra and some of the most commonly used symbols shows that it took mankind's finest mathematicians 3,000 years to develop a compact, efficient symbolic algebra. Learning to use the algebraic symbol system is a challenge for good reason. The origins of some commonly used symbols provides interesting glimpses into a mathematics that is an evolving human endeavor, rather than the compact and highly abstract presentation of most textbooks. Seeing that mankind struggled for 3,000 years to produce it may encourage students to persist a little longer in their attempts to understand it. The reference list is an excellent way to continue exploring how the modern notation system developed.