A brief history of algebraic notation

School Science and Mathematics, May 2000 by Stallings, Lynn

This paper traces three major stages in the development of algebraic notation: rhetorical or prose, syncopated, and symbolic. The development of algebra began in Babylonia and Egypt around 1700 BC. Examples of rhetorical algebra by al-Khowdrizmi are used to illustrate potential difficulties that arise when algebraic problems are worked using words without symbols. Greek mathematician Diophantus was one of the pioneers of syncopated algebra. In this stage of notation, some shorthand was used along with prose. Indian mathematicians developed a syncopated algebraic notation independently of Diophantus. Around 1500 BC, symbolic algebra began to develop. The process of the development of a standardized, efficient symbol system is illustrated by tracing the evolution of some common symbols, including the symbols for equals, addition, subtraction, multiplication, and division.

Exploration of the historical development of algebraic notation shows mathematics to be a creation of individuals from all parts of the world. Our elegant, efficient symbol system arose from cumbersome, even awkward, beginnings. Learning more about the history of algebraic notation or doing problems using rhetorical or syncopated algebra can give students a different perspective on the modern notation and so broaden their understanding and appreciation of mathematics. This article traces the three major stages of development of algebraic notation, from the earliest stage of rhetorical, or prose, algebra to the intermediate stage of syncopated algebra and, finally, to symbolic algebra (Nelson, 1993). This article also traces the development of some of the most common mathematical symbols.

The historic development of algebra began in both Egypt and Babylonia about 1650 BC, although it is uncertain whether ideas traveled between Egypt and Babylonia. Famous relics such as the Rhind (or Ahmes) papyri and the Hammurabi clay tablets (c. 1800-1600 BC) show that Egyptian algebra was less sophisticated than the Babylonian algebra (Gullberg, 1997). One reason may be that their numeral system was more primitive. It is thought that the Babylon influences spread to Greece (c. 500 BC to 300 BC), then to the Arabian Empire and India (c. 700 AD), and on to Europe (c. 1100 AD) (Baumgart, 1969).

The types of equations and notations used today were first introduced around 1700 BC and stabilized by about 1700 AD. The two factors playing the largest role in standardizing mathematical symbols were the invention of the printing press (c.1450) and economies strong enough to encourage the travel of scholars, resulting in the transmission of ideas. Even today, some differences in use of notation persist in different areas of the world. For example, many Europeans use a comma where Americans use a period, so they would write 3,14 as an approximation for n, while Americans would write 3.14 (Baumgart, 1969).

Rhetorical Algebra

Rhetorical algebra refers to the stage when algebra was written in words without the use of mathematical symbols. The early Babylonian and Egyptian algebras were both rhetorical (Baumgart, 1969). An early algebra book (AD 825) written in rhetorical or prose algebra included the following problem involving multiplication of binomials: "ten and thing to be multiplied by thing less ten."

In symbols, today's student would write (x 10) (x - 10) = x^sup 2^ - 100. Compare that with the following translation of the solution from that text:

If the instance be, "ten and thing to be multiplied by thing less ten," then this is the same as if it were said thing and ten by thing less ten. You say, therefore, thing multiplied by thing is a square positive; and ten by thing is ten things positive; and minus ten by thing is ten things negative. You now remove the positive by the negative, then there only remains a square. Minus ten multiplied by ten is a hundred, to be subtracted from the square. This, therefore, altogether, is a square less a hundred... (Nelson, 1993, p. 33)

It is cumbersome to write out each step in words. Room for ambiguity exists without the precision of our standardized, efficient symbol system. Some of the difficulties encountered in reading that problem in rhetorical algebra will show why mathematicians continued to abbreviate and refine their notation. Examples of rhetorical algebra can be found on Babylonian cuneiform clay tablets written in the sexagesimal (base 60) number system from about 1700 BC (Baumgart, 1969).

The Arab mathematician Mohammed ibn Musa al-Khowarizmi (c. 825) wrote the early algebra book that was the source of the previous problem. The book title, Hisab al jabr w'al-muqabala, translates into Latin as Liber algebrae et almucabala (Boyer, 1991), but is also referred to as Liber algorismi (Book of alKhowarizmi) (Baumgart, 1969). In English, it might be translated as Science of Transposition and Cancellation (Eves, 1983) or Science of Reduction and Confrontation (Struik, 1967), but the subject is the same: the science of equations (e.g., Eves, 1983; Menninger,1992). Following its translation into Latin in the 12th century, al-Khowarizmi's book became very influential in Europe (Menninger, 1992). The word algebra is derived from the Latin translation of the word al-jabr in the title of the book. AlKhowarizmi's name provides the basis for the word algorithm.