The birth of symbolic notation

Writing $-5a^3b^2$ feels natural to us, yet the ability to represent quantities with letters and manipulate them with a handful of symbols is a surprisingly recent achievement. For most of its history algebra was rhetorical: equations were written out in full, in sentences of ordinary language.

From “word” algebra to symbolic algebra

Medieval and Renaissance algebraists — from al-Khwārizmī all the way to Cardano and Tartaglia in the sixteenth century — still stated their problems almost entirely in words, using at most a few abbreviations (the syncopated algebra of Diophantus). A sentence such as “the square of a quantity increased by six times the quantity equals seven” was the normal way of stating what we now write as $x^2 + 6x = 7$.

The turning point came with the Frenchman François Viète who, in his In artem analyticem isagoge (1591), introduced the idea of denoting the unknown quantities by vowels ($A$, $E$, $I$, …) and the known ones by consonants ($B$, $C$, $D$, …). For the first time one could write a general formula, independent of the specific numbers: this is the origin of the “literal coefficient” that lies at the foundation of all literal calculus (Boyer; Katz).

The symbols we still use today

Viète’s language was still cumbersome (powers were written out in words, as A quadratum). It was the mathematicians of the following generations who gave it its modern form:

• the Englishman Robert Recorde, in his The Whetstone of Witte (1557), introduced the equals sign $=$, justifying it with the remark that nothing can be more equal than two parallel segments of the same length;
• his countryman Thomas Harriot, in the posthumous Artis analyticae praxis (1631), spread the inequality symbols $<$ and $>$ and the habit of writing powers as repeated products ($aaa$ for $a^3$), a step towards the exponent notation that Descartes would soon make definitive.

So when in this chapter you write a monomial or expand a special product, you are using a mathematical alphabet built up, piece by piece, over several centuries (Boyer; Katz).

Links

Topics: Algebraic calculus
Concepts: Monomial
People: François Viète (Vieta) · Robert Recorde · Thomas Harriot