An Arabic Finger-reckoning Rule Appropriated for Proofs in Algebra

Introduction

Precalculus students are typically subjected to three different methods of solving quadratic equations: factoring, completing the square, and the quadratic formula. The latter two are related in that the quadratic formula is regularly proven by completing the square in the general case. And completing the square, for its part, relies on the formula for squaring a binomial: \((a+b)^2=a^2+2ab+b^2\). This rule, in one form or another, lies behind many derivations and proofs for the rules to solve quadratic equations throughout the history of algebra. (See Note 1, below.)

The binomial formula is of course not the only rule that has served this purpose. Since the ninth century many algebraists have appealed instead to propositions II.5 and II.6 of Euclid's Elements, sometimes in an arithmetical form, to justify the solutions to quadratic equations. (See Note 2, below.) In this article I describe yet another way. In a 1301 book the Moroccan scholar Ibn al-Bannāʾ based one set of proofs on a common shortcut for mental multiplication. But before plunging into those proofs I should first describe their background with respect to arithmetic, algebra, and geometry. And this, in turn, must be prefaced with an account of just what numbers and magnitudes were for medieval authors.

Note 1. In Arabic algebra the most famous example is the second of al-Khwārizmī's proofs for his type 4 equation. Others are the derivations given in al-Karajī's al-Fakhrī that he called the "method of Diophantus", and the set of proofs in Ibn al-Bannāʾ's algebra book (described below).

Note 2. The earliest Arabic authors we know to do this are Abū Kāmil and Thābit ibn Qurra. Arithmetical versions of these propositions appear in books at least as early as the late twelfth century [Oaks 2018].