Introduction

Algebra lets us state a relationship once, using variables (symbols standing for real numbers), and then read off its truth for every particular number by substitution -- this is what makes it so much more powerful than working with one numerical example at a time. This chapter's variables always denote real numbers.

The real number system itself took centuries to pin down. The Pythagoreans already knew that $\sqrt2$ could not be written as a ratio of whole numbers, even though they had no name yet for 'irrational'. Constructions of irrational lengths appear in the Indian Shulba Sutras (c. 800 BCE); Aryabhata (476-550) approximated $\pi$; Brahmagupta (598-670) solved the general quadratic for both positive and negative roots; Bhaskaracharya (1114-1185) handled quadratics in more than one unknown and allowed negative/irrational solutions; and the number zero itself is an Indian contribution to mathematics. In Europe, Rene Descartes (1596-1650) coined the word real to distinguish genuine roots of a polynomial from imaginary ones, and it was Richard Dedekind (1831-1916) who finally gave the real number system a rigorous construction.