Before Abel and Galois

Why do students struggle at school with those famous "quadratic equations" and their discriminant, "delta"? For centuries, scholars sought to solve algebraic equations, believing that general formulas must exist for expressing their solutions in terms of the coefficients. With degree 2, they seemed to be off to a good start. But mathematical reality does not always bend to our wishes…

An equation is called algebraic if it involves only integer powers of the unknown, multiplied by coefficients. In modern notation (see box), such an equation of degree n is written *a_nx ⁿ + a_n*₋₁*x ⁿ*⁻¹+ … + a₁x + a₀ = 0, where the coefficients *a_n, a_n*₋₁… a₁ and a₀ are real numbers, with *a_n* nonzero.

Thus, *x ³ − 3*x ² + 4x − 1.34 = 0 is an algebraic equation of degree 3. Similarly, x⁷ − πx + 1 = 0 is one of degree 7, whereas *x ² + x − sin(x) = 0 and x ^2/3 – 3 = 0 are not algebraic equations.

The value of good notation ---------------------------

Writing an algebraic expression such as *x ² + 2x + 3 seems entirely natural to us. But it took a long time to arrive at this form, which now appears elementary. In the time of Diophantus of Alexandria (before the 3^rd century), it was written Δ^Yα ς β Ṁ γ, where Δ^Y denotes the square x², ς (the arithmos, "the number") denotes the unknown x, and Ṁ indicates that the following number contains no unknown; the lowercase letters encode numerical values (α = 1, β = 2, γ = 3…). There is no minus sign. We would undoubtedly struggle to do algebra with this notation today!

In the 9^th century, the Arab mathematician al-Khwarizmi (c. 780–c. 850) called this unknown shay, meaning "the thing." In equations, it was represented by the word’s first letter: "sh." In Spain, however, non-Arabic-speaking Andalusians wrote it in Latin characters, producing "xay."

After various attempts to simplify notation, notably by Nicolas Chuquet (c. 1445–c. 1488), Luca Bartolomes Pacioli (c. 1445–1517), and François Viète (1540–1603), René Descartes shortened "xay" to its initial letter, "x." He also established the convention of using lowercase letters from the end of the Latin alphabet for unknowns and letters from the beginning for known quantities (coefficients, data, parameters…).