Zero: The Biography of a Dangerous Idea

by Charles Seife

Descartes thought that these numbers were even worse than negative numbers; he came up with a scornful name for the square roots of negatives: imaginary numbers. The name stuck, and eventually, the symbol for the square root of –1 became i. Algebraists loved i. Almost everyone else hated it. It was wonderful for solving polynomials—expressions like x3 + 3x + 1 that have x raised to various powers. In fact, once you allow i into the realm of numbers, every polynomial becomes solvable: x2 + 1 suddenly splits into (x – i)(x + i)—the roots of the equation are +i and –i. Cubic expressions like x3 – x2 + x – 1 split three ways, such as (x – 1)(x + i)(x – i). Quartic expressions—ones with a leading x4 term—always split into four terms, and quintics—ones with a leading x5 term—split five ways. All polynomials of degree n—those that have a leading term of xn—split into n distinct terms. This is the fundamental theorem of algebra. As early as the sixteenth century, mathematicians were using numbers with i included—the so-called complex numbers—to solve cubic and quartic polynomials. And while many mathematicians saw the complex numbers as a convenient fiction, others saw God. Leibniz thought that i was a bizarre mix between existence and nonexistence, something like a cross between 1 (God) and 0 (Void) in his binary scheme. Leibniz likened i to the Holy Spirit: both have an ethereal and barely substantial existence. But even Leibniz didn’t realize that i would finally reveal the rel...