Bombelli and the roots of negative numbers

When we impose the existence condition $a\ge 0$ for $\sqrt{a}$, we are saying that, among the real numbers, the square root of a negative number does not exist. For most of the history of mathematics this was enough: $\sqrt{-1}$ was regarded as a meaningless symbol. The first to take it seriously, out of sheer computational necessity, was Rafael Bombelli (1526–1572).

The problem arises from cubic equations

The impetus came not from radicals themselves, but from third-degree equations. The solution formula published by Gerolamo Cardano in the Ars Magna (1545) — also drawing on the work of Niccolò Tartaglia and of Scipione del Ferro — displayed a baffling phenomenon, the so-called casus irreducibilis: certain equations with three real solutions, all perfectly ordinary, could be obtained from the formula only by passing through square roots of negative numbers.

A classic example is $x^3 = 15x + 4,$ which has the obvious solution $x=4$. Yet Cardano’s formula gives $x = \sqrt[3]{2 + \sqrt{-121}} + \sqrt[3]{2 - \sqrt{-121}}.$ The quantity $\sqrt{-121}$ appears: an “impossible” quantity that seemed to doom the formula.

Bombelli’s bold idea

In his Algebra (1572) Bombelli had the insight of treating $\sqrt{-1}$ as a new kind of quantity, with its own sign rules that he called “plus of minus” and “minus of minus”. He conjectured that the two cube-root terms had the form $2\pm\sqrt{-1}$ (that is, $2\pm i$ in today’s language) and verified by direct computation that $(2+\sqrt{-1})^3 = 2 + \sqrt{-121}, \qquad (2-\sqrt{-1})^3 = 2 - \sqrt{-121}.$ Adding the two terms, the imaginary parts cancel and there remains exactly $x = (2+\sqrt{-1}) + (2-\sqrt{-1}) = 4,$ the expected real solution (Boyer; Katz).

Why it matters

Bombelli did not really “believe” in the existence of these numbers — he called them sophistic quantities — but he showed that, handled with consistent rules, they led to correct real results. This was the first step towards the complex numbers, which only with Euler and Gauss, two centuries later, would gain full legitimacy. The condition $a\ge 0$ remains valid for the real radicals you study in this chapter; Bombelli’s story shows, however, that by enlarging the number system, even the “impossible” $\sqrt{-1}$ can become a powerful tool.

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Topics: Radicals Concepts: Radical · Existence conditions People: Rafael Bombelli · Gerolamo Cardano · Niccolò Tartaglia