Fermat's Enigma

by Simon Singh

The problem began by asking, What is the square root of one, ? The obvious answer is 1, because 1 × 1 = 1. The less obvious answer is –1. A negative number multiplied by another negative number generates a positive number. This means –1 × –1 = +1. So, the square root of +1 is both +1 and –1. This abundance of answers is fine, but then the question arises, What is the square root of negative one, ? The problem seems to be intractable. The solution cannot be +1 or –1, because the square of both these numbers is +1. However, there are no other obvious candidates. At the same time completeness demands that we must be able to answer the question. The solution for Bombelli was to create a new number, i, called an imaginary number, which was simply defined as the solution to the question, What is the square root of negative one? This might seem like a cowardly solution to the problem, but it was no different to the way in which negative numbers were introduced. Faced with an otherwise unanswerable question, the Hindus merely defined –1 as the solution to the question, What is zero subtract one? It is easier to accept the concept of –1 only because we have experience of the analogous concept of “debt,” whereas we have nothing in the real world to underpin the concept of an imaginary number. The seventeenth-century German mathematician Gottfried Leibniz elegantly described the strange nature of the imaginary number: “The imaginary number is a fine and wonderful recourse of the divi...