The Number Nobody Believed In

Here is the sequence of events, and it is the opposite of how the subject is taught.

Cardano, in 1545, is working on cubic equations. The formula he has produces, for certain cubics with perfectly ordinary real solutions, square roots of negative numbers partway through the calculation. They cancel by the end. The answers come out right and can be checked by substitution.

So there is a quantity in the working that nobody can interpret, on the route to an answer that is unquestionably correct. Cardano, in a different problem, called such a thing manifestly impossible and then wrote: nevertheless, we will operate.

Rafael Bombelli, in 1572, did the work of writing down how to calculate with them. Rules for multiplying expressions involving roots of negatives, consistent and usable, produced by a man who described the objects as sophistical, meaning worthless.

Descartes, in 1637, gave them the name imaginary. He meant it as a dismissal.

Then nothing much, for about a century and a half, except continued use. Euler worked with them extensively and introduced the i notation in the middle of the eighteenth century. Nobody had any account of what they were.

The meaning arrived around 1800, and it arrived twice independently. Caspar Wessel, a Norwegian surveyor, presented a memoir to the Danish Academy; Jean-Robert Argand, a Parisian bookkeeper, published a pamphlet. Neither was a professional mathematician and both had the same idea: put the numbers on a plane. Real numbers along one axis, imaginary along the other, and then multiplication by the imaginary unit is a rotation by a quarter turn.

That is a meaning, not a permission. Once the units are directions, the question of whether they exist dissolves: of course a quarter turn exists. And the strange fact that the imaginary unit squared gives minus one becomes obvious, because two quarter turns are a half turn, which is exactly what multiplying by minus one does.

Gauss introduced the word complex in 1831, and in the same work said the whole confusion would have been avoided if the units had been given names describing their behaviour. The usual rendering of his suggestion is direct, inverse and lateral: along the line, backwards along the line, and sideways off it. The exact wording depends on which translation you read, and the point survives the translation.

We are stuck with Descartes's insult instead, and it still does damage. Every year a large number of students conclude that one branch of mathematics is about numbers that are made up, on the authority of a name coined by somebody who was trying to be rude about them.

Look at the structure of all three lessons in this book, because it is one structure.

Somebody needs an answer. A quantity appears that nobody can interpret. The arithmetic with it turns out to be consistent, so it gets used while being disparaged, often by the same person in the same paragraph. Decades or centuries later a model arrives, debts for the negatives, a plane for the imaginaries, and the quantity is retroactively declared to have been respectable all along.

Use first, meaning afterwards. That is not a failure of rigour, it is how the subject actually grows, and it is a decent reason to be slow about dismissing whatever is currently being called meaningless.