Imagining Numbers

The square roots of negative numbers are called 'imaginary'. But do they really require any more imagination than other types of numbers, or other types of things. In Imagining Numbers Particularly √-15 Barry Mazur tries to find out.

Mazur compares the way we think of numbers with the way we might imagine something from a line of poetry. He discusses how an imagined 'yellow of the tulip' still seems to have more substance than an abstract number, and goes on to consider how mathematicians come to accept things like imaginary numbers. Do they just get used to them, or is it the case that new visualisations make them seem more acceptable. And who decides that such things are 'allowed'. Of course, consistency with the rest of mathematics is an important feature, and Mazur explains how this leads to something which puzzles many schoolchildren - why minus times minus equals plus. Later in the book Mazur looks in more detail at the origins of imaginary numbers - how they were useful in solving cubic equations but how a couple of centuries had to pass before the idea of extending the number line to the complex plane took hold.

All in all it's a strange sort of a book, obviously aimed at those without much experience of mathematics - probably those of a more literary frame of mind. But I wouldn't say that it's really for those wanting to understand imaginary numbers better, rather for those wishing to better understand how mathematicians get to understand such numbers.