08 Imaginary numbers
We can certainly imagine numbers. Sometimes I imagine my bank account is a million
pounds in credit and there’s no question that would be an ‘imaginary number’. But the
mathematical use of imaginary is nothing to do with this daydreaming.
The label ‘imaginary’ is thought to be due to the philosopher and
mathematician René Descartes, in recognition of curious solutions of equations
which were definitely not ordinary numbers. Do imaginary numbers exist or not?
This was a question chewed over by philosophers as they focused on the word
imaginary. For mathematicians the existence of imaginary numbers is not an
issue. They are as much a part of everyday life as the number 5 or π. Imaginary
numbers may not help with your shopping trips, but go and ask any aircraft
designer or electrical engineer and you will find they are vitally important. And
by adding a real number and an imaginary number together we obtain what’s
called a ‘complex number’, which immediately sounds less philosophically
troublesome. The theory of complex numbers turns on the square root of minus
- So what number, when squared, gives −1?
If you take any non-zero number and multiply it by itself (square it) you
always get a positive number. This is believable when squaring positive numbers
but is it true if we square negative numbers? We can use −1 × −1 as a test case.
Even if we have forgotten the school rule that ‘two negatives make a positive’ we
may remember that the answer is either −1 or +1. If we thought −1 × −1
equalled −1 we could divide each side by −1 and end up with the conclusion that
−1 = 1, which is nonsense. So we must conclude −1 × −1 = 1, which is
positive. The same argument can be made for other negative numbers besides
−1, and so, when any real number is squared the result can never be negative.
This caused a sticking point in the early years of complex numbers in the 16th
century. When this was overcome, the answer liberated mathematics from the
shackles of ordinary numbers and opened up vast fields of inquiry undreamed of
previously. The development of complex numbers is the ‘completion of the real
numbers’ to a naturally more perfect system.