Professor Steven Strogatz of Cornell has published another installment in his mathematics series in the “Opinionator” blog at the New York Times. In this post, he talks about solving for X, as he promised last time; this is just another way of referring to the process of finding the roots of an algebraic equation. That is, if we have:
f(x) = x² – 4
we want to find the values of x that make f(x) = 0. In this case, there are two such values: 2 and -2. But there is another, very similar function that will give us a bit of a problem:
g(x) = x² + 4
In order for the value of g(x) to be zero, x must be a number which, when squared, gives -4. This is a problem, of course, because we know that a negative number multiplied by another negative number gives a positive number. So it would seem that there is no solution to our second equation. For many centuries, mathematicians thought that was that; the square root of a negative number simply did not exist.
Of course, if you have been reading Prof. Strogatz’s earlier postings, you know how this is going to turn out (I also more or less gave the game away in an earlier post on this series):
We’ve seen crises like this before. They occur whenever an existing operation is pushed too far, into a domain where it no longer seems sensible. Just as subtracting bigger numbers from smaller ones gave rise to negative numbers and division spawned fractions and decimals, the free-wheeling use of square roots eventually forced the universe of numbers to expand…again.
This expansion, the inclusion of the imaginary numbers, of the form b i, where b is areal number and i is defined such that i ² = -1, was in many ways the most disturbing expansion of all. Augustus DeMorgan, one of the pioneers of modern algebra, acknowledged the possibility of imaginary numbers, but called them “unintelligible”. There is some evidence that Lewis Carroll (who in real life was Charles Dodgson, a tutor in mathematics at Christ Church college, Oxford) used part of Alice’s Adventures in Wonderland as a satirical attack on some of this new-fangled algebra.
Yet, as Prof. Strogatz points out, imaginary numbers, and the more important complex numbers that they allow, have turned out to be a very handy tool. Engineers, particularly electrical engineers, find them incredibly useful for the analysis of things like alternating electric current and the lift on aircraft wings. Complex numbers, which can be represented as a + b i, where a and b are real numbers, and i is defined as above, encompass the complete number menagerie. The set of complex numbers is closed under all arithmetic operations, meaning that the result of doing arithmetic on two complex numbers is always a complex number. (It should be obvious that all the other sets of numbers — integers, rational numbers, and real numbers, for example — are subsets of the complex numbers.)
Once again, we have an example of what Prof. Eugene Wigner called “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” in his famous essay. In a sense, the complex numbers were always there, just waiting for us to discover that we needed them.
It seems fitting to close by mentioning Euler’s Identity, which has been called the most beautiful equation in mathematics:
e i π + 1 = 0
This ties together our friend i with π, the ratio of the circumference of a circle to its diameter, and with e, the transcendental number that is the base of natural logarithms. We have come rather a long way from counting rocks, but somehow it all hangs together — and we study it not least because it is interesting.