Friday, February 6, 2009

Imaginary Numbers - a metaphor

One of my favourite mathematical metaphors comes from an area called "complex analysis".

I can't resist the metaphor, because of the nature of the mathematical language involved.

I've always loved mathematics. It is enchanting, beautiful, yet infinitely challenging. There is no area within mathematics that cannot be developed into an almost impossibly esoteric branch of its own. Its theoretical abstraction must surely exceed the complexity of the physical universe (unless we consider abstract mathematical ideas to actually be part of the physical universe).

The appreciation of mathematics as an art form or as a form of esthetics has, unfortunately, been hampered by an educational approach which often leads people to experience mathematics with dread, anxiety, or despair.

Anyway, here is the mathematics:

1) A "square root" of a number is another, smaller number, which, when multiplied by itself, gives the first number. So, for example, the square root of 25 is 5--since 5 times 5 equals 25. I suppose we could add that 25 in fact has 2 square roots, since (-5) times (-5) also equals 25. This idea of a "second" square root already involves a higher degree of abstraction.

2) There are some numbers which do not seem to have any possible square root. For example, what would be the square root of (-25)? There does not seem to be any number which, when multiplied by itself, yields a negative number.

3) So, how about if we create such a number, imaginatively? Such a number has been invented, and it is called the "imaginary number", signified as "i". The imaginary number i is an abstraction, with the property that i times i equals -1.

4) What is the use of having such an imaginary number? What application could it have? Well, as it turns out, it is enormously useful in understanding and solving problems in physics and engineering. And, I think, it demonstrates a very beautiful link between phenomena that might initially seem completely different.

The exponential functions are phenomena which, if represented graphically, appear to represent rapidly accelerating growth. If something keeps doubling regularly, the growth is "exponential". Many phenomena in nature can be described using exponentials.

The trigonometric functions are phenomena which, if represented graphically, appear to represent waves, which oscillate regularly; in the case of the "sine" function, we have a "sine wave", which fluctuates, forever, between -1 and 1. Many other phenomena in nature can be described using the trigonometric functions.

There appears, at first sight, to be no obvious relationship between the exponentials, which represent unbridled growth (e.g. population growth); and the sine wave, which represents continuous, regular, well-bounded waves (e.g. the swinging of a pendulum).

But if we figure out a formula which can calculate an exponential, and a formula which can calculate a sine wave function, we find that if "imaginary numbers" are allowed, the two types of functions are variations of the same thing. Hence we have the mathematical fact:
exponential (ix) = cos(x) + i sine(x).

So here is the psychological metaphor:

The link between something which rises, escalates, explodes upwards towards infinity, and something which is stable, repetitious, and finite -- is "imagination". They are variants of the same, larger, thing, as long as you can expand your perspective of understanding.

The introduction of imagination may transform a problem of unbridled excess into one which could include stable regularity. Similarly, imagination could transform the monotony of a "sine wave" type of life into something more excitedly or wildly "exponential".

In approaching seemingly impossible life problems, I think it is important to be able to step back, and sometimes to allow an entirely new perspective or way of thinking.

No comments: