Wednesday, December 16, 2009

Understanding "Imaginary" Numbers

In grade school, when I learned about negative numbers, I thought they made a lot of sense. Until I learned that multiplying two negatives make a positive number:

-1 * -1 = 1

I recall this seemed completely backwards, intuitively. And most the class agreed. But the teacher explained to everyone that they were wrong. However, the beauty of math is that it really is just a set of rules, made up, out of thin air.

This is just one definition of multiplication. We can imagine a different system of numbers, with a different system of multiplication rules, where the opposite is true:

-1 * -1 = -1

And as long as we can make the rules consistent, each system of numbers is just as valid.

However, if we have two number systems, we of course can't have the same symbol mean two different things or it's hard to read. So we can define the more intuitive '-1' with a special symbol like 'i' (perhaps to stand for something like "intuitive -1").

Then we can connect the two number systems together like:

i * i = -1

Then we can treat 'i' like any other unit of measure. As in: 1i, 2i, 3i, ...

A 'complex number' makes use of both number systems, and can be visualized the same way as two-dimensional XY coordinate system. The difference is that each axis has a different number system attached to it. These can written as a sum:

x = a + bi

The term 'imaginary' is a horribly confusing term though, since these numbers are no more or less imaginary than ordinary negative numbers. If anything, they are based in more intuitive rules about negative numbers.

If anything, you can just think of i as "a more intuitive -1" which follows an alternative multiplication rule: two "intuitive" negative numbers multiplied together equal a "regular" negative number.  This is entirely consistent with the standard rule: two "regular" negatives multiplied together equal a "regular" positive number.


KW said...

I remember my teacher taught me an easy way to spell ARITHMETIC:

I think that's the only thing I remember from all my years of math schooling.

sevkeifert said...

Not a bad trick. I don't think I learned to spell that correctly till high school or later. :)