Complex numbers

what is the square root of -1?

Well, there is no real number that could be it, but there IS a solution when we go to imaginary numbers.

Square root of negative 1 is denoted by i. In other words, i is such a number that i² = -1. This i is called the imaginary unit.

Imaginary numbers are of the form a + bi, where a and b are real numbers. For example 2 + 9i, 7 - 82i, or -15.5 + 3/4i are imaginary numbers.

They have a real part and an imaginary part. For example, in 2 + 5i, the real part is 2 and the imaginary part is 5.

Imaginary (or complex) numbers are often plotted on the complex plane, which is just like your normal coordinate plane, except that the axes are different. Now the usual x-axis is the real part axis, and the usual y-axis is the imaginary axis.

You can plot them as points...



or as vectors.



It's easy to add or subtract complex numbers; you just add/subtract the real parts and imaginary parts separately.

For example,

(4 + 9i) + (-5 - 3i) = -1 + 6i

This corresponds to usual vector addition.

Multiplication is also very easy: you do it like you'd multiply algebraic expressions, just remembering that i² = -1. For example,

(3 - 4i)(2 + 5i) = 6 + 15i - 8i - 20i² = 6 + 7i + 20 = 26 + 7i.

But what does that sort of multiplication mean?


Let's first multiply some complex numbers by i:

i * (1 + i) = i + i² = - 1 + i.



i * (4 + i) = 4i + i² = -1 + 4i



i * (2 - 3i) = 2i - 3i² = 3 + 2i.



Look! It seems that the number, when multiplied by i, gets rotated 90 degrees counterclockwise.

That's something interesting: multiply a number, and it gets turned!

Here's some more:
(1 + i)*( 2 - 3i) = 2 - 3i + 2i - 3i² = 5 - i.


The multiplier (1 + i) is shown in the picture too. This time the number (2-3i) got rotated 45 degrees and it got longer.

And the last one:
(-0.5 + 0.5i)*(2 - 3i) = -1 + 1.5i + i - 1.5i² = 0.5



This time our number (2-3i) got rotated 135 degrees and got shorter.

It turns out there is an interesting geometric interpretation of multiplication of complex numbers.

If you multiply two complex numbers, (a + bi) and (c + di), you can get the result number by taking (c + di) and rotating it as many degrees as is between (a + bi) vector and the positive real axis (the usual x-axis). Also the length of the result vector is the product of the lengths of the two.

The calculations are much easier done when one represents complex numbers in the polar form. But that's beyond this blogpost for now.

You know, we are so used to multiplying real numbers. Sometimes it's good to expand one's horizons. In mathematics, there are all sorts of "multiplication" operations defined between all sorts of "objects" that aren't even always numbers.

Multiplication of complex numbers isn't the only example by far (multiplication of matrices, or dot and cross products of vectors come to mind as easy examples).

Grade school mathematics never gets to any of that. But believe me, there is a world of fascinating mathematics awaiting those students who get to study beyond arithmetic and beginning algebra.


And NOW... you can... get ready for Dave's Short Course on Complex Numbers or Introduction to the Mandelbrot Set!

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P.S. If you wonder how I drew the pictures, I used the good ol' Microsoft Word with its 'gridlines' turned on and with the option "snap objects to grid", and took a screen capture using Irfanview.

Tags: math, geometry