Visualizing Complex Numbers

Date: 03/19/2001 at 22:30:39
From: Anonymous
Subject: Complex Numbers

What exactly are imaginary numbers and how are they used?  I do not 
quite understand how the square root of -1 is possible.

Thank you.

Date: 03/20/2001 at 09:23:20
From: Doctor Floor
Subject: Re: Complex Numbers

Hi,

Thanks for writing.

This is a difficult question!

Imaginary numbers are all numbers that are the product of a real 
number and the square root of -1. The set of complex numbers is the 
set of numbers that can be found as the sum of a real and an imaginary 
number.

First of all, to understand complex numbers we have to get rid of the 
number line alone. Instead we consider a number plane, with a grid. 
The usual real numbers are found along the x-axis, so that's our usual 
number line. So the point (x,0) in the plane we take for the real 
number x.

To find the square root of -1, we first try to find out what 
multiplying by the square root of -1 should do. We do that by 
geometric interpretation. By such interpretation, multiplying by -1 is 
the same as rotating about the origin through 180 (or -180) degrees. 

Multiplying by the square root of -1 should do this halfway, so that 
after doing it twice it would be the same as multiplying by -1. And 
halfway rotating through 180 (or -180) degrees, is not difficult: 
that's the same as rotating through 90 (or -90) degrees. 

So we can take rotating through 90 degrees as "multiplying by the 
square root of -1" (through -90 degrees is then muliplying by minus 
the square root of -1).

Now, the square root of -1 as a number in the plane is found as 1 
multiplied by the square root of -1, or (1,0) after rotation through 
90 degrees about (0,0), i.e. the point (0,1).

And we see that imaginary numbers will all be found on the y-axis. 
Therefore, when we talk about the plane of complex numbers, the x-axis 
is called the real axis, and the y-axis the imaginary axis.

I hope this helps you.

Best regards,
- Doctor Floor, The Math Forum
  http://mathforum.org/dr.math/