[ExI] imaginary numbers

John Clark jonkc at bellsouth.net
Sun Nov 28 05:02:06 UTC 2010


On Nov 27, 2010, at 3:39 PM, Mike Dougherty wrote:
> 
> "if you put real numbers on X and real numbers on Y then the product is the number of unit squares that cover the area.  So a 5 x 5 square is literally a 25 unit square. A 5i x 5i square is a negative 25 unit square?"

On the complex plane, (also called the Argand plane) where the horizontal axis is the real numbers and the vertical axis is the imaginary numbers and you multiply 2 complex numbers together, you don't get an area you get another complex number, or to use another name for the same thing, you get another vector. In your example of 5i times 5i is indeed negative 25, and that is just a vector pointing straight down with the magnitude of 25.

In your example the real part is zero but suppose we want to multiply a more interesting complex number by itself, 3+4i for example. You can do the formal vector arithmetic but even before you do that you can get a feel for what the resulting vector of the multiplication must look like, all you need to do is add the angles and multiply the magnitude of the 2 vectors. 

The magnitude of 3+4i is 5 because the square root of (3^2 + 4^2) is 5, so the magnitude of the vector we are looking for, the magnitude of the vector that results from multiplying 3+4i by itself, must be 25 because 5*5=25. 

It's only slightly more difficult to quickly and intuitively estimate the angle. The angle of 3+3i would be 45 degrees so the angle 3+4i must be more than 45 degrees but less than 90, so the angle of the vector we're looking for must be twice that, or more than 90 degrees but less than 180. 

We can now do the actual arithmetic and see if our estimate was correct.
(3+4i)*(3+4i) = 9+16i^2 +24i = −7 +24i because i^2=-1. So −7 +24i is the complex number, or vector we want, and the magnitude of that is the square root of (−7 * -7 plus 24 * 24) and that is equal to the square root of 625, and that is equal to 25. So that part of our guess was correct.

To calculate the angle of the vector −7 +24i we use trigonometry and the arctangent of 24/-7, and we get 106.2 degrees, so that part of our quick estimate was correct too.

 John K Clark






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