Complex numbers


#1

I explained a little bit about complex numbers a while back and why they are plotted in 2d.

After reading this, how about we calculate the square root of i? Mentally of course.
Here we go. The “square root of i” is a process, when executed 4 times equals -1:

x^4=-1.

In a geometric plane, this is easy: it is 1, rotated by 45 degrees! Do this rotation 4 times and we go from 1 to -1.

From the Pythagorean theorem we know that cos(45) = \sqrt{\frac{1}{2}}.
x=y, so if we create a complex number from this x and y, we get:

\sqrt{\frac{1}{2}} + \sqrt{\frac{1}{2}}i

This is the square root of i. Wolfram Alpha confirms our answer.


#2

Let’s do another one.
(1+i)^2.

(a+b)^2 = a^2 + 2ab + b^2, so:

(1+i)^2 = 1^2 + 2*1*i + i^2 = 1 + 2i - 1 = 2i

Now geographically:

(1+i) x = 1, y = 1
Mentally draw a line from (0,0) to (1,1).
Length = \sqrt 2
angle = 45 degrees

Now double the angle and square the length:
angle = 90 and length = 2.
Imagine a line of length 2 (from (0,0) to (0,2)) and rotate it 90 degrees.
Now we are at 2i.

Try to see this calculation geographically.
See how easy this is?
We double the angle and square the length. Done.


(Josh Cohen) #3

There is an interesting video about complex number in Dimensions Math:


#4

It is indeed a interesting video, Josh.
Robert, this should answer your questions about plotting complex numbers in a plane.
Let me know if something is still unclear.


#5

Thanks as always Kinma,

It may be a weekend or two before I sit down and give this a bit of a think and do some graphing. I haven’t really thought about how this is used when people talk about “rotation” or how it generalizes to circles, ellipses, spirals etc. As a method of representing functions I’m going to have to play with it quite a bit until it becomes more natural to think about. My math study has been fairly nonexistent that last few weeks. I’m back at practicing calculation but haven’t climbed back on the study horse yet.


#6

The main point to understand is that an operation - performed twice - to get from +1 to -1 cannot be found on a number line.

However; if you extend the line to a plane, the operation can be found easily; either rotate clockwise by 90 degrees or rotate anti clockwise by 90 degrees. We call the first ‘i’ and the second ‘-i’.