1. Complex Numbers

Hey,

Had a lecture about complex numbers today, and I've been trying to figure the whole thing out and visualize it graphically. Please tell me if I am correct:

Real numbers are included in the complex numbers and can be thought of as vectors of varying length coinciding with the x-axis (ReZ-axis). So all real numbers are sort of one dimensional (along one line), while complex numbers have two dimensions - one real dimension and one imaginary.

This means that it is possible to get a real number even though you're adding, subracting, multiplying or dividing complex numbers that have both real and imaginary parts. But that will only happen when the imaginary coordinate is zero and the resulting vector coincides with the x-axis.

2. All that sounds good to me.

3. Great. One question though. What is the significance of two complex numbers, drawn as vectors, having the same length?

If you draw a circle with its center in the origin and draw a radius to some random point in the circle, it will correspond to a complex number given as the x-coordinate plus the y-coordinate (or z=a+bi). But at the same time there will be an infinite number of other complex numbers (corresponding to the points of the circle) that have the same length vector. And we get those by altering the the angle.

In addition, for any given vector there will be three others around the circle that have the same coordinates (in absolute value).

I'm sure this must mean something, I just don't know what exactly.

4. There'll actually be uncountably infinitely many vectors of the same length as a given vector. You can rotate through any angle in the interval $[0,2\pi].$

As for significance, that would probably depend on the application. Rotations are important. For example: if your energy function doesn't depend on a particular angular variable, then the angular momentum corresponding to that angle is conserved.

5. Originally Posted by TwoPlusTwo
Hey,

Had a lecture about complex numbers today, and I've been trying to figure the whole thing out and visualize it graphically. Please tell me if I am correct:

Real numbers are included in the complex numbers and can be thought of as vectors of varying length coinciding with the x-axis (ReZ-axis). So all real numbers are sort of one dimensional (along one line), while complex numbers have two dimensions - one real dimension and one imaginary.

This means that it is possible to get a real number even though you're adding, subracting, multiplying or dividing complex numbers that have both real and imaginary parts. But that will only happen when the imaginary coordinate is zero and the resulting vector coincides with the x-axis.
Yes, (4+ 3i)+ (2- 3i)= 6, a real number. (4+ 3i)- (2+ 3i)= 2, a real number. (2- 3i)(2+ 3i)= 4- 6i+ 6i+ 3= 7, a real number.
$\frac{2- 3i}{\frac{2}{7}- \frac{3}{7}i}= \frac{2- 3i}{\frac{1}{7}(2- 3i)}= 7$, a real number.