Complex Variables - Lesson 002 - Complex Numbers
For anyone who actually read Complex Variables - Lesson 001, this might be confusing, but idk. I'm trying to go by the book now, but technically both lessons are correct. Let me know what doesn't make sense and I will fix it ASAP.
Using Complex Variables and Applications, 8th Ed, by Brown and Churchill as a basis for these posts until further notice...
Complex numbers are described as being rectangular ordered pairs of real numbers (x,y) that are plotted in the complex plane. In the complex plane, the horizontal or x-axis is the real axis. The vertical or y-axis is the imaginary axis. Points of the form (x,0) correspond to pure, real values and are plotted on the real axis. Points of the form (0,y) correspond to pure, imaginary values when y ≠ 0 (because if y were to equal 0, our values for y would be plotted on the real axis) and are plotted on the imaginary axis.
Complex numbers are generally represented by z, where z = (x,y). Because x represents the real component of the complex number, and y represents the imaginary component of a complex number, we say that:
x = Re z, y = Im z
And any complex number can be written as:
z = (x,0) + (0,y)
And (0,y) = (0,1)(y,0). We replace this because y is part of a real ordered pair, where both x and y are given real values, and we stated that the left value in an ordered pair is the real value. So z becomes:
z = (x,0) + (0,1)(y,0)
If we let (0,1) represent the pure imaginary number i on the complex plane, z becomes:
z = (x,0) + i(y,0)
And at this point, because we are big kids, we can drop the 0's:
z = x +iy
This shows how we can use real values for y and yet are still able to plot them on an imaginary axis - an i is attached to the values. *Note: This is also why Im z = y, not iy*
Oh God no one is going to understand this crap. If you have any questions, don't hesitate to ask.