Review of Complex Numbers

Overview

Complex numbers are defined as numbers that have a real component and an imaginary component. The real component is simply a real number, which you should already be familiar with. The imaginary component, on the other hand, is a real number multiplied by a constant i, which represents \sqrt{-1}. The real component is usually represented by a, while the imaginary component is represented by bi, the real number b multiplied by i. Thus, complex numbers take the form a+bi.

Complex Arithmetic

Addition and Subtraction

Adding and subtracting complex numbers is straightforward. Given x=a_1 + b_1 i and y=a_2 + b_2 i, we get:

x+y = (a_1 + a_2) + (b_1 + b_2)i
x-y = (a_1 - a_2) + (b_1 - b_2)i

Multiplication

Multiplication of two complex numbers is similar to multiplying two binomials using the FOIL method. In short, if x and y are defined as above, then x \times y = (a_1 a_2 - b_1 b_2) + (b_1 a_2 + a_1 b_2)i.

Plotting Complex Numbers

When plotting complex numbers on a 2-dimensional plane, a direct mapping to normal x-y coordinates is used. That is, if you draw an x- and y-axis, you can plot a+bi by plotting the point (a,b) (basically, the x- and y- axes become a- and b- axes).

Magnitude of Complex Numbers

The magnitude of a complex number is defined by the distance between it and zero when plotted on a plane. Therefore, the magnitude of a+bi is \sqrt{a^2 + b^2}.