Geometry of Complex Numbers

Complex numbers can be represented in both rectangular and polar coordinates. All complex numbers can be written in the form a + bi, where a and b are real numbers and i 2 = −1. Each complex number corresponds to a point in the complex plane when a point with coordinates ( a, b) is associated with a complex number a + bi. In the complex plane, the x‐axis is named the real axis and the y‐axis is named the imaginary axis.

Example 1: Plot 4− 2 i −3 + 2 i, and −5 − 3 i in the complex plane (see Figure  1).


Figure  1
             Complex numbers plotted in the complex plane.

Complex numbers can be converted to polar coordinates by using the relationships x = r cos θ and y = r sin θ. Thus, if z is a complex number:

Sometimes the expression cos θ + sin θ is written as cis θ. The absolute value, or modulus, of z is . The angle formed between the positive x‐axis and a line drawn from the origin to z is called the argument or amplitude of z. If z = x + iy is a complex number, then the conjugate of z is written as z = xiy

Example 2: Convert the complex number 5 − 3 i to polar coordinates (see Figure  2).

Figure  2
               Drawing for Example 2.  

Reference angle θ ≈ 31°.

Since θ is in the fourth quadrant,


To find the product of two complex numbers, multiply their absolute values and add their amplitudes.

To find the quotient of two complex numbers, divide their absolute values and subtract their amplitudes.

Example 3: If z = a(cosα + isinα) and w = b(cosβ +isinβ), then find their product zw.

Example 4: If z = a(cosα + isinα) and w = b(cosβ + isinβ), then find their quotient z/w.

Example 5: If z = 4(cos 65° + i sin 65°) and w = 7(cos 105° + i sin 105°), then find zw and z/w.