15.4 Complex Numbers

The numbers you are most familiar with are called real numbers. These include numbers like 4, 275, -200, 10.7, ½, π, and so forth. All these real numbers can be plotted on a number line. For example, if we wanted to show the number 3, we plot a point:

A number line marked from -5 to 5 in steps of 1, with a single blue dot plotted on 3.

To solve certain problems like $x^{2} = - 4$ , it became necessary to introduce imaginary numbers.

To plot a complex number like $3 minus 4 i$ , we need more than just a number line since there are two components to the number. To plot this number, we need two number lines, crossed to form a complex plane.

Because this is analogous to the Cartesian coordinate system for plotting points, we can think about plotting our complex number $z = a + b i$ as if we were plotting the point $(a, b)$ in Cartesian coordinates. Sometimes people write complex numbers as $z = x + y i$ to highlight this relation.

Arithmetic on Complex Numbers

Before we dive into the more complicated uses of complex numbers, let’s make sure we remember the basic arithmetic involved. To add or subtract complex numbers, we simply add the like terms, combining the real parts and combining the imaginary parts.

When we add complex numbers, we can visualize the addition as a shift, or translation, of a point in the complex plane.

We can also multiply complex numbers by a real number, or multiply two complex numbers.

Example 5

Multiply: $4 open parenthesis 2 plus 5 i close parenthesis$

Show solution

To multiply the complex number by a real number, we simply distribute as we would when multiplying polynomials.

$\begin{matrix} 4 (2 + 5 i) & Distribute \\ = 4 \cdot 2 + 4 \cdot 5 i & Simplify \\ = 8 + 20 i \end{matrix}$

Example 6

Multiply: $(2 + 5 i) (4 + i)$

Show solution

To multiply the complex number by a complex number, we simply distribute as we would when multiplying polynomials.

$\begin{matrix} (2 + 5 i) (4 + i) & Expand \\ = 8 + 20 i + 2 i + 5 i^{2} & Since i = \sqrt{- 1}, i^{2} = - 1 \\ = 8 + 20 i + 2 i + 5 (- 1) & Simplify \\ = 3 + 22 i \end{matrix}$

To understand the effect of multiplication visually, we’ll explore three examples.

Example 8

Complex plane with axes labelled real and imaginary, both running from about -4 to 4. Dashed red rays join the origin to the plotted points 1 plus 2i and -2 plus i, and a solid red arc with an arrowhead sweeps counter-clockwise from the first ray to the second, showing that multiplying by i turns the point 90 degrees about the origin while its distance from the origin is unchanged. Visualize the product $i open parenthesis 1 plus 2 i close parenthesis$

Show solution

Multiplying, we’d get

$i \cdot 1 + i \cdot 2 i = i + 2 i^{2} = i + 2 (- 1) = - 2 + i$

In this case, the distance from the origin has not changed, but the point has been rotated about the origin, 90° counter-clockwise.

Example 9

Complex plane with axes labelled real and imaginary, both running from -6 to 6, carrying four plotted points each joined to the origin by a dashed red ray: 1 plus 2i, then -1 plus 3i, then -4 plus 2i, then -6 minus 2i. Each point is the one before it multiplied by 1 plus i, so every step swings the ray a further 45 degrees counter-clockwise and lengthens it. Visualize the result of multiplying $1 plus 2 i$ by $1 plus i.$ Then show the result of multiplying by $1 plus i$ again.

Show solution

Multiplying $1 plus 2 i$ by $1 plus i$ ,

$(1 + 2 i) (1 + i) = 1 + i + 2 i + 2 i^{2} = 1 + 3 i + 2 (- 1) = - 1 + 3 i$

Multiplying by $1 plus i$ again,

$(- 1 + 3 i) (1 + i) = - 1 - i + 3 i + 3 i^{2} = - 1 + 2 i + 3 (- 1) = - 4 + 2 i$

If we multiplied by $1 plus i$ again, we’d get $– 6 – 2 i$ . Plotting these numbers in the complex plane, you may notice that each point gets both further from the origin, and rotates counterclockwise, in this case by 45°.

In general, multiplication by a complex number can be thought of as a scaling, changing the distance from the origin, combined with a rotation about the origin.

import cmath
import math
import matplotlib.pyplot as plt

Z = complex(1, 2)      # EDIT: the starting point, 1 + 2i
W = complex(1, 1)      # EDIT: what we keep multiplying by, 1 + i
STEPS = 4              # EDIT

def show(c):
    a, b = c.real, c.imag
    sign = "+" if b >= 0 else "-"
    return f"{a:g} {sign} {abs(b):g}i"

print(f"start z = {show(Z)},  multiplier w = {show(W)}")
print(f"|w| = {abs(W):.4f}  so every multiplication stretches distance by {abs(W):.4f}")
print(f"arg(w) = {math.degrees(cmath.phase(W)):.1f} degrees  so every multiplication turns "
      f"the point by that angle\n")

print(f"{'k':>3}{'z * w^k':>18}{'distance from 0':>18}{'angle (deg)':>14}")
points = [Z]
for k in range(STEPS + 1):
    p = points[-1] if k == 0 else points[-1] * W
    if k:
        points.append(p)
    print(f"{k:>3}{show(p):>18}{abs(p):>18.4f}{math.degrees(cmath.phase(p)):>14.1f}")

print(f"\naddition instead of multiplication - a slide, not a turn:")
shift = complex(-1, 5)          # EDIT
print(f"   ({show(Z)}) + ({show(shift)}) = {show(Z + shift)}"
      f"   (moved {shift.real:g} across and {shift.imag:g} up)")

fig, ax = plt.subplots(figsize=(6.0, 5.4))
xs = [p.real for p in points]
ys = [p.imag for p in points]
ax.plot(xs, ys, "o-", color="#b03a2e")
for k, p in enumerate(points):
    ax.annotate(f" w^{k}", (p.real, p.imag), fontsize=10)
    ax.plot([0, p.real], [0, p.imag], ":", color="gray", lw=0.8)
ax.axhline(0, color="black", lw=1); ax.axvline(0, color="black", lw=1)
ax.set_xlabel("real axis"); ax.set_ylabel("imaginary axis")
ax.set_title(f"multiplying {show(Z)} by {show(W)}, over and over")
ax.set_aspect("equal"); ax.grid(alpha=0.3)
plt.show()
# Try it: set W = complex(0, 1). Pure rotation - the distance never changes.

Adapted from Math in Society by David Lippman, hosted on LibreTexts (math.libretexts.org) and licensed under CC BY-SA 3.0. Changes were made. License: CC-BY-SA-3.0.