Complex Numbers

Complex Number

A complex number is an extension of a real number whose properties can be used to solve previous unsolvable problems and that model 2D geometry. Complex numbers appear in many engineering and math equations such as the Fundamental Theorem of Algebra, Euler’s Formula and the Mandelbrot Set.

Definition

There are two assumptions, shown below, that form the definition of a complex number. These two assumptions and the properties they exhibit when combined with arithmetic and other math operators form the basis of a complex number.

  1. Assume there is some number so that .
  2. Give a home in the complex plane.

The notation for a complex number is shown below, where and are real numbers and .

Formally, when talking about the set of complex numbers, the symbol is used.

Complex Constant

The first assumption is that there exists some number that satisfies the equation . This by itself can be used in combination with the rules of algebra to solve previously unsolvable problems. However, a lot of the usefulness of a complex number comes from its application to two dimensional geometry in the coordinate plane. So first, we will give the notion of a complex number a home in the complex plane.

Complex Plane

The second assumption is that a complex number has a “home” in the complex plane and can be drawn as an arrow from the origin to some point in the plane. The vertical axis marks the complex part of the number and the horizontal axis marks the real part of the number. For example, the complex number is drawn as an arrow starting from the origin and going units to the right and units up as shown in the figure below.

This figure illustrates the complex plane. The horizontal axis represents the real part of a complex number and the vertical axis represents the complex part of a complex number.
Figure 1: Complex Plane

Now that we have established the foundation of a complex number, we will look at some of the basic operations using complex numbers from an algebraic and geometric persepctive.

Complex Addition

The properties of addition for two complex numbers follow the rules of algebra. The real parts of the number are added together and the complex parts of the number are added together.

Visually, addition is represented by drawing the two arrows corresponding to each of the complex numbers “tail-to-tail” as shown below. The result of the addition is the arrow drawn from the origin to the tip of the second arrow.

Complex plane addition

Complex Multiplication

The properties of complex number multiplication demonstrate why complex numbers are able to elegantly express an area of mathematics. Conceptually, multiplying by a complex number corresponds to stretching and rotating a notation of the complex plane.

Identity Property

Multiplying any complex number by results in . Visually, this correspond with stretching and rotating the number so that it lies on top of the complex number in the complex plane. Conceptually, you can imagine stretching and rotating all of the horizontal and vertical axis lines in the first plane.

This figure illustrates visual representation multiplying 1 by a complex number
Figure 2: Complex Plane (Identity Property)

Complex Property

The second property of complex number mutliplication is multiplying a number by is equivalent to rotating by a quarter rotation. For example, the complex number multiplied by is equal to the rotation shown below:

This figure illustrates visual representation of multiplying a complex number by the complex number i.
Figure 3: Complex Plane (Complex Property)

Let’s verify this geometric property using algebra. Start with the complex number and multiply by .

Distribute the multiplication across the real and complex part of the number.

Substitute for in the expression.

Simplify and rearrange the expression.

This verifies the rotation property for the example complex number. This property is where the usefulness of complex numbers starts to appear.

Distributive Property

The distributive property combines the first two properties and represents the operation of multiplying two complex numers together. Multiplication can be broken into two steps. (1) Stretch and rotate the coordinate system using the first complex number. (2) Use the transformed coordinate system to draw the second vector whose coordinates represent the result of the multiplication.

For example, let’s multiply the complex numbers and together as written below.

First visualize the stretch and rotation of the coordinate system using the first number . This gives us the blue coordinate system shown below.

Complex Number Multiplication Step 1

Then, draw the complex number arrow using this new coordinate system. So, using the components of the second complex number, travel units along horizontal axis and units along the vertical axis in the transformed coordinate system. This is shown in red below.

This figure illustrates visual representation of multiplying two complex numbers together.
Figure 4: Complex Plane Multiplication

The end result in the original coordinate system is the complex number .

Let’s verify this result using algebra. Start by distributing the multiplication across the terms.

Combine like terms.

Substitute for in the expression.

Evaluate the addition.

This result verifies the geometric interpretation of complex multiplication for this example.

Trigonometry

The fact that multiplying two complex numbers together results in a stretch and rotation in the complex plane suggestively hints that complex numbers may be useful when applied to trigonometry. Recall, a points position on the unit circle can be defined in terms of the cosine and sine of the radian angle. A point on the unit circle corresponding to a radian angle then takes the form:

This piont is visualized by the figure below.

A point on the unit circle in the complex plane.
Figure 5: Complex Unit Circle

Trigonometric Identities

The trigonometric identities are a set of equations that are used to transform and manipulate math expressions. Assuming that you have had exposure to trigonometry and these identities, this is one of the areas where the usefulness of complex numbers shines. The complex plane represents an elegant way to think about and transform trigonometric expressions.

Derive Sum of Two Angles Identities (Complex Plane)

The sum of two angles identities can be derived using the properties of the complex plane and Euler's formula.

Derive Difference of Two Angles Identities (Complex Plane)

Derive the difference of two angle identities using the properties of complex numbers and Euler's formula in the complex plane.

Derive Half Angle Identities (Complex Plane)

This example derives the trig. half angle identities in the complex plane.

Derive Double Angles Identities (Complex Plane)

The trigonometric double angle identites can be derived using the properties of the complex plane.

Euler’s Formula

Perhaps the most notorious formula in math and physics is Euler’s Formula as shown above.

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