Complex Numbers

The complex number (3 + 2i) in the complex plane.

Complex numbers are an extension of the real number system with useful properties that model two-dimensional space and trigonometry. A complex number is written as and is visualized as a point in the complex plane with the horizontal component of and a vertical component of . The complex constant satisfies the equation and, when combined with the rules of algebra, is what gives the number system useful properties.

Definition

There are two assumptions that define complex numbers.

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

Complex Constant

The complex constant gives the complex number system its properties and can be treated like a variable when performing algebra. When complex numbers were invented (or discovered), assuming such a number existed allowed for the roots of previously unsolvable polynomials to be found. Instead of discussing polynomials and their roots, this page focuses on a visual interpretation of complex numbers in the complex plane.

Complex Plane

The complex number (2 + 1i) in the complex plane.

A complex number represents a point in the complex plane with a horizontal component equal to the real part of the number and a vertical component equal to the complex part of the number. Complex numbers can be drawn as both points and as arrows.

By interpretting complex numbers this way, a single complex number encodes magnitude and direction. This fact, when combined with the complex constant and the rules of algebra is why the complex number system elegantly describes rotation and trigonometry.

Example 1 The complex number (-4+3i) in the complex plane.
Example 2 The complex number (-4+3i) in the complex plane.
Example 3

Complex Addition

Adding two complex numbers together follows the rules of algebra; the complex constant is treated like a variable. The real parts of the numbers are added together and the complex parts are added together. For example, to add and together the real parts and are added to get and the complex parts and are added to get .

Complex addition is visualized by drawing the arrows that represent the two numbers “tail-to-tail”. The sum of the two numbers is equal to the complex number starting from the origin and ending at the tip of the second number.

Complex plane addition
Example Complex plane addition
Example Complex plane addition
Example Complex plane addition

Complex Multiplication

Complex number multiplication interpretation.

Multiplying a complex number by another complex number can be interpreted as stretching and rotating the coordinate system by the first complex number and then traveling the distance of the second complex number in the new system. For example, the result of multiplying the complex number by is visualized below.

Stretching and rotating the original coordinate system by and then traveling the distance in the new system returns the complex number in the original coordinate system. Geometrically, this is the same as adding their angles and multiplying their radii together. The same result is calculated using algebra and the distributive property of multiplication.

Complex multiplication expanding the expression (2+1i) multiplied by (4+3i).

When distributing the multiplication, if an expression is multiplied by the complex constant its direction is rotated in the counter-clockwise direction. For example, the expression scales the complex number by a factor of and rotates it by .

Shown below are some examples that go through the full process of complex multiplication. Otherwise, the next sections go through different properties of complex multiplication building up to the full process.

Example Complex Multiplication Example (4 + 3i)(-1 + 2i)
Example Complex Multiplication Example (2 + 1i)(4 + 2i)
Example Complex Multiplication Example (2 + 1i)(4 + 3i)

Identity Property

Multiplying any complex number by one is the same as stretching and rotating the coordinate system by and traveling one unit in the positive real direction in the new coordinate system. For example, when multiplying imagine stretching and rotating the coordinate system so the vector unit vector sits on top of the complex number .

Complex plane identity property

Here are a couple more examples.

Example Complex identity property example
Example Complex identity property example
Example Complex identity property example

It is important to recognize the positive and negative directions in the new coordinate system are rotated according to the number . For example, traveling one unit in the positive real direction represented by the multiplication down and to the left.

Example Complex plane identity property example.

The identity property also demonstrates what it means to travel mutliple units and in the negative direction of this new coordinate system.

Example Complex plane identity property example.
Example Complex plane identity property example.
Example Complex plane identity property example.

Rotation Property

Multiplying a complex number by rotates the number degrees counter-clockwise in the complex plane. For example, multiplying the number by is equal to the complex number which is the same as rotating the number degrees about the origin.

Complex multiplication multiply number by i.

Using the visualization technique of stretching and rotating the coordinate system by and then traveling units in the new coordinate system we can see why this is true. Traveling one unit in the direction is orthogonal to traveling one unit in the positive real direction and so the result is a degree rotation.

Complex multiplication multiply number by i.

The reason this mutliplication results in a pure rotation and no stretching is because the magnitute of the number we are multiplying by is one. The rotation that the complex property encompasses can be extended to common rotations like , , and degrees.

Complex plane rotation.

As we will see later, using some trigonometry, we can rotate a complex number by any angle we want using complex multiplication.

Example Complex Rotation Property Example

For example, the complex number multiplied by is equal to the rotation shown below:

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.

Example Complex Rotation Property Example
Example 3 Complex Rotation Property Example

Distributive Property

The distributive property combines the two properties above together and shows how the rules of algebra describe the operation of multiplying two complex numbers together. For example, when multiplying and together the first number is distributed according to the rules of algebra to the real and imaginary part of the second complex number.

The resuts in the complex addition of two complex numbers shown as the red and yellow arrows below.

Complex multiplication expanding the expression (2+1i) multiplied by (4+3i).

Trigonometry

Complex addition and multiplication strongly relates to trigonometry. For example, multiplying two complex numbers together is the same thing as adding their angles together and multiplying their radii together to form the resulting complex number.

This fact, when combined with trigonometric concepts like the functions sine and cosine, allows for ideas relating to rotation and space to be elegantly expressed. For example, you can easily perform transformations like rotations, derive trigonometric identities and convert between polar and cartesian coordinates.

Complex Unit Circle

Recall from trigonometry, a point on the unit circle corresponding to some angle (theta) has a horizontal component equal to and a vertical component equal to . This means that a point in the complex plane on the unit circle is equal to .

This fact can be leveraged along with the properties of complex multiplication. For example, to rotate a complex number around the origin by an angle (theta) we can multiply the complex number by .

The complex number system can also be used to answer many more trigonometric questions. For example, the sum of two angles trig identities can be derived using complex numbers with relative ease, because the point corresponding to the sum of two angles can be both expressed as the complex number and as the multiplication of the numbers and [1].

All of the the trigonometric identities can be derived this way using the properties of the complex plane[2].

Exponential Function

The other place that complex numbers notoriously show up is Euler’s Formula. When given the complex constant and an angle (theta) as input, the exponential function returns a complex number on the unit circle corresponding to the angle.

Euler's formula relating the exponential function to a point on the unit circle in the complex plane.

To convert a point in polar coordinates in the form of to a complex number in the complex plane the following formula can be used.

Natural Logarithm

As a final note here, to compute the angle corresonding to the complex number the inverse of the exponential function, the natural logarithm function, is used. When given a complex number as input, the natural logarithm function returns a complex numbers whose imaginary part is equal to the angle corresponding to the input number.

References

  1. Derive Sum of Two Angles Identities (Complex Plane)
    Wumbo (internal)
  2. Derive the Trigonometric Identities (Complex Plane)
    Wumbo (internal)