Hello. In this mathematics post, I cover graphing complex numbers. For those who know some trigonometry, graphing complex numbers is not that much different.

## Real & Imaginary Parts of Complex Numbers

A complex number `z`

is of the form:

where , `a`

is the real number part of `z`

and `b`

is the imaginary number part of `z`

.

## Argand Diagram

A complex number can be represented graphically with the use of a Argand diagram. This diagram is for plotting complex numbers as points. The x-axis is the real axis and the y-axis is the imaginary axis. (Screenshot)

Reference: https://mathworld.wolfram.com/ArgandDiagram.html

## Modulus & Radius

Similar to trigonometry with Cartesian co-ordinates and `(x, y)`

points, we can find the radius of the point `(a, b)`

that represents a complex number. This radius in the complex numbers setting is called the modulus. This modulus is a distance measure that is related to the Pythagorean Theorem. Wolfram uses `z = x + iy`

while I use `z = a + bi`

in this post.

## Theta Angle

After drawing the point for `(a, b)`

, we can draw two lines. The first line is the radius from the origin to the point. Line two is a horizontal line from the origin to `(a, 0)`

. The angle theta is the angle between these two lines.

**Determine The Angle Theta**

When the angle theta is unknown, trigonometry is used to find the angle. The tangent of the unknown angle is the opposite side length `b`

divided by the adjacent side length of `a`

.

Obtaining the angle involves the use of the inverse tangent function (arctan).

## A Few Examples

**Example One**

The complex number that we are working with in this example is . Five is the real part of the complex number and 2 is the imaginary part. The co-ordinate would be (5, 2) on the Argand diagram. (Screenshot with MathisFun drawing website.)

The angle associated with the right angled triangle is as follows.

**Example Two**

Consider a different complex number of `z = -2 -4i`

. The co-ordinate here is (-2, -4) on the Argand diagram.

Using the formula for finding the angle we have:

The angle theta is for the angle in the triangle. If we want the angle associated with the complex number `z`

, we need `arg(z)`

which is the angle from the terminal arm to the hypotenuse of the triangle.

In this case, the argument for `z`

is 180 degrees + 63.43 degrees which is 243.43 degrees.

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