# Complex Coordinate Plane

How to graph complex numbers on a complex coordinate plane: examples and their solutions.

## Complex Coordinate Plane

A complex coordinate plane has

an *x*-axis and an *yi*-axis.

The *x*-axis is the real number axis.

The *yi*-axis is the imaginary number axis.

So (*a*, *bi*) is the point

that shows *x* = *a* and *yi* = *bi*.

The [complex number] means

[real number] ± [imaginary number].

So the complex coordinate plane is used

to point complex numbers

on the complex coordinate plane.

## Example 1: Graph 4 + 3*i*

4 + 3*i* is the point (4, 3*i*).

So draw (4, 3*i*)

on the complex coordinate plane.

Start from the origin.

The *x* value is 4.

So move 4 units to the right.

The *yi* value is 3*i*.

So move 3 units upward.

This point is (4, 3*i*).

It's like drawing (4, 3)

on a regular coordinate plane.

Coordinate plane

## Example 2: Graph -5*i*

-5*i*, which is 0 - 5*i*, is the point (0, -5*i*).

So draw (0, -5*i*)

on the complex coordinate plane.

Start from the origin.

The *x* value is 0.

So don't move either to the left or to the right.

The *yi* value is -5*i*.

So move 5 units downward.

This point is (0, -5*i*).

## Example 3: Graph (3 + 2*i*)*i*

Expand (3 + 2*i*)*i*.

3⋅*i* = 3*i*

+2*i*⋅*i* = +2⋅(-1)

Powers of *i*

+2⋅(-1) = -2

Write the real number part first.

Then (given) = -2 + 3*i*.

-2 + 3*i* is the point (-2, 3*i*).

So draw (-2, 3*i*)

on the complex coordinate plane.

Start from the origin.

The *x* value is -2.

So move 2 units to the left.

The *yi* value is 3*i*.

So move 3 units upward.

This point is (-2, 3*i*).

Let's see the meaning of multiplying *i*.

(3 + 2*i*)*i* = -2 + 3*i*

By multiplying *i*,

(3, 2*i*) moved to (-2, 3*i*).

(3, 2*i*) → (-2, 3*i*) shows

the rotation of 90º counterclockwise.

Rotation of 90º counterclockwise

So, by multiplying *i*,

the point on a complex coordinate plane

rotates 90º counterclockwise.