Complex Coordinate Plane

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.

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 + 3i

Graph the given expression on a complex coordinate plane. 4 + 3i

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

So draw (4, 3i)
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 3i.
So move 3 units upward.

This point is (4, 3i).

It's like drawing (4, 3)
on a regular coordinate plane.

Coordinate plane

Example 2: Graph -5i

Graph the given expression on a complex coordinate plane. -5i

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

So draw (0, -5i)
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 -5i.
So move 5 units downward.

This point is (0, -5i).

Example 3: Graph (3 + 2i)i

Graph the given expression on a complex coordinate plane. (3 + 2i)i

Expand (3 + 2i)i.

3⋅i = 3i
+2ii = +2⋅(-1)

Powers of i

+2⋅(-1) = -2

Write the real number part first.

Then (given) = -2 + 3i.

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

So draw (-2, 3i)
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 3i.
So move 3 units upward.

This point is (-2, 3i).

Let's see the meaning of multiplying i.

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

By multiplying i,
(3, 2i) moved to (-2, 3i).

(3, 2i) → (-2, 3i) shows
the rotation of 90º counterclockwise.

Rotation of 90º counterclockwise

So, by multiplying i,
the point on a complex coordinate plane
rotates 90º counterclockwise.