Complex Numbers

We cannot plot complex numbers on a number line as we might real numbers. However, we can still represent them graphically. To represent a complex number, we need to address the two components of the number. We use the complex plane, which is a coordinate system in which the horizontal axis represents the real component and the vertical axis represents the imaginary component. Complex numbers are the points on the plane, expressed as ordered pairs $(a, b)$, where $a$ represents the coordinate for the horizontal axis and $b$ represents the coordinate for the vertical axis.

Let's consider the number $-2+3 i$. The real part of the complex number is $-2$ and the imaginary part is 3. We plot the ordered pair $(-2,3)$ to represent the complex number $-2+3 i$, as shown in Figure 2.

Figure 2

COMPLEX PLANE

In the complex plane, the horizontal axis is the real axis, and the vertical axis is the imaginary axis, as shown in Figure 3.

Figure 3

EXAMPLE 2

Plotting a Complex Number on the Complex Plane

Plot the complex number $3-4 i$ on the complex plane.

Solution

The real part of the complex number is $3$, and the imaginary part is $-4$. We plot the ordered pair $(3,-4)$ as shown in Figure $4 .$

Figure 4

 

TRY IT #2

Plot the complex number $-4-i$ on the complex plane.