Visualizing a complex number

Just as a real number can be visualized on the number line, a complex number can be visualized on the complex plane in which the horizontal axis is the real axis and the vertical axis the imaginary axis.

Let \(z = -2+i\) and \(w = 1+2i\). They can be plotted on the complex plane as follows:

The dotted yellow line represents all the complex numbers having real part equal to \(-2\) and the dotted blue line represents all the complex numbers having imaginary part equal to \(1\). Therefore, the intersection of these two lines represent the complex number with real part equal to \(-2\) and imaginary part equal to \(1\), which is the complex number \(z\).

In general, a vertical line that crosses the real axis at the value \(a\) represents all the complex numbers with real part equal to \(a\). A horizontal line that crosses the imaginary axis at the value \(b\) represents all the complex numbers with imaginary part equal to \(b\).

Rectangular form

A complex number written in the form \(a+bi\) where \(a\) and \(b\) are real numbers is said to be in rectangular form (or standard form). But as we will see, the rectangular form is not the only way to specify a complex number.

Exercises

Plot each of the following on the complex plane.

1. \(3\)

2. \(3+4i\)

3. \(3-4i\)

4. \(\displaystyle-2+\sqrt{3}i\)

5. \(2i\)