Problem 93 - Algebra and Trigonometry

Question

Explain how to plot a complex number in the complex plane. Provide an example with your explanation.

Step-by-Step Solution

Answer

A complex number like \(3+4i\) can be represented visually in a complex plane by treating it as a point in two dimensions, where '3' the real part is the x-coordinate and '4' the imaginary part is the y-coordinate. To plot, move 3 units to the right along the real (horizontal) axis from the origin and then move 4 units upwards along the imaginary (vertical) axis. Mark this point, which represents \(3+4i\).

Step 1 : Understand Complex Plane

Before plotting complex numbers, it is important to understand the structure of a complex plane. Unlike the Cartesian plane which has X and Y axes, a complex plane is slightly different. It has a real axis (horizontal) and an imaginary axis (vertical). Each complex number can be represented as a point in this plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate.

Step 2 : Identify Real and Imaginary Parts

Next step is identifying the real and imaginary parts of a complex number. Suppose we have an example of a complex number, say \(3 + 4i\), the real part will be '3' and the imaginary part, removed from the imaginary unit 'i', will be '4'.

Step 3 : Plot the point in Complex Plane

Next, plot the complex number on the complex plane. As identified, the real part '3' is the x-coordinate and the imaginary part '4' is the y-coordinate. Starting from the origin, move 3 units to the right along the real axis (horizontal axis), and then move 4 units upwards along the imaginary axis (vertical axis). Mark this point, which represents the complex number \(3+4i\) on the complex plane.