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Representing Distances on the Complex Plane

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Read this lesson to learn the formula you need to use to help you find distances on the complex plane. You'll learn that all you need to know are the coordinates of the beginning and end of your distance line.

The Complex Plane

Remember your complex numbers? You know, the numbers that look like this?

  • a + bi

The i tells you that the number b is the imaginary part and the a is the real part.

Well, you can plot these numbers as (a, b) on the complex plane or the Argand Diagram. The complex plane is just like the coordinate plane, except you have the imaginary axis for the y-axis and the real axis for the x-axis.

For example, the complex number -6 + 2i plotted as (-6, 2) on the complex plane looks like this:


complex plane distances


It looks just like the Cartesian coordinate plane where you plot your (x, y) points.

Distances

When you have two plotted points, just like you do when you have two points on the Cartesian plane, you can find the distance between these two points.


complex plane distances


The process is very similar. You use a formula with a square root for finding distances on the Cartesian plane. And you'll use a similar formula for finding distances on the complex plane.

The Formula

The formula for finding the distance between two points (a, b) for complex number a + bi and (c, d) for complex number c + di on the complex plane is:


complex plane distances


It really doesn't doesn't matter which point you subtract from as long as you keep the order the same. For example, if you subtract the real part of the first number from the second, you also need to subtract the imaginary part of the first number from the second.

Let's take at this formula in action.

Example 1

Let's use the numbers that you saw in the last graph, -6 + 2i and 4 + 3i. This means that your two points are (-6, 2) and (4, 3).

Label them with your a, b, c, and d. You'll label the first point as (a, b) and your second point as (c, d). So, your a is equal to -6 and your b is equal 2. Your c is equal to 4 and your d is equal to 3.

Now go ahead and plug these numbers into your formula. Then you'll evaluate to find your answer.


complex plane distances


From this calculation, you see that your distance between these two points is 10.05 units. And you are done!

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