How to Add Complex Numbers in the Complex Plane Video

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  • 0:00 A Complex Number
  • 1:26 How to Add Complex Numbers
  • 2:05 Example 1
  • 2:28 Example 2
  • 2:55 Lesson Summary
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Lesson Transcript
Instructor: Yuanxin (Amy) Yang Alcocer

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

Watch this video lesson to see how you can use the complex plane to help you add your complex numbers. See how similar this is to the very familiar Cartesian coordinate plane.

A Complex Number

Look at these numbers:

6 + 10i
7 - 8i

These are called complex numbers. A complex number is a number with both a real part and imaginary part. The real part is the number by itself, and the imaginary part is the number with the i attached to it. In math, we also have a definition for this imaginary i. We define it as the square root of -1.

The cool thing about these numbers is that we can plot them on a plane called the complex plane. Even though it's called the complex plane, it's more like the Cartesian plane with different labels for the x and y axes. The x-axis becomes your real axis and the y-axis becomes your imaginary axis. So the number 6 + 10i plots to the point (6, 10) on the plane since the real part is the number 6 and the imaginary part is number 10 attached to the i. The number 7 - 8i plots to the point (7, -8) since the real part is 7 and the imaginary part is the -8 attached to the i.

add complex number

Now that we have our points, we can also add an arrow to each point starting from the origin, the (0, 0) point.

add complex number

When we draw an arrow like we just did, we call this the vector form of the complex number.

How to Add Them

With this vector form and our complex plane, we can now very easily add our complex numbers together. All we have to do is to move one of our vectors so that it begins where the other ends. Let's move the blue arrow so that it begins where the red arrow ends.

add complex number

When we have done this, our answer is the point that the blue arrow is pointing at. This point is (13, 2). This gives us a complex number of 13 + 2i.

We could also have moved the red arrow so that it begins where the blue arrow ends. When we do this, the red arrow will be pointing at our answer.

Let's look at a couple more examples.

Example 1

Add -6 + 10i and 4 + 2i.

First, we plot the points (-6, 10) and (4, 2) on our complex plane. Then we draw arrows from the origin.

add complex number

Then we move one of the arrows so that it begins where the other ends.

add complex number

We see that it is pointing at (-2, 12). So our answer is -2 + 12i.

Example 2

Add 1 + 3i and 4 + 5i

We are going to do the same here. We first plot the points (1, 3) and (4, 5). We then draw arrows to each.

add complex number

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