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Add and Subtract Complex Numbers

Add and Subtract Complex Numbers

Author: Colleen Atakpu

This lesson covers adding and subtracting complex numbers. 

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Today, we're going to talk about adding and subtracting complex numbers. So we're going to go over the general procedure for adding and subtracting complex numbers and then we'll do some examples.

So let's start by reviewing complex numbers. Complex numbers are in the form a plus bi, where a is the real part of a complex number, b is the imaginary part of the complex number, and i is the imaginary unit.

We define i to be equal to the square root of negative 1, and we say that it's imaginary number. And the reason that we say it's an imaginary number is because no real number squared can equal a negative number.

All right, so let's start by looking at how we would add two complex numbers together. So I want to add 3 plus 5i to 2 minus 3i.

So I'm going to do this by first adding together my real number components, the 3 and the 2, and then by adding together my imaginary number components, the 5i and the negative 3i.

So if I add my real number components, 3 plus 2, that will give me 5. And if I add my imaginary number components, 5i plus negative 3i, that's going to give me 2i. So I found that my answer, these two complex numbers together equals 5 plus 2i.

And in general, when we're adding two complex numbers together-- let's say we wanted to add a plus bi plus c plus di. We can, again, add our real number components so it will be equal to a plus c plus our imaginary number components, b plus di.

Let's look at subtracting complex numbers. I've got 8 minus 2i and I want to subtract 4 plus 5i. So I'm subtracting these two complex numbers from each other.

And similarly to addition, we're going to first subtract our real number components from each other and then subtract our imaginary number components together-- or from each other.

So I'm going to start by subtracting 8 minus 4. And that's going to give me 4 for my real number component. And then I'll subtract negative 2i minus 5i, which will give me a negative 7i for my imaginary component. So putting those together, I have this complex number for my answer.

So again in general when we are subtracting complex numbers, for example, a plus bi minus c plus di, we again are going to subtract our real number components. So that will be equal to a minus c. Plus. And then we will subtract our imaginary number components. So we will have b minus di.

So let's do some more examples adding and subtracting complex numbers. So for my first example, I've got negative 5 plus 2i plus 1 minus 3i.

I'm going to start by adding my real number components. Negative 5 plus 1 will give me negative 4. And 2i plus negative 3i is the same as 2i minus 3i, which will give me a negative 1i, or just negative i.

Before we do our second example, if you're feeling pretty confident, go ahead and pause and then check back with us later to see if you got the right answer. So for this example, my first complex number doesn't have a real number component, or we can think of it as a 0. So when I'm combining it with the real number component here, I really have 0 minus 2, which will give me a negative 2.

And then, 3i minus a positive 4i is the same as 3 minus 4i, which will give me, again, a negative 1i, or just negative i. So I have negative 2 minus i.

For my third example, I've got negative 8 minus 6i minus 5 minus 2i. So I have a lot of negative signs and subtraction, so I need to be careful when I'm subtracting. So I'm going to start with negative 8 minus 5. That's going to give me a bigger negative number. That will give me negative 13.

And then I have negative 6i minus a negative 2i. Subtracting a negative 2i is the same as adding a positive 2i. So I have negative 6i plus positive 2i, which is going to give me negative 4i. So negative 13 minus 4i.

And finally, I've got 7 plus i plus negative 3. So here, I notice that I don't have an imaginary component to this complex number. So I'm going to start by adding my real number component. 7 plus a negative 3 is the same as 7 minus 3. So that will give me 4. And then I don't have an imaginary number again here to combine, so I'll just bring down my plus i.

So let's go over our key points from today. Complex numbers consist of a real part and an imaginary part. The square root of negative 1 is imaginary because no real number squared results in a negative number. And when adding or subtracting two complex numbers, you first combine the real numbers, and then combine the imaginary numbers.

So I hope that these key points and examples helped you understand a little bit more about adding and subtracting complex numbers. Keep using your notes and keep on practicing and soon you'll be a pro. Thanks for watching.

Formulas to Know
Imaginary Number

i equals square root of negative 1 end root

i squared equals negative 1