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Video: Performing Arithmetic with Complex Numbers

Lauren McNaughten

Learn how to add, subtract, and multiply complex numbers, dealing with the real and imaginary parts separately with addition and subtraction and combining them as required with multiplication. Also, learn about using the fact that i^2 = -1 to simplify the result.

08:07

Video Transcript

In this video, we’re going to look at how to perform arithmetic with complex numbers. So we’re going to look at how to add, subtract, and multiply complex numbers. Division of complex numbers will be covered as part of another video.

So let’s begin by looking at addition of complex numbers. And the question I’ve got on the screen here, I want to add together the complex number two plus four 𝑖 and the complex number three plus seven 𝑖. Now before we do this, just a quick reminder about the structure of a complex number, so complex number is made up of a real part and an imaginary part. And if I look at this first complex number here, two plus four 𝑖, then the real part of this complex number is two, and the imaginary part of a complex number is the plus four 𝑖 here. So adding together these two complex numbers is actually very straightforward. And what we have to remember is that we just need to deal with the real and the imaginary parts separately. So first of all, if I look at the real parts, then I have two for the first complex number, and I’m adding that to three for the second complex number, giving me an overall real part of five. Now if I look at the imaginary parts, I have positive four 𝑖 for the first complex number and positive seven 𝑖 for the second. And if I add those together, it gives me an overall imaginary part of plus eleven 𝑖. So the answer to this addition sum is five plus eleven 𝑖.

Now let’s consider an example of how to subtract two complex numbers, and it works in exactly the same way as with the addition. So the question I want to look at, five plus six 𝑖, and then I am subtracting the complex number three minus two 𝑖. So in exactly the same way as we did with addition, first look at the real parts. So for the first complex number, I have five and for the second I have three, so I’m just doing five subtract three which gives me two as the real part of this complex number. Now if I look at the imaginary part, I have positive six 𝑖, and then I am subtracting negative two 𝑖. So if I’m subtracting negative two 𝑖, I will actually be adding two 𝑖 overall, giving me an imaginary part of positive eight 𝑖. So dealing with the real and imaginary parts of the complex number separately gives me a result of two plus eight 𝑖 for this subtraction sum.

Now let’s look at a slightly more complicated question. So what I have here are two complex numbers to be added together, but they both involve brackets. So first one two lots of one plus three 𝑖 and then I’m adding it to four lots of negative two plus six 𝑖. So the first step here is I just need to expand the brackets. So if I do that carefully, I will have two plus six 𝑖 for that first complex number, and then I will have negative eight plus twenty-four 𝑖 for that second one. When I expand the bracket, the four and the negative two make a negative eight there.

Having expanded the brackets, I now just need to combine the real and the imaginary parts as we did in the previous examples. So if I look at the real parts, I have two and then negative eight, giving an overall real part of negative six. And if I look at the imaginary parts, positive six 𝑖 plus twenty-four 𝑖, I have an overall imaginary part of positive thirty 𝑖, giving an overall result of negative six plus thirty 𝑖. So what we did there, just to remind you, we expanded the single brackets first and then we grouped the real and the imaginary terms as we’ve done in previous examples.

Now let’s look at an example of how to multiply two complex numbers. So we’re gonna look at how to multiply four plus two 𝑖 by the complex number one plus three 𝑖. Now when we do this, we’re going to need to use a key fact that’s absolutely fundamental to complex numbers, and it’s the fact that 𝑖 squared is equal to negative one. So I’ve written that over on the right-hand side of the screen in a red box. I will need to refer to that during our working out.

So the first step is just to expand this pair of double brackets, and there’re various different methods that you might use for this. Some people use the acronym β€œfoil”, standing for firsts, outers, inners, and lasts to remind them about all the different pairs. It’s up to you how you do this, but you just need to make sure you multiply every term in the first bracket by every term in the second bracket.

So that’s what I’ve done here. I’ve expanded the brackets, and I’ve got four terms: four plus twelve 𝑖 plus two 𝑖 plus six 𝑖 squared. Now thinking about the key fact that 𝑖 squared is equal to negative one, that means I will be able to simplify this last term here. Because if 𝑖 squared is equal to negative one, then six 𝑖 squared must be equal to negative six. So I can actually replace that last term with negative six, so I’ll have four plus twelve 𝑖 plus two 𝑖. And now I will have negative six. Now it just becomes a case of tidying up the results. So as we did in the previous examples, combining the real and imaginary parts, so real parts four take away six, I will end up with negative two. And positive twelve 𝑖 plus two 𝑖, I will have positive fourteen 𝑖 overall, giving me a final result of negative two plus fourteen 𝑖 for this multiplication.

So just a reminder of what we did. First, we expanded the double brackets using whichever method you feel comfortable with, then you remember this key fact that 𝑖 squared is equal to negative one and use that to simplify the final term. And then finally, we combined the real parts and combined the imaginary parts to give an overall result for this multiplication.

One final way in which you might use multiplication with complex numbers is if you are squaring a complex number. So an example, we can look at we have a complex number three plus 𝑖, and we want to square that. Now when squaring a complex number, or in fact any expression with two terms, the best way to do this is to write the bracket out twice. So you remember that you are not just squaring the individual terms, but you are actually just expanding a pair of double brackets. So three plus 𝑖 squared is equivalent to three plus 𝑖 multiplied by three plus 𝑖 again.

Now I can apply exactly the same method as we’ve talked about in the previous example. So I can expand these brackets to give me four terms. And there they are, nine plus three 𝑖 plus three 𝑖 plus 𝑖 squared. Then I need to remember that this 𝑖 squared here can be simplified to negative one by using that key result for complex numbers. So replacing that 𝑖 squared with negative one, I now have nine plus three 𝑖 plus three 𝑖 subtract one. And then I can just combine the real parts and the imaginary parts as in previous examples, giving me a final result of eight plus six 𝑖 for this multiplication sum.

So key step to remember was this stage here where I wrote the bracket out twice in order to make sure I remembered that I needed to multiply everything in the first bracket by everything in the second bracket. So there you have it. There is a summary of how we can do arithmetic with complex numbers. We’ve looked at how to add, how to subtract, and how to multiply. Dividing complex numbers will be covered in another video.