Video: Adding and Subtracting Complex Numbers

What is (βˆ’7 βˆ’ 𝑖) βˆ’ (3 βˆ’ 4𝑖) + (2 βˆ’ 7𝑖)?

02:12

Video Transcript

What is negative seven minus 𝑖 minus three minus four 𝑖 plus two minus seven 𝑖?

Okay, we’re actually gonna simplify this expression. And we’re gonna do that in the same way that we’d actually simplify an algebraic expression. And I’m gonna do that by collecting like terms. The difference is that, with our expression, our like terms are the real parts, which I’ve underlined here in orange, and the imaginary parts that I’ve underlined in pink.

Okay, great! So now we know how we’re gonna approach this question. Let’s get on and simplify. The first step I’m going to do is group together the imaginary and real parts.

I’m gonna start with the real parts. So I will have negative seven and then minus three β€” and the reason it’s minus three is because we’ve actually got the negative sign before the parentheses β€” and then plus two. Okay, great! That’s our real parts.

Let’s move on to our imaginary parts. So first of all, we have negative 𝑖. And then we have plus four 𝑖 β€” and the reason we have plus four 𝑖 is because we’ve got a negative in front of the parentheses, so it’s negative negative four 𝑖, and a negative and negative make it positive β€” and then finally minus seven 𝑖.

Okay, great! So we now have our terms grouped. So we have negative seven minus three plus two minus 𝑖 plus four 𝑖 minus seven 𝑖. Okay, great! So now we can collect our terms.

So first of all, we have negative eight. That’s because negative seven minus three gives us negative 10, plus two leaves us with negative eight. And then it’s negative eight minus four 𝑖. We get minus four 𝑖 because if we have negative 𝑖 plus four 𝑖, it’s gonna give us positive three 𝑖. And then subtract another seven 𝑖 is gonna leave us with negative four 𝑖.

Okay, great! So we can say that negative seven minus 𝑖 minus three minus four 𝑖 plus two minus seven 𝑖 is equal to negative eight minus four 𝑖.

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