Hi and welcome. This is Anthony Varela, and today we're going to be adding and subtracting complex numbers. So we're going to review our complex numbers a plus bi. We're going to be adding two complex numbers, and then subtracting two complex numbers. So first let's review complex numbers. It is a plus bi generally, and it has a real part and an imaginary part.
So a is the real part of the complex number, b is the imaginary part, and then we have i, which is the imaginary unit. And i is the square root of negative 1. It's imaginary because any real number squared is a positive number. So to take the square root of a negative number would be imaginary.
Now when we're adding two complex numbers, I like to align them vertically. So here we have our real components, here we have our imaginary components, and the process is pretty simple. It's not complex at all. So to add complex numbers, first what we do is we add the real numbers. So 3 plus 2 is 5. Then, we add the imaginary numbers. So we have 4 plus 7 gives us 11. So 4i plus 7i is 11i. So this sum is 5 plus 11i.
So in general, if we're adding a plus bi to c plus di, we add those real parts together first, so we have a plus c. And then we add the imaginary parts b plus d, and that is a multiple of i. So very simple, just add the real numbers, then add the imaginary numbers. Now when we're subtracting complex numbers, we're doing pretty much the same thing, we'll just be subtracting instead of adding.
So here I'd like to subtract 2 plus 7i from 3 plus 4i. So once again, I'm writing it vertically, and the only thing then I need to pay attention to is that we're subtracting numbers instead of adding them. So when we're subtracting complex numbers, let's first subtract our real parts. So 3 minus 2 that gives me 1. Then I'm going to subtract the imaginary parts. So 4 minus 7 is a negative 3. So 4i minus 7i is minus 3i. So here the difference is 1 minus 3i.
So in general, if we have a complex number, a plus bi, and we're going to subtract another complex number, we'll call it c plus di, we subtract our real part, so a minus c, and then we subtract the imaginary parts, so b minus d multiple of i. And probably the trickiest thing is that you still see this plus here, so let's talk about this for a minute. Well, our general complex number is a plus bi, and so we're still going to generally write a real part plus the imaginary part. And you might see this b negative in the case where b minus d is negative. We saw that over here, 4 minus 7 was negative 3. But in general, we connect them with a plus sign with our complex numbers.
So now let's go ahead and practice then adding and subtracting complex numbers. So here I'd like to add negative 5 plus 2i to 2 minus 3i. So adding my real numbers, negative 5 plus 2, I get negative 3. And now let's add our imaginary numbers, 2i plus negative 3i, so I can think of this as 2i minus 3i. That's going to be a negative 1i, so I'll say minus i. Let's go through a subtraction problem.
Here we have 8 minus 4i, and we're subtracting 6 plus 3i. So let's subtract the real parts, 8 minus 6, that gives us 2. And now we're going to subtract the imaginary part, so here's negative 4 minus 3. So negative 4 minus 3 is negative 7, so negative 4i minus 3i is negative 7i, or minus 7i. So that's the difference to this subtraction problem.
So let's review adding and subtracting complex numbers. We've talked about a complex number containing a real part, an imaginary part, and then i, the square root of negative 1. When adding complex numbers, first add the real components, and then add the imaginary components. So very straightforward. Same thing with subtracting complex numbers. You'll subtract the real numbers and then subtract your imaginary numbers. Probably the most difficult thing is keeping your sign straight, so what's positive and what's negative.
So thanks for watching this tutorial in adding and subtracting complex numbers. Hope to see you next time.