>> Now, we're going to go through operations involving complex numbers, and we're going to add, subtract, multiply, and divide them, and just see how those operations work. And they will be pretty much as expect. We treat I as though it were a letter, as, as though it were a variable. It's not a variable. It's a constant. It's the square root of negative 1, but in, in terms of performing operations, a lot of times it's handy to think of it like a variable and treat it that way. Here, for example, if we're adding, excuse me, if we're subtracting these two complex numbers, then we know exactly what we do, and it would be a pretty trivial situation if this were simply X and X and these positions were I. Well, treat the I in the same way that we treat the X, and we have no problem here. We have, well, let's see, 4 plus 3I, and then minus understood 1 times all of this or just a minus sign before the parenthesis changes all the signs. So we have minus 7 plus 2I, and we add the like terms. Well, let's see. The constants for minus 7 is negative 3, and then 3I plus 2I is 5I. So we're adding two complex, or we're collecting two complex numbers, and we get, get a complex result. Now, a lot of times it's, it's important to put the answers in problems like this in the form of a complex number so that we can identify distinctly the A and B parts. Now, we've already written this answer in that form. This is the A plus BI form, you see. In other situations, we may have to manipulate a little bit to arrive at this form. Well, here we have a, a real number, and more specifically, it's an integer that we're collecting with a complex number, and let's see what happens when we collect those. We get, let's see. This is negative 3 plus 3 plus 5I collecting here. The 3's go out, and we have 5I. Gee. Five I is an imaginary number. So it looks like when we collect a real number with a complex number, we could get a purely imaginary number, and actually all varying combinations will, will turn out to be true. That, that we can add different kinds of numbers and end up with a, a totally different kind of number than we started with although this isn't totally different. It, this is part of the complex number system because it can be written in the form A plus BI where A is 0. You see, 0 plus 5I, and in fact, all of the numbers that we've ever talked about, all of the real numbers that we've talked have been part of or a subset of the complex numbers because, oh, a number as simply as, let's say, 3 can be written in complex form as 3 plus 0I. So it's a complex number.