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# Complex number conjugates

Sal finds the conjugate of (7-5i). Created by Sal Khan and Monterey Institute for Technology and Education.

Video transcript

We're asked to find the conjugate of the complex number seven minus five i. And what you're going to find in this video is finding the conjugate of a complex number is shockingly easy. It's really the same as this number, I should say - be a little bit more particular- It has the same real parts, so the conjugate of this is going to have the exact same real part, but its imaginary part is going to have the opposite sign. So, instead of having a negative five i, it will have a positive - a positive- 5 i. So that right there is the complex conjugate of seven minus five i. And some times the notation for doing it that is -you'll take seven minus five i- If you have seven minus five i, and you put a line over it like that, that means you want the conjugate of seven minus five i. And that will equal seven plus five i. Or sometimes someone will write, -you'll see z is the variable people often use for complex numbers- if z is seven minus five i, then they'll say complex conjugate of z, -you put that line over the z- is going to be seven plus five i. Now you're probably saying ok, fairly straightforward to find a conjugate of a complex number, but what is it good for? And the simplest reason, or the most basic place where this is useful is when you multiply any imaginary, any complex number times its conjugate, you're going to get a real number. And I want to emphasize, this right here is the conjugate seven plus five, is the conjugate of seven minus five i. But seven minus five i is also the conjugate of seven plus five i, for obvious reasons. If you started with this, and changed the sign of the imaginary part, you would get seven minus five i. They're conjugates of each other. But let me show you when I multiply complex conjugates, that I get a real number. So lets multiply seven minus five i times seven plus five i. And I will do that in blue. Seven minus five i times seven plus five i. And remember, whenever you multiply these expressions, you really just have to multiply every term times each other. You could do the distributive property twice. You could do something like foil, to remind yourself every part of this complex number times every part of this complex number. So let's just do it any which way. So you have seven times seven, which is forty-nine. Seven times five i, which is thirty-five i. Then you have negative five i times seven, which is negative thirty-five i. You can see the imaginary part is canceling out. Then you have negative five i time positive five i. Well that's negative twenty-five i-squared. And negative twenty-five i-squared, -remember i-squared is negative one- So negative twenty five i-squared, -let me write this down- negative five i, times five i, is negative twenty-five times i-squared. i-squared is negative one, so negative twenty-five times negative one is positive twenty-five. And these two guys over here cancel out, and we're just left with forty-nine plus twenty-five. Let's see, fifty plus twenty-five is seventy-five. So, this is just seventy-four. So, we are just left with the real number seventy-four. Another way to do it, -you don't even have to do all this distributive property- You might just recognize this looks like. This is something plus something times -or something minus something- times that same thing plus something. And we know this pattern from our early algebra. That a plus b -a plus b- times a minus b is equal to a-squared minus b-squared. Is equal to a diference of squares. And so in this case a is seven, a-squared is forty-nine. And b, b in this case is five i. b-squared is five i-squared, which twenty-five i-squared, which is negative twenty-five, and we're subtracting that. So it's going to be positive plus-twenty-five. You add 'em together, you get seventy-four.