Complex number conjugates

We're asked to find the conjugate of the complex number 7 minus 5i. 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-- or I should be a little bit more particular. It has the same real part. 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 5i, it will have a positive 5i. So that right there is the complex conjugate of 7 minus 5i. And sometimes the notation for doing that is you'll take 7 minus 5i. If you have 7 minus 5i, and you put a line over it like that, that means I want the conjugate of 7 minus 5i. And that will equal 7 plus 5i. Or sometimes someone will write-- you'll see z is the variable that people often use for complex numbers. If z is 7 minus 5i, then they'll say the complex conjugate of z-- you put that line over the z-- is going to be 7 plus 5i. 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 complex number times its conjugate, you're going to get a real number. And I want to emphasize. This right here is the conjugate. 7 plus 5i is the conjugate of 7 minus 5i. But 7 minus 5i is also the conjugate of 7 plus 5i, for obvious reasons. If you started with this and you change the sign of the imaginary part, you would get 7 minus 5i. They're conjugates of each other. But let me show you that when I multiply complex conjugates that I get a real number. So let's multiply 7 minus 5i times 7 plus 5i. And I will do that in blue-- 7 minus 5i times 7 plus 5i. 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 to multiply every part of this complex number times every part of this complex number. So let's just do it any which way. So you'd have 7 times 7, which is 49. 7 times 5i, which is 35i. Then you have negative 5i times 7, which is negative 35i. You can see the imaginary part is canceling out. Then you have negative 5i times positive 5i. Well, that's negative 25i squared. And negative 25i squared-- remember, i squared is negative 1. So negative 25i squared-- let me write this down. Negative 5i times 5i is negative 25 times i squared. i squared is negative 1. So negative 25 times negative 1 is positive 25. And these two guys over here cancel out. And we're just left with 49 plus 25-- let's see, 50 plus 25 is 75. So this is just 74. So we are just left with the real number 74. Another way to do it-- you don't even have to do all this distributive property. You might just recognize that this looks like this is something minus something times that same something plus something. And we know this pattern from our early algebra, that a plus b times a minus b is equal to a squared minus b squared, is equal to a difference of squares. And so in this case, a is 7. a squared is 49. And b, in this case, is 5i. b squared is 5i squared, which is 25i squared, which is negative 25. And we're subtracting that. So it's going to be positive plus 25. You add them together. You get 74.