Complex number conjugates

we're asked to find the conjugate of the complex number seven 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 I should say 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 a positive 5i so that right there is the complex conjugate of seven minus 5i and sometimes the notation for doing that is you'll take seven minus 5i if you have 7 minus 5i 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 if Z is 7 minus 5i then they'll say the complex conjugate of Z and you put that line over the z is going to be 7 plus 5i now you're probably saying okay 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 or any complex number times its conjugate you're going to get a real number and I want to emphasize this this right here is the conjugate seven plus 5i is the conjugate of seven minus 5i but seven minus 5i is also the conjugate of seven plus 5i for obvious reasons if you started with this and you change the sign of the imaginary part you would get seven 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 multiply every every part of this complex number times every part of this complex number so let's just let's just do it any which way so you'd have 7 times 7 which is 49 7 times 5i which is 7 times 5i which is 35 I then you have negative 5i times 7 which is negative 35 I you can see the imaginary part is cancelling out then you have negative 5i times positive 5 I well that's negative 25 I squared and negative 25 I squared remember I squared is negative 1 so negative 25 I 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 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 this is something plus something times or something minus something times that same something 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 difference of squares and so in this case a is 7 a squared is 49 and B B in this case is 5i B squared is 5i squared which is 25 I 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