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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 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.