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# Intro to complex number conjugates

Video transcript

I want to make a
quick clarification and then add more tools in
our complex number toolkit. In the first video, I said that
if I had a complex number z, and it's equal to a
plus bi, I used a word. And I have to be
careful about that word, because I used in
the everyday sense. But it also has a
formal reality to it. So clearly, the real part
of this complex number is a. Clearly, that is the real part. And clearly, this
complex number is made up of a real number plus
an imaginary number. So just kind of talking
in everyday terms, I called this the
imaginary part. I called this imaginary
number the imaginary part. But I want to just
be careful there. I did make it clear
that if you were to see the function
the real part of z, this would spit out the a. And the function the
imaginary part of z, this would spit
out-- and we talked about this in the first
video-- it would spit out the number that's scaling the i. So it would spit out the b. So if someone is talking
in the formal sense about the imaginary
part, they're really talking about the
number that is scaling the i. But in my brain, when I
think of a complex number, I think of it having a real
number and an imaginary number. And if someone were to say,
well, what part of that is the imaginary number? I would have given
this whole thing. But if someone says just
what's the imaginary part, where they give you this
function, just give them the b. Hopefully, that
clarifies things. Frankly, I think the word
"imaginary part" is badly named because clearly,
this whole thing is an imaginary number. This right here is not
an imaginary number. It's just a real number. It's the real number
scaling the i. So they should call
this the number scaling the imaginary part of z. Anyway, with that
said, what I want to introduce you to is the
idea of a complex number's conjugate. So if this is z, the
conjugate of z-- it'd be denoted with z
with a bar over it. Sometimes it's z with a little
asterisk right over there. That would just be
equal to a minus bi. So let's see how they
look on an Argand diagram. So that's my real axis. And then that is
my imaginary axis. And then if I have
z-- this is z over here-- this height
over here is b. This base, or this
length, right here is a. That's z. The conjugate of
z is a minus bi. So it comes out a
on the real axis, but it has minus b as
its imaginary part, so just like this. So this is the conjugate of z. So just to visualize it, a
conjugate of a complex number is really the mirror image of
that complex number reflected over the x-axis. You can imagine if this
was a pool of water, we're seeing its
reflection over here. And so we can
actually look at this to visually add the complex
number and its conjugate. So we said these are just
like position vectors. So if we were to add
z and its conjugate, we could essentially
just take this vector, shift it up here,
do heads to tails. So this right here, we are
adding z to its conjugate. And so this point right
here, or the vector that specifies that point,
is z plus z's conjugate. And you can see right
here, just visually, this is going to be 2a. And to do that algebraically. If we were to add z--
that's a plus bi-- and add that to its
conjugate, so plus a minus bi, what are we going to get? These two guys cancel out. We're just going to have 2a. Or another way to
think about it-- and really, we're just
playing around with math-- if I take any complex number,
and to it I add its conjugate, I'm going to get 2
times the real part of the complex number. Oh, and this is also going
to be 2 times the real part of the conjugate because they
have the exact same real part. Now with that said,
let's think about where the conjugate could be useful. So let's say I had
something like 1 plus 2i divided by 4 minus 5i. So it's no real obvious way
to simplify this expression. Maybe I don't like having
this i in the denominator. Maybe I just want to write
this as one complex number. If I divide one complex
number by another, I should get another
complex number. But how do I do that? Well, one thing to
do is to multiply the numerator and
the denominator by the conjugate
of the denominator, so 4 plus 5i over 4 plus 5i. And clearly, I'm just
multiplying by 1, because this is the same
number over the same number. But the reason why
this is valuable is if I multiply a number
times its conjugate, I'm going to get a real number. So let me just
show you that here. So let's just multiply this out. So we're going to get 1
times 4 plus 5i is 4 plus 5i. And then 2i times 4 is plus 8i. And then 2i times
5i-- that would be 10i squared, or negative 10. And then that will
be over-- now, this has the form a
minus b times a plus b. Well, a plus b times a minus b
is a squared minus b squared. So it's going to be equal
to 4 squared, which is 16, minus 4 squared-- oh, why
did I have 4 plus 4i here? This should be 4 plus 5i. What am I doing? 4 plus 5i, the same number
over the same number. This was a 10 right over there. This is the conjugate. I don't know. My brain must have
been thinking in 4's. So obviously, I don't want
to change the number-- 4 plus 5i over 4 plus 5i. So let's multiply it. This is a minus b times
a plus b, so 4 times 4. So this is going to be 4
squared minus 5i squared. And so this is going to
be equal to 4 minus 10. Let's add the real parts. 4 minus 10 is negative 6. 5i plus 8i is 13i. Add the imaginary parts. And then you have
16 minus 5i squared. Well, 5i squared-- i
squared is negative 1. 5 squared's just going
to be negative 25. The negative and the
negative cancel out, so you have 16 plus 25. So that is 41. So we can write this
as a complex number. This is negative
6/41 plus 13/41 i. We were able to divide
these two complex numbers. So the useful thing
here is the property that if I take any
complex number, and I multiply it
by its conjugate-- and obviously, the
conjugate of the conjugate is the original number. But I would take
any complex number and I multiply it
by its conjugate, so this would be a plus
bi times a minus bi. I'm going to get a real number. It's going to be a squared
minus bi squared, difference of squares, which is
equal to a squared-- now, this is going to be
negative b squared. But we have a negative
sign out here, so they cancel out-- a
squared plus b squared. And just out of curiosity,
this is the same thing as the magnitude of our
complex number squared. So this is a neat property. This is what makes
conjugates really useful, especially when you
want to simplify division of complex numbers. Anyway, hopefully,
you found that useful.