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### Course: Precalculus > Unit 3

Lesson 3: Complex conjugates and dividing complex numbers# Complex number conjugates

Through a guided example with 7 - 5i, this video explains how to find the conjugate of a complex number, which is simply changing the sign of the imaginary part. Multiplying a complex number by its conjugate results in a real number. This is useful for simplifying complex numbers and is similar to the difference of squares. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

- I have learned on school that the conjugate from 7-5i is -7+5i here he say it's 7+5i.

What's correct or is it both correct?(18 votes)- When multiplying a number by its conjugate you should end up with a real number. You can check which 2 complex numbers, multiplied, give you a real number. Let's start with your school's answer. If you do (7-5i)*(-7+5i), you get 49 +70i-25i^2. This, in simplified form, is equal to 74+70i, which is a complex number, not a real number. Therefore, 7+5i has to be the conjugate of 7-5i.(6 votes)

- What grade do you have to know this by I'm in 7th grade how much time do i have(1 vote)
- Susan,

You have all the time you need to learn any math concept at Khan Academy.

At Khan Academy, you can slow down and keep working on a concept until you really grasp it and not worry about missing out on what else is being taught.

And because at Khan Academy lets you learn at your own pace you also don't need to wait until everyone else also learns a concept before you can continue on.

Probably 90% of the population of adults never learned about Complex Conjugates. It is taught in high school algebra, but often is skipped over because there is never enough time to teach everything.

But at Khan Academy, the lessons and practice session with step-by-step help (via the "Id like a Hint" button) are just waiting for you whenever you need them.(94 votes)

- What do you use that real number answer for? (Real Life Example)(8 votes)
- It is mainly used in electricity to obtain the answer of circuit to a sinusoidal signal(47 votes)

- A complex number, say 3 + 4i......

It's conjugate will be 3-4i as per definition....

So, we can also write/represent 3 + 4i as 4i + 3......then it's conjugate will become 4i-3.......

So, is it correct that a complex number can have 2 conjugates?

Because 3+4i and 4i+3 are the same thing.....it's just that when i am writing it in a different manner, I get two different conjugates?

For 3+4i, it is 3-4i.....and For 4i+3, it is 4i-3.....

But 3+4i = 4i+3......

This is implying that one complex number can have 2 conjugates.......

So, is it correct? Please help(6 votes)- No, the complex conjugate is defined as switching the sign of the imaginary term not whichever term happens to be 2nd.

Your example would be a conjugate in the binomial sense, but it is not a "complex conjugate". I don't believe there is any special name for it. (real conjugate?)(10 votes)

- Why Sal always writes a dashed line in the middle of the letter Z? I have never seen it before except in the lessons of complex numbers.(3 votes)
- The dashed line is used so that there is no mistaking a written z for a 2.

They can often look the same if written in a hurry!(11 votes)

- What is a conjugate in general ?(5 votes)
- A conjugate is when we take an expression like (x + 2) and make the resulting conjugate of (x - 2). Notice that the second term in the second expression has been negated or, in other words, has had its sign flipped to the opposite. So, the conjugate of (x - 2) would be (x + 2)--they are conjugates of each other.(6 votes)

- is -2i the conjugate of 2i?(4 votes)
- Yep! you can think of -2i as 0 - 2i, so if the conjugate is a - bi from a + bi then 0 - 2i has a conjugate of 0 + 2i which is just 2i(4 votes)

- What is the use of complex and imaginary numbers?(5 votes)
- You'll find the answer here:

http://mathforum.org/library/drmath/view/53606.html

Hope it helps!(2 votes)

- What is a complex conjugate?(3 votes)
- Google says that complex conjugate is when "Each of two complex numbers having their real parts identical and their imaginary parts of equal magnitude but opposite sign."

------------------------------ If you still don't get it, look below --------->

For example:

1) 5 – 3i

Its Complex Conjugate is: 5 + 3i

2) 4i + 8

Its Complex Conjugate is: –4i + 8

3) –6i

Its Complex Conjugate is: 6i

4) 17

Its Complex Conjugate is: 17 (ONLY if you look at this as 17 + 0i )

5) 2 + sqr(7)

Its Complex Conjugate is: 2 + sqr(7) (This is NOT a Complex Number so the rule of complex conjugate does not apply here)

6) 3 - sqr(4)i

Its Complex Conjugate is: 3 + sqr(4)i

p.s. Look at the comments if you still don't get it!(6 votes)

- So can I describe the complex conjugate as the additive inverse?(2 votes)
- No. An additive inverse is the opposite or negative value of a number. The additive inverse of a+bi is -a-bi, which is not a conjugate. We take a-bi as the conjugate of a+bi because it allows us to completely eliminate the imaginary part of the complex number when multiplied. (a+bi)(a-bi)=a^2-(bi)^2=a^2+b^2.

Remember difference of squares factoring? This is multiplying to get a sum of squares, which can only be done with complex numbers.(4 votes)

## 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.