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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?
• 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.
• 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.
• What do you use that real number answer for? (Real Life Example)
• It is mainly used in electricity to obtain the answer of circuit to a sinusoidal signal
• 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.......
• 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?)
• 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.
• 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!
• What is a conjugate in general ?
• 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.
• is -2i the conjugate of 2i?
• 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
• What is the use of complex and imaginary numbers?
• What is a complex conjugate?
• 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!