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## Algebra (all content)

### Course: Algebra (all content)>Unit 16

Lesson 6: Complex conjugates & dividing complex numbers

# Intro to complex number conjugates

Sal explains what is the conjugate of a complex number, and shows how the product of a complex number and its conjugate is always a real number. Created by Sal Khan.

## Want to join the conversation?

• why does sal uses a Z wih a dashed line? is it a rule for complex no.
• Do you mean the Z with the bar over the top, or the dash across the middle of the Z?

The bar over the top (in this case) means that we are talking about the conjugate of Z, not Z itself.

If you talking about the dash across the middle of the Z, that's a way to make it clear that it's a Z, and not a 2, which with messy handwriting in math can be a problem.
• At , why is a^2 + b^2 = |z|^2 ?
• general form of complex number is a+ib and we denote it as z
z=a+ib. but |z|=[a^2+b^2]^1/2.
|z|^2=a^2+b^2
• at the end of the video. why did sal say that any complex number multiplied by its conjugate is equal to the magnitude of the complex number squared??
• think distance. Imagine you stand in the corner of a room. Now you choose say, your right as the x- axis and up, front of you as y-axis.

If you now move 4 steps to the x-axis (your right), and 4 steps in the y-axis( infront of you). then you have moved from (0,0,) to (4,4).

The magnitude is the direct line length from the corner (start position) to your new position.
Pythagoras gives us that length of line from (0,0) to (4,4) is equal to the square(4^2 + 4^2)
• At the very last part of this video, Can the magnitude of vector be negative? Or does it always stays positive?
• The magnitude of a vector is always nonnegative. The direction can be negative, but not the magnitude.
• is the equation at the Pythagorean theorem?
• This equation, along with the Pythagorean theorem, leads to the conclusion that for any complex number z, it is always true that z times the conjugate of z equals the square of the magnitude of z.
• Can Sal add one or two short videos explaining conjugates? I dont understand why conjugate of a+b is a-b but not -a-b?
• The conjugates of complex numbers are the same as used for radicals. You need to create a difference of 2 squares which in factored form is: (a+b)(a-b), or with complex numbers (a+bi)(a-bi).

If you multiply your version: (a+bi)(-a-bi), you get:
-a^2-abi-abi+b = -a^2-2abi+b
The whole purpose of using the conjugate is the create a real number rather than a complex number. Your version leaves you with a new complex number. If you use Sal's version, the 2 middle terms will cancel out, and eliminate the imaginary component.

Hope this helps.
• At time stamp why does the speaker state that I squared is negative one for five I sqared. He did not stat that for the numberator.
• i^2 is indeed (-1). You'll notice that earlier on he quickly mentions at that 2i times 5i equals 10*i^2, or -10. This is because i^2 is (-1), and 10*(-1) is (-10).
• What's the "magnitude"?
• the magnitude is the absolute value of z, which represents the distance between the point and the origin. if you have z= a + bi, the magnitude r=|z|= sqrt(a^2+ b^2)
• What does conjugate mean?
• If you have a number like 2+3√3, the conjugate is what you get by flipping the sign on the irrational or imaginary term.

So the conjugate of 2+3√3 is 2-3√3.
The conjugate of π-4i is π+4i.
The conjugate of 5 is 5 again.

If you add or multiply a number with its conjugate, you get a rational or real number back.