So let me do that.

Well, we have a tool in our toolkit that can make sure that we don't have an imaginary or complex number in the denominator.

So, instead of having a negative five i, it will have a positive a positive 5 i.

They're conjugates of each other.

And the sign for conjugate is that line over the top.

So the conjugate of ztwo is c minus d i.

And so what do we do?

And that's just good terminology to know.

Let's multiply the numerator and the denominator by the conjugate of this.

And when you scale them.

Complex number

Complex number

Because the conjugate is the tool we're going to use to divide this.

Or you could say it the other way around.

So c minus d i over c minus d i.

And the simplest reason, or the most basic place where this is useful is when you multiply any imaginary, any complex number times its conjugate, you're going to get a real number.

So let's multiply the numerator and denominator of this expression by the conjugate of the denominator.

So the real part is a, the imaginary part is bi.

So that right there is the complex conjugate of seven minus five i.

So let's do that.

In future videos I'll actually do examples where we'll make more use of complex numbers.

But seven minus five i is also the conjugate of seven plus five i, for obvious reasons.

If we multiply both the numerator and the denominator of this expression by the complex conjugate of the denominator, then we will have a real number in the denominator.

Or we could write that as if we wanted to write that in the traditional complex form is eight thirteen plus one over thirteeni.

That's pretty much every number outside of the complex numbers.

Move the orange dot to 2 2i.

So when you add these two complex numbers you get 8 5i.

Or sometimes someone will write,

And our vertical axis is going to be the imaginary part.

I didn't have to erase what I had before.

That's interesting.

This could be z sub two.

That equals a squared minus b squared.

Let's say I have one plus two i.

But complex 0's come in pairs of two, the two complex conjugates.

And what you're going to find in this video is finding the conjugate of a complex number is shockingly easy.

Can we take a square root of a negative number?

It's not in the set of reals that's a complex number.

And so a number that has a real part, like the five, and an imaginary part, like the twoi, this is called a complex number.

And just as a little review.

Where I left off in the last video was, how do we divide two complex numbers.

you'll see z is the variable people often use for complex numbers if z is seven minus five i, then they'll say complex conjugate of z,

And so if I multiply a complex number.

Now you're probably saying ok, fairly straightforward to find a conjugate of a complex number, but what is it good for?

be a little bit more particular