Here, on the horizontal axis that's going to be the real part of our complex number.

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

Can we take a square root of a negative number?

Zero times i is zero, so it's still just going to be negative twenty one.

And just as a little review.

Actually, let me make that clear has a real part and an imaginary part.

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.

Complex number

Complex number

In complex numbers the convention is to use z, the letter z.

That's five.

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

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

I have the complex number 3 minus 4i. I've plotted it on the complex plane.

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

3 Imaginary Part

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

Or sometimes someone will write,

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

We have to simplify the principal square root of 52 and we're going to assume because we have a negative 52 here inside of the radical then this is the principal branch of the complex square root function that we can actually input negative numbers in the domain of this function that we can actually get the imaginary or complex result.

5 Squared Imaginary Part

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.

Once again, the real part is 5.

The real part is 4.

Let's say I have one plus two i.

That's interesting.

This part right here is the real part.

Actually, algebraically, multiply this out and just realize you can only add real parts to real parts and imaginary parts to imaginary parts.

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

It has a real part 2.

So in in this example, this complex number, our real part is 2.

That's that, and that.

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

And we're distributing the 1 74 times both of these, I guess is one way to think about it.

And that's how we got this thing over here, where we have a nice real part and a nice imaginary part.

And we are done.

One is a plus bi.

You get another complex number.

This could be z sub two.

That equals a squared minus b squared.

And that's the new real part.

You could say, well, this is just going to be equal to take the real part squared.

And that's just good terminology to know.