Introduction to complex numbers | Imaginary and complex numbers | Precalculus | Khan Academy

Voiceover:Most of your mathematical lives

you've been studying real numbers.

Real numbers include
things like zero, and one,

and zero point three
repeating, and pi, and e,

and I could keep listing real numbers.

These are the numbers that you're

kind of familiar with.

Then we explored something interesting.

We explored the notion of what if

there was a number that if I squared it

I would get negative one.

We defined that thing
that if we squared it

we got negative one, we
defined that thing as i.

So we defined a whole new class of numbers

which you could really view as multiples

of the imaginary unit.

So imaginary numbers
would be i and negative i,

and pi times i, and e times i.

This might raise another
interesting question.

What if I combined
imaginary and real numbers?

What if I had numbers
that were essentially

sums or differences of
real or imaginary numbers?

For example, let's say
that I had the number.

Let's say I call it z,

and z tends to be the most used variable

when we're talking about

what I'm about to talk
about, complex numbers.

Let's say that z is equal to,

is equal to the real number five plus

the imaginary number three times i.

So this thing right over here

we have a real number
plus an imaginary number.

You might be tempted to
add these two things,

but you can't.

They won't make any sense.

These are kind of going in different,

we'll think about it visually in a second,

but you can't simplify this anymore.

You can't add this real number

to this imaginary number.

A number like this, let me make it clear,

that's real and this is
imaginary, imaginary.

A number like this we
call a complex number,

a complex number.

It has a real part and an imaginary part.

Sometimes you'll see notation like this,

or someone will say what's the real part?

What's the real part of
our complex number, z?

Well, that would be the
five right over there.

Then they might say,

"Well, what's the imaginary part?

"What's the imaginary part
of our complex number, z?

And then typically the
way that this function

is defined they really want to know

what multiple of i is this imaginary part

right over here.

In this case it is going to
be, it is going to be three.

We can visualize this.

We can visualize this in two dimensions.

Instead of having the traditional

two-dimensional Cartesian plane

with real numbers on the horizontal

and the vertical axis,

what we do to plot complex numbers

is we on the vertical axis we plot

the imaginary part, so
that's the imaginary part.

On the horizontal axis
we plot the real part.

We plot the real part just like that.

We plot the real part.

For example, z right over here

which is five plus three i,

the real part is five so we would go

one, two, three, four, five.

That's five right over there.

The imaginary part is three.

One, two, three, and so
on the complex plane,

on the complex plane we would visualize

that number right over here.

This right over here is how we

would visualize z on the complex plane.

It's five, positive five
in the real direction,

positive three in the imaginary direction.

We could plot other complex numbers.

Let's say we have the complex number a

which is equal to let's
say it's negative two

plus i.

Where would I plot that?

Well, the real part is negative two,

negative two,

and the imaginary part is going to be

you could imagine this as plus one i

so we go one up.

It's going to be right over there.

That right over there
is our complex number.

Our complex number a
would be at that point

of the complex,

complex, let me write that,

that point of the complex plane.

Let me just do one more.

Let's say you had a complex number b

which is going to be,

let's say it is, let's say
it's four minus three i.

Where would we plot that?

Well, one, two, three, four,

and then let's see minus one, two, three.

Our negative three gets
us right over there.

That right over there would be

the complex number b.