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## What are the complex numbers?

# Classifying complex numbers

CCSS.Math:

## Video transcript

Now that we know a little bit
about the imaginary unit i, let's see if we can simplify
more involved expressions, like this one right over here. 2 plus 3i plus 7i squared plus
5i to the third power plus 9i to the fourth power. And I encourage you to pause
the video right now and try to simplify this on your own. So as you can see here, we
have various powers of i. You could view this as
i to the first power. We have i squared here. And we already
know that i squared is defined to be negative 1. Then we have i to
the third power. I to the third power would just
be i times this, or negative i. And we already reviewed
this when we first introduced the imaginary
unit, i, but I'll do it again. i to the fourth power is
just going to be i times this, which is the same
thing as negative 1 times i. That's i to the third
power times i again. i times i is negative 1. So that's negative 1 times
negative 1, which is equal to 1 again. So we can rewrite this
whole thing as 2 plus 3i. 7i squared is going
to be the same thing, so i squared is negative 1. So this is the same thing
as 7 times negative 1. So that's just
going to be minus 7. And then we have 5i
to the third power. i to the third
power is negative i. So this could be
rewritten as negative i. So this term right over here
we could write as minus 5i, or negative 5i, depending on
how you want to think about it. And then finally, i to the
fourth power is just 1. So this is just equal to 1. So this whole term
just simplifies to 9. So how could we
simplify this more? Well we have several terms
that are not imaginary, that they are real numbers. For example, we have
this 2 is a real number. Negative 7 is a real number. And 9 is a real number. So we could just add those up. So 2 plus negative 7
would be negative 5. Negative 5 plus 9 would be 4. So the real numbers add up to 4. And now we have these
imaginary numbers. So 3 times i minus 5 times i. So if you have 3 of
something and then I were to subtract 5 of that
same something from it, now you're going to have
negative 2 of that something. Or another way of thinking
about it is the coefficients. 3 minus 5 is negative 2. So three i's minus
five i's, that's going to give you negative 2i. Now you might say, well, can
we simplify this any further? Well no, you really can't. This right over here
is a real number. 4 is a number that
we've been dealing with throughout our
mathematical careers. And negative 2i, that's
an imaginary number. And so what we really consider
this is this 4 minus 2i, we can now consider
this entire expression to really be a number. So this is a number
that has a real part and an imaginary part. And numbers like this
we call complex numbers. It is a complex number. Why is it complex? Well, it has a real part
and an imaginary part. And you might say, well,
gee, can't any real number be considered a complex number? For example, if I have
the real number 3, can't I just write the
real number 3 as 3 plus 0i? And you would be correct. Any real number is
a complex number. You could view this right
over here as a complex number. And actually, the
real numbers are a subset of the complex numbers. Likewise, imaginary
numbers are a subset of the complex numbers. For example, you could
rewrite i as a real part-- 0 is a real number-- 0 plus i. So the imaginaries are a
subset of complex numbers. Real numbers are a subset
of complex numbers. And then complex
numbers also have all of the sums and differences,
or all of the numbers that have both real
and imaginary parts.