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## Modulus (absolute value) and argument (angle) of complex numbers

# Absolute value of complex numbers

CCSS.Math:

## Video transcript

I have the complex
number 3 minus 4i. I've plotted it on
the complex plane. We see that the real part
is 3, so we've gone 3 along the horizontal
axis, or the real axis. And the imaginary
part is negative 4, so we've gone down 4
along the vertical axis. So this right here is
the point 3 minus 4i. Now what I want
to think about is what the absolute
value of 3 minus 4i is. And just as a reminder,
absolute value literally means-- whether we're talking
about a complex number or a real number, it literally
just means distance away from 0. So the absolute
value of 3 minus 4i is going to be the
distance between 0, between the origin on the
complex plane, and that point, and the point 3 minus 4i. So this distance
right over here is going to be the absolute
value of 3 minus 4i. So how can we think about that? Well, we could literally
just set up a right triangle and then use the
Pythagorean theorem. So let's think about it. If we wanted to set up a right
triangle, the height here, the distance between 0
and negative 4, well, that distance is going to be 4. And then the base
of this triangle, the distance between 0 and
3, is just going to be 3. And this is definitely
a right angle. This is a horizontal line. This is a vertical line. We can now use the Pythagorean
theorem to figure out the absolute value
of 3 minus 4i. The distance between
this point and 0-- it's the hypotenuse of
this right triangle. So we just use the
Pythagorean theorem. This side squared, 3 squared,
plus this side squared, plus 4 squared, is going to
be equal to the absolute value of 3 minus 4i squared, the
absolute value squared. So 3 squared plus 4 squared,
that's 9 plus 16, which is 25. So you get 25 is equal to the
absolute value of 3 minus 4i squared. And we know if you take the
absolute value of something, this is just a distance. It's going to be positive. So we want to take the
positive square root, the principal square root,
of both sides of this. And so we're going to
be left with-- well, the principal square root, the
positive square root of 25, is 5, is equal to the
absolute value of 3 minus 4i. So another way of saying it,
this thing right over here is going to be equal to 5. This distance right
over here is equal to 5. Now, without having
to draw it, one way you could just think about
this is I'll have my real part. I have my imaginary part. I could literally take
each of those parts, square them, take the sum,
and take the square root. So another way of
taking it, if you didn't want to visualize
all this-- but this is really what we're doing. You could say,
well, this is just going to be equal to take
the real part squared. Take the imaginary part
squared-- so let me write this. Add them together, and
then take the square root, or the principal root. The principal root's just
the positive square root. So that's going to be
the square root of 9 plus 16, which, once
again, is equal to 5.