Associated Topics || Dr. Math Home || Search Dr. Math

### The Absolute Value of a Complex Number

```
Date: 05/06/98 at 21:00:16
Subject: Complex Numbers

In our Alg II class, we were working with complex numbers (a + bi
format). Our textbook and graphing calculators said that the absolute
value of a + bi was the same thing as sqrt(a^2 + b^2). We couldn't
understand where the i went. Could you help us?
```

```
Date: 05/11/98 at 10:56:02
From: Doctor Nick
Subject: Re: Complex Numbers

Hi Russell -

One way of understanding absolute value is the following: the absolute
value of a number is the distance from that number to zero. That is,
if we think of a number as a point in the x-y plane, the absolute
value of that number is the distance between that point and the
origin.

Now, if the number is real, as a point it lies on the x-axis, so the
distance from the origin to the number is just the usual absolute
value.

But, if we take a complex number a + bi, it sits at the point (a,b) in
the x-y plane. How far is (a,b) from (0,0)? Well, we can make a
triangle to help us out:

draw a line (the hypotenuse) between the origin and (a,b);
draw a line from (a,0) to (a,b);
draw a line (a,0) to (0,0)

That gives us a right triangle that lies on the x-axis.

Now, since it's a right triangle, we know that the length of the
hypotenuse is the square root of the sum of the squares of the other
sides.

How long are these other sides?

Well, one is the absolute value of a, and the other is the absolute
value of b. (If a and b are positive, you don't need the absolute
value, but we need to allow a and b to be negative, so that a + bi can
be any complex number.)

Now, the Pythagorean theorem says that the distance from (a,b) to
(0,0) is the square root of |a|^2 + |b|^2. But, |a|^2 = a^2 for all
real numbers a. (Why? If a >= 0, then |a| = a, so it's true. If a < 0,
then |a| = -a, so |a|^2 = (-a)^2 = ((-1)^2)*a^2 = (1)a^2 = a^2.)

So the distance from (a,b) to (0,0) is the square root of a^2 + b^2,
and that's why it's the absolute value of a + bi.

Notice that this formula works for real numbers as well as complex
ones. If x is real, then x = x + 0i, and the absolute value of x is
the square root of x^2 + 0^2, which is just the square root of x^2.
What's the square root of x^2? The usual absolute value of x.

Have fun,

-Doctor Nick, The Math Forum
Check out our web site! http://mathforum.org/dr.math/
```
Associated Topics:
High School Imaginary/Complex Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search