The Absolute Value of a Complex Number

Date: 05/06/98 at 21:00:16
From: Russell Nadel
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/