Why are the absolute values of a complex number and its conjugate always equal?

If you understand what the absolute value of a complex number is, skip to the tl;dr part at the bottom.

The absolute value can be thought of as a sorts of 'norm', because it assigns a positive value to a number, which represents that number's "distance" from zero (except for the number zero, which has an absolute value of zero).

For real numbers, the "distance" from zero is merely the number without it's sign.

For complex numbers, the "distance" from zero is the length of the line drawn from 0 to the number plotted on the complex plane.

In order to see why, take any complex number of the form a + b*i, where 'a' and 'b' are real numbers and 'i' is the imaginary unit. In order to plot this number on a complex plane, just simply draw a normal graph. The number is located at (a,b).

In order to determine the distance from 0 (0,0) to our number (a,b) we draw a triangle using these three points:

(0,0)

(a,0)

(a,b)

Where the points (0,0) and (a,b) form the hypotenuse. The length of the hypotenuse is also the "distance" of a + b*i from zero. Because the legs run parallel to the x and y axes, the lengths of the two legs are 'a' and 'b'.

By using the Pythagorean theorem, we can find the length of the hypotenuse as (a2 + b2)(1/2).

Because the length of the hypotenuse is also the 'distance' of the complex number from zero on the complex plane, we have the definition:

|a + b*i| = (a2 + b2)(1/2)

ALRIGHT, almost there.

tl;dr:

Remember that the complex conjugate of a complex number a + b*i is a + (-b)*i. By plugging this into the Pythagorean theorem, we have:

b2 = (-b)2

So:

(a2 + (-b)2)(1/2) = (a2 + b2)(1/2)

QED.