Why the absolute value of a complex number is (a^2+b^2)^1/2 ?

Generally, the absolute value of a number represents the
distance from that number to the origin of the cartesian system of
coordinates.

A complex number z = a + bi is represented in
the complex plane by the point that has the coordinates
(a,b).

The absolute value of z is the distance form (a,b)
to origin (0,0).

To determine the distance from the origin
to the point (a,b), we'll draw a triangle that has:

- OA:
hypothenuse: the line that joins (0,0) and the point
(a,b).

- AB: cathetus: the line from (a,0) to
(a,b)

- OB: cathetus: the line from (a,0) to
(0,0).

 We'll apply Pythagorean
theorem:

hypothenuse^2 = cathetus^2 +
cathetus^2

OA^2 = AB^2 +
OB^2

OA = sqrt (AB^2 +
OB^2)

AB = b and OB = a

OA =
sqrt (b^2 + a^2)

But OA is the distance from the point
(a,b) to (0,0), namely the absolute value of the complex number
z.

|z| = sqrt(a^2 +
b^2)