By definition, the module of a number is 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 identity:
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 module of the complex number z.
|z| = sqrt(a^2 + b^2)