A complex number is of the type z = x+iy, where x and y are real and i is sqrt(-1).
z bar = x-iy is called the conjugate of complex number x+iy . Also x+iy is the complex conjugate number of the complex number x-iy. Thus (zbar)bar = z.
The the modulus of the complex number is defined as |z |= sqrt(z* cojugate of z) = sqrt(z*zbar).
Therefore |z| = sqrt(z*zbar) = sqrt((x+iy)(x-iy)) .
|z| = sqrt [(x+iy)(x-iy)].
|z| = sqrt(x^2 -ixy+iyx - i^2*y)
|z| = sqrt(x^2 + y^2) , as -i^2*y^2 = -(-1)y^2 = y^2.
|z| = |x+iy| =sqrt( x^2+y^2).
Thus if a+ib is complex number , then |a+ib| = sqrt(a^2+b^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)