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

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neela | High School Teacher | (Level 3) Valedictorian

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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).

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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)