Module of complex number.Find why the absolute value of a complex number is (a^2+b^2)^1/2 ?

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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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)

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