What are the argument and the modulus of the complex number z=1+i*3^1/2 ?

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neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

If z = x+iy,

Then we can write this in the polar form as r (cosp +isinp)

where  r = sqrt(x^2+y^2).

x = rcosp

y = r sin p.

So tan p = y/x.

Therefore p = arctan (y/x) is called argument of x+iy.

Also r = sqrt(x^2+y^2) is the modulus of z or (x+iy).

Given z = 1+i*3^(1/2).

Therefore modulus of z = |z| = sqrt{1^2+(3^1/2)^2}

|z| = sqrt(1+3) = sqrt4 = 2.

Therefore modulus of z = |z| = 2.

Argument of 1+i*3^(1/2) =  arc tan (3^1/2)/1 = arc tan (sqrt3)= pi/3.

Therefore argument of z = argument of (1+i*3^1/2) = pi/3 or 60 degree.

giorgiana1976's profile pic

giorgiana1976 | College Teacher | (Level 3) Valedictorian

Posted on

The modulus of the complex number can be found from rectangular form;

z = x + i*y

|z| = sqrt(x^2 + y^2)

We'll identify the real part and the imaginary part of z:

x = Re(z) = 1

y = Im(z) = sqrt 3

Now, we'll calculate the modulus:

|z| = sqrt[1^2 + (sqrt3)^2]

|z| = sqrt (1+3)

|z| = sqrt 4

|z| = 2

The modulus of the given complex number is |z| = 2.

The argument of the complex number is the angle to x axis.

arg(z) = a

tan a = y/x

tan a = sqrt 3/1

tan a = sqrt 3

a = pi/3 + k*pi

arg(z) = pi/3 + k*pi

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