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

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The absolute value of the complex number can be evaluated when we know the rectangular form of z:

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 absolute value:

|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 made by the vector of position of the complex number.

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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