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

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neela | High School Teacher | (Level 3) Valedictorian

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Argument of a complex number z = x+iy arg(z) = arc tan (y/x)

And the absolute value of z= x+iy is |z| = sqrt (x^2+y^2).

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

Let z = x+iy.

Then x = 2 and y = (3^(1/2)/2

 x^2 = 2^2 = 4 and y^2 = {(3^1/2)/2}^2 = 3/4.

Therefore x^2+y^2 = 4+3/4 = 19/4.

|z| = sqrt(x^2+y^2) = sqrt(19/4) = (1/2)sqrt(19).

Absolute value of z = |z| = (sqrt19)/2.

Arg z = arc tan (y/x) = sqrt(3/4) /2 =  arc tan {(sqrt 3)/ 4} =  23.41 deg or  = 0.130073pi = 0.40864 rad nearly.

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giorgiana1976 | College Teacher | (Level 3) Valedictorian

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The absolute value of a complex number is also called the modulus of the complex number and it can be found from rectangular form:

z = x + i*y (rectangular form)

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

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

x = Re(z) = 2

y = Im(z) = (sqrt 3)/2

Now, we'll calculate the modulus:

|z| = sqrt[2^2 + (sqrt3)^2/4]

|z| = sqrt (4+3/4)

|z| = sqrt (19/4)

|z| = sqrt (19)/2

The modulus of the given complex number is |z| = sqrt (19)/2.

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

arg(z) = a

tan a = y/x

tan a = (sqrt 3)/4

a = arctan[(sqrt 3)/4] + k*pi

arg(z) = arctan[(sqrt 3)/4] + k*pi

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