What is the modulus and principal argument of the complex number z = -1 + i * sqrt(3)?
Answer and explanation
Correct answer: A. Modulus r = 2, argument theta = 2 pi / 3
- Step 1: Compute modulus: r = sqrt((-1)^2 + (sqrt(3))^2) = sqrt(1 + 3) = sqrt(4) = 2.
- Step 2: Identify quadrant: Re(z) = -1 < 0, Im(z) = sqrt(3) > 0, so z lies in the Second Quadrant.
- Step 3: Compute argument: theta = pi - arctan(sqrt(3) / 1) = pi - pi/3 = 2pi/3.
- Step 4: Confirm: r = 2 and theta = 2pi/3.
Answer
Modulus r = 2, argument theta = 2 pi / 3.
Why not the others: 'Modulus r = 2, argument theta = pi / 3' places z in the First Quadrant; 'Modulus r = 4' forgets the square root; 'argument theta = -pi / 3' places z in the Fourth Quadrant.
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