Questions: Modulus and Argument of Complex Numbers

Question 1

The complex number z = 1 + i is multiplied by i. What is the geometric effect on z?

• A. z is scaled by a factor of √2 with no change in direction
• B. z is rotated 90° counterclockwise about the origin, with its modulus unchanged
• C. z is reflected across the imaginary axis
• D. z is rotated 45° counterclockwise, since i is at 45° from the real axis

Answer: (B)

Multiplication by a complex number with modulus r and argument θ scales by r and rotates by θ. The number i has modulus 1 and argument π/2 (it lies on the positive imaginary axis, 90° from the real axis). Multiplying any complex number by i therefore scales it by 1 (no change in size) and rotates it by π/2 = 90° counterclockwise. For z = 1 + i: z has modulus √2 and argument π/4. After multiplying by i, the modulus stays √2 and the argument becomes π/4 + π/2 = 3π/4, which corresponds to the point −1 + i. The common error in option D is confusing the argument of i (90°) with the argument of z (45°) — multiplication adds the arguments.

Question 2

What is the principal argument Arg(z) of z = −√3 + i?

• A. −π/6
• B. π/3
• C. 5π/6
• D. −5π/6

Answer: (C)

The point −√3 + i lies in the second quadrant (negative real part, positive imaginary part). Its modulus is √(3 + 1) = 2. The reference angle is arctan(1/√3) = π/6. Since the point is in the second quadrant, the argument is π − π/6 = 5π/6. The principal argument Arg(z) is the unique value in (−π, π], so 5π/6 is correct — it lies in (0, π), well within the principal range. Option A applies the formula arctan(y/x) without adjusting for quadrant. Option D gives a negative angle appropriate for the third quadrant.

Question 3

The argument of a product of two complex numbers equals the sum of their individual arguments — but this sum may need to be adjusted by adding or subtracting 2π to bring it within the chosen standard range.

• T. True
• F. False

Answer: True

The rule arg(z₁z₂) = arg(z₁) + arg(z₂) holds exactly, but 'argument' here is multi-valued (defined only up to multiples of 2π). If we use principal arguments Arg(z) ∈ (−π, π], the sum Arg(z₁) + Arg(z₂) may fall outside (−π, π]. For example, Arg(−1 + 0i) = π and Arg(−1 + 0i) = π, but (−1)(−1) = 1 has Arg = 0, not 2π. So we must reduce: π + π = 2π ≡ 0 (mod 2π). The adjustment is real and necessary — this is exactly why the multi-valuedness of arg matters.

Question 4

Nearly every complex number has a unique argument, just as most positive real number has a unique absolute value.

• T. True
• F. False

Answer: False

The argument is defined only up to integer multiples of 2π — any angle θ and θ + 2πk (for integer k) point in the same direction and represent the same complex number. The principal argument Arg(z) ∈ (−π, π] is a unique representative chosen by convention, but the argument itself is inherently multi-valued. This multi-valuedness has serious consequences in complex analysis: when defining log z = ln|z| + i·arg(z), the multi-valuedness of arg makes log itself multi-valued, requiring a choice of 'branch' to work with a single-valued function. This is distinct from the absolute value |z|, which genuinely has a unique value for each z.

Question 5

Why does multiplying a complex number by i rotate it by 90°, and how does this explain why i² = −1?

Model answer: The number i sits on the unit circle at argument π/2 (90° counterclockwise from the positive real axis) and has modulus 1. When you multiply any complex number z by i, the moduli multiply (1 × |z| = |z|, so the distance from the origin is unchanged) and the arguments add (arg(z) + π/2). Multiplication by i is therefore a 90° counterclockwise rotation. Applying this twice — computing i² — rotates by 90° + 90° = 180°. A 180° rotation maps any point to its antipodal point on the opposite side of the origin. For the real number 1 (which lies at argument 0 on the unit circle), a 180° rotation lands at −1. So i × i = i² = −1. This is not a coincidence or an arbitrary rule — it is the geometric content of what it means to rotate the complex plane by 90° twice.