Two complex numbers z₁ = 2(cos(π/6) + i sin(π/6)) and z₂ = 3(cos(π/3) + i sin(π/3)) are multiplied. What is z₁z₂ in polar form?
A5(cos(π/2) + i sin(π/2))
B6(cos(π/2) + i sin(π/2))
C6(cos(π/18) + i sin(π/18))
D6(cos(π/9) + i sin(π/9))
In polar form, multiplication multiplies moduli and adds arguments. The moduli are 2 and 3, so the product modulus is 2 × 3 = 6. The arguments are π/6 and π/3, so the product argument is π/6 + π/3 = π/6 + 2π/6 = π/2. Result: 6(cos(π/2) + i sin(π/2)) = 6i. Option A (modulus 5) is the classic error of adding moduli instead of multiplying them. Options C and D use incorrect combinations of the arguments.
Question 2 Multiple Choice
What geometric transformation does multiplying any complex number z by i correspond to in the complex plane?
AReflection across the imaginary axis
BScaling by a factor of 1 (no change)
CRotation by π/2 counterclockwise about the origin
DReflection across the line y = x
The number i has modulus 1 and argument π/2. Multiplying z by i multiplies z's modulus by 1 (unchanged) and adds π/2 to z's argument — a 90° counterclockwise rotation. For example: 1 (angle 0) → i (angle π/2) → −1 (angle π) → −i (angle 3π/2) → 1. This four-step rotation is the geometric picture behind i⁴ = 1. The polar form makes this rotation interpretation immediate; in Cartesian form, multiplying (a + bi)(0 + i) = −b + ai is mechanically correct but geometrically opaque.
Question 3 True / False
Raising a complex number z = r(cos θ + i sin θ) to the nth power gives rⁿ(cos(nθ) + i sin(nθ)) — the modulus is raised to the nth power and the argument is multiplied by n.
TTrue
FFalse
Answer: True
This is De Moivre's theorem, which follows directly from the multiplication rule for polar form. Multiplying z by itself multiplies the moduli (r × r = r²) and adds the arguments (θ + θ = 2θ). Repeating n times gives rⁿ(cos(nθ) + i sin(nθ)). Geometrically: raising a complex number to the nth power scales its distance from the origin by rⁿ and rotates it by n times its original angle. For example, (1+i)⁸ has modulus (√2)⁸ = 16 and argument 8 × π/4 = 2π, so (1+i)⁸ = 16.
Question 4 True / False
Polar form is mainly a notational convenience — computing with it is about the same difficulty as using the Cartesian form a + bi.
TTrue
FFalse
Answer: False
This underestimates polar form. For multiplication and division, polar form is dramatically simpler: multiply moduli and add arguments vs. expanding (a+bi)(c+di) = (ac−bd) + (ad+bc)i and simplifying. For powers and roots, the advantage is overwhelming — computing (1+i)¹⁰ in Cartesian form requires nine multiplications; in polar form it is immediate: modulus (√2)¹⁰ = 32, argument 10 × π/4 = 5π/2 = π/2. Polar form is not a notational choice; it is the form that makes multiplication, division, powers, and roots geometrically transparent.
Question 5 Short Answer
Why does multiplying two complex numbers in polar form reduce to multiplying their moduli and adding their arguments? Explain the geometric meaning of this rule.
Think about your answer, then reveal below.
Model answer: The rule follows from the cosine and sine addition formulas. Expanding z₁z₂ = r₁r₂[(cos θ₁ cos θ₂ − sin θ₁ sin θ₂) + i(sin θ₁ cos θ₂ + cos θ₁ sin θ₂)] gives r₁r₂(cos(θ₁ + θ₂) + i sin(θ₁ + θ₂)) by the angle addition identities. Geometrically, multiplying by z₂ performs two operations: it scales the entire complex plane by r₂ (stretching z₁'s distance from the origin) and rotates by angle θ₂. Complex multiplication is scaling + rotation. This geometric picture — that multiplication in the complex plane combines distance and direction independently — is the key insight polar form provides.
The mathematical derivation uses angle addition formulas, but the geometric interpretation is the important takeaway. Every complex number z = r(cos θ + i sin θ) is completely characterized by how far it is from the origin (r) and what direction it points (θ), and multiplication combines these by the simplest possible rules. This interpretation leads directly to De Moivre's theorem and to the connection e^(iθ) = cos θ + i sin θ that polar form anticipates.