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What is the modulus r of z = a + bi?
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All Flashcards in Topic 1.13
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1.13.18 cards
What is the modulus r of z = a + bi?
r = |z| = √(a² + b²) — its distance from the origin on the Argand diagram.
What is the argument of z?
The angle θ from the positive real axis (anticlockwise positive); fix the quadrant by sketching the point.
Write the three equivalent polar/exponential forms of z.
z = r(cos θ + i sin θ) = r cis θ = r e^(iθ).
How do you multiply two complex numbers in polar form?
Multiply the moduli and add the arguments: z₁z₂ = r₁r₂ e^(i(θ₁+θ₂)).
How do you divide two complex numbers in polar form?
Divide the moduli and subtract the arguments: z₁/z₂ = (r₁/r₂) e^(i(θ₁−θ₂)).
Convert z = 1 + √3 i to exponential form.
r = √(1+3) = 2, θ = arctan(√3) = π/3, so z = 2 e^(iπ/3).
Convert 4 e^(iπ/6) to a + bi.
4(cos π/6 + i sin π/6) = 4(√3/2 + i/2) = 2√3 + 2i.
Geometrically, what does multiplying by r e^(iθ) do to the Argand diagram?
It scales (stretches) by r and rotates by angle θ — that's why complex numbers model rotations and phase shifts.
1.13.28 cards
State De Moivre's theorem.
(r cis θ)ⁿ = rⁿ cis(nθ) = rⁿ e^(inθ): power the modulus, multiply the argument by n.
Evaluate (1 + i)⁸ using De Moivre.
r = √2, θ = π/4; (√2)⁸ cis(8·π/4) = 16 cis(2π) = 16.
When is zⁿ a (positive) real number?
Real when nθ is a multiple of π; positive real when nθ is a multiple of 2π.
What is the impedance of an AC circuit in complex form?
Z = R + iX, where R is resistance and X is reactance; |Z| is the total opposition and arg Z is the phase angle.
How do impedances combine in series?
They add as complex numbers: Z_total = Z₁ + Z₂ + …
How do you add two sinusoids of the same frequency?
Represent each as a phasor A e^(iφ), add the phasors, then read off the resultant amplitude (modulus) and phase (argument).
Find |Z| for Z = 3 + 4i Ω.
|Z| = √(3² + 4²) = √25 = 5 Ω.
Find z⁵ for z = 1 − √3 i.
r = 2, θ = −π/3; z⁵ = 32 cis(−5π/3) = 32 cis(π/3) = 16 + 16√3 i.
Topic 1.13 study notes
Full notes & explanations for Complex numbers: continued (HL only)
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