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What is polar (modulus-argument) form?
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All Flashcards in Topic 1.13
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1.13.18 cards
What is polar (modulus-argument) form?
z = r(cosθ + i sinθ) = r cisθ, where r = |z| (modulus) and θ = arg z (argument).
How do you find the modulus and argument from a + bi?
r = √(a² + b²); θ = arctan(b/a), then adjust for the quadrant.
How do you convert polar back to Cartesian?
a = r cosθ and b = r sinθ, then write a + bi.
Why must you check the quadrant for the argument?
arctan(b/a) only returns quadrant 1 or 4 angles; quadrant 2 or 3 points need ±π added.
Write 1 + √3 i in polar form.
r = 2, θ = π/3, so 2 cis(π/3).
Write −√3 + i in polar form.
r = 2; quadrant 2 so θ = π − π/6 = 5π/6; 2 cis(5π/6).
Convert 2 cis(π/6) to Cartesian.
a = 2cos30° = √3, b = 2sin30° = 1, so √3 + i.
What does the argument θ mean geometrically?
The angle the line from 0 to z makes with the positive real axis.
1.13.28 cards
How do you multiply complex numbers in polar form?
Multiply the moduli and add the arguments: r₁r₂ cis(θ₁ + θ₂).
How do you divide complex numbers in polar form?
Divide the moduli and subtract the arguments: (r₁/r₂) cis(θ₁ − θ₂).
What does multiplying by r cisθ do geometrically?
Scales the point by r and rotates it by θ about the origin.
What does multiplying by i do on the Argand diagram?
Rotates 90° anticlockwise (i = cis(π/2), modulus 1).
(2 cis(π/6))(3 cis(π/4)) = ?
6 cis(5π/12) — moduli 2×3 = 6, arguments π/6 + π/4 = 5π/12.
(12 cis(2π/3))/(4 cis(π/4)) = ?
3 cis(5π/12) — moduli 12÷4 = 3, arguments 2π/3 − π/4 = 5π/12.
Common mistake multiplying in polar form?
Adding the moduli or multiplying the arguments. It's moduli × and arguments +.
Why is polar form good for products?
Multiplying just scales and rotates, so it needs only one multiply and one add — no FOIL.
1.13.38 cards
What is exponential (Euler) form?
z = r e^(iθ), with r the modulus and θ the argument in radians.
What is Euler's formula?
e^(iθ) = cosθ + i sinθ — it links the exponential form to the polar form.
How do you multiply in exponential form?
Multiply the moduli and ADD the exponents: r₁r₂ e^(i(θ₁+θ₂)) (an index law).
How do you divide in exponential form?
Divide the moduli and SUBTRACT the exponents: (r₁/r₂) e^(i(θ₁−θ₂)).
Write 4 + 4i in exponential form.
r = 4√2, θ = π/4, so 4√2 e^(iπ/4).
What is e^(iπ)?
−1 (so e^(iπ) + 1 = 0, Euler's identity).
The three forms of a complex number?
Cartesian a + bi, polar r cisθ, exponential r e^(iθ) — all the same number.
Why is the exponent written iθ, not θ?
Because e^(iθ) = cosθ + i sinθ; the i is essential — e^(θ) would be a real exponential.
Topic 1.13 study notes
Full notes & explanations for Complex numbers: polar (HL only)
Math AA exam skills
Paper structures, command terms & tips
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