Complex Numbers in Polar Form
Master conversions, operations, and De Moivre's Theorem for complex numbers in polar form.
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Questions Covered in This Set
10 cards to master
What is the polar form of a complex number?
z = r(cos θ + i sin θ) or z = r e^(iθ), where r is the modulus and θ is the argument
How do you find the modulus r of z = a + bi?
r = |z| = √(a² + b²)
How do you find the argument θ of z = a + bi?
θ = arctan(b/a), adjusting for the correct quadrant
How do you multiply complex numbers in polar form?
Multiply the moduli and add the arguments: z₁·z₂ = r₁r₂ e^(i(θ₁ + θ₂))
How do you divide complex numbers in polar form?
Divide the moduli and subtract the arguments: z₁/z₂ = (r₁/r₂) e^(i(θ₁ - θ₂))
State De Moivre's Theorem.
[r(cos θ + i sin θ)]ⁿ = rⁿ(cos(nθ) + i sin(nθ)) or (r e^(iθ))ⁿ = rⁿ e^(inθ)
What is the polar form of z = 1 + i?
z = √2 e^(iπ/4) or √2(cos(π/4) + i sin(π/4))
How do you find the n distinct nth roots of z = r e^(iθ)?
z^(1/n) = r^(1/n) e^(i(θ + 2πk)/n) for k = 0, 1, 2, ..., n-1
What is Euler's formula for complex exponentials?
e^(iθ) = cos θ + i sin θ
What geometric interpretation does the complex plane (Argand diagram) provide?
The horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number
