Convert complex numbers between rectangular (a+bi) and polar form (r·eiθ = r(cosθ + i·sinθ)) — find modulus r and argument θ.
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Enter θ in degrees. Any angle is accepted (wraps to [−180°, 180°]).
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Polar Form of Complex Numbers
z = r(cosθ + i sinθ) = r·eiθ
Any complex number z = a + bi can be written in polar form where:
r = |z| = √(a² + b²) is the modulus — the distance from the origin to z in the complex plane.
θ = arg(z) = atan2(b, a) is the argument — the angle from the positive real axis, measured counterclockwise.
By Euler's formula, eiθ = cosθ + i sinθ, so the two polar representations are equivalent.
Multiplication in Polar Form & De Moivre's Theorem
z₁ · z₂ = r₁r₂ · ei(θ₁+θ₂)
Multiplying two complex numbers in polar form: multiply the moduli and add the arguments. This is far simpler than expanding (a+bi)(c+di).
This leads directly to De Moivre's Theorem:
[r(cosθ + i sinθ)]ⁿ = rⁿ(cos nθ + i sin nθ)
To raise a complex number to a power, raise the modulus to that power and multiply the argument by n. Polar form makes this intuitive.
Work through polar form, De Moivre's theorem, and complex roots with a live tutor — step by step, at your pace.