Multiply, divide, raise to powers, and extract roots of complex numbers in polar form using De Moivre's theorem with an interactive polar plot.
Polar form turns complex-number multiplication into simple arithmetic: you multiply the moduli and add the arguments. This property makes polar notation the natural language for powers, roots, and rotations in the complex plane. De Moivre's theorem (r∠θ)ⁿ = rⁿ∠nθ is the key identity and extends effortlessly to finding the n distinct nth roots of any complex number.
This calculator handles four polar-form operations — multiplication, division, exponentiation, and root extraction. Enter moduli and arguments for one or two complex numbers, choose an operation, and get instant results in polar, rectangular, and Euler form. A live polar plot shows all the points and their relationship, while a reference table summarises the modulus and argument rules for every operation.
When computing nth roots, the tool lists every root with its own polar and rectangular representation, evenly spaced around a circle of radius r^(1/n). The preset buttons let you explore classic configurations — two equal phasors multiplied, cube roots of unity, and more. Engineers, scientists, and students will find this indispensable for AC circuit analysis, signal processing, or any application where complex exponentials dominate.
Multiplying, dividing, or raising complex numbers to a power in rectangular form requires tedious FOIL expansions and trig identities. Polar form reduces these operations to simple arithmetic on moduli and angles, and this calculator shows every step: multiply moduli, add (or subtract) arguments, and optionally apply De Moivre's theorem for powers and roots. It lists all n distinct nth roots equally spaced around the circle and converts every result back to rectangular form, so you can verify your work in whichever representation your course expects.
Multiply: r₁r₂ ∠ (θ₁+θ₂) | Divide: (r₁/r₂) ∠ (θ₁−θ₂) | Power: rⁿ ∠ nθ | Root k: r^(1/n) ∠ (θ+2πk)/n
Result: 10 ∠ 83°
Multiply moduli: 5 × 2 = 10. Add arguments: 53° + 30° = 83°. Result is 10∠83°.
In rectangular form, multiplying (a + bi)(c + di) requires four products and grouping real/imaginary parts. In polar form, the same operation is r₁r₂ ∠ (θ₁ + θ₂) — two simple operations. Division is equally clean: (r₁/r₂) ∠ (θ₁ − θ₂). This advantage compounds for powers: (r∠θ)ⁿ = rⁿ ∠ nθ by De Moivre's theorem, whereas expanding (a + bi)ⁿ rectangularly is impractical for n > 2.
Every nonzero complex number z = r∠θ has exactly n distinct nth roots, given by r^(1/n) ∠ (θ + 360°k)/n for k = 0, 1, …, n−1. These roots are evenly spaced on a circle of radius r^(1/n), separated by 360°/n. For example, the cube roots of 8∠270° lie at 2∠90°, 2∠210°, and 2∠330°. This calculator lists every root with both polar and rectangular forms.
Polar-form arithmetic is the backbone of AC circuit analysis (impedance, phasors), signal processing (Fourier coefficients), and control theory (pole-zero plots). Multiplying phasors gives the combined amplitude and phase shift; dividing them gives the transfer function. Roots of unity — the nth roots of 1∠0° — appear in the Fast Fourier Transform (FFT) and cyclotomic polynomials.
It states that (r∠θ)ⁿ = rⁿ∠nθ, allowing you to raise a complex number to any integer power by raising the modulus and multiplying the argument. Use this as a practical reminder before finalizing the result.
Multiply the moduli and add the arguments: (r₁∠θ₁)(r₂∠θ₂) = r₁r₂ ∠ (θ₁+θ₂). Keep this note short and outcome-focused for reuse.
Every nonzero complex number has exactly n distinct nth roots, evenly spaced 360°/n apart on a circle of radius r^(1/n). Apply this check where your workflow is most sensitive.
Polar form simplifies multiplication, division, powers, and roots to basic operations on moduli and arguments, which is cumbersome in rectangular form. Use this checkpoint when values look unexpected.
Yes — toggle the angle unit between degrees and radians with the dropdown. Validate assumptions before taking action on this output.
Division by zero is undefined; the calculator shows an error instead of a result. Review this only if your environment differs from defaults.