Complex Number Calculator
Add, subtract, multiply or divide two complex numbers; find the modulus, argument, conjugate and polar form of one; or compute integer powers and nth roots with De Moivre's theorem — with a plot on the complex plane and every step shown.
Written by Suhaib Hassan. Every identity is re-derived by hand. Our methodology.
Arithmetic uses the standard algebraic rules (i² = −1); powers and roots use De Moivre's theorem on the polar form, r(cos θ + i sin θ).
It plots every number on the Argand plane, so the geometry behind the algebra is never hidden.
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Argand plane
Each number plotted as a point (Re, Im).
Step-by-step working
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Worked examples
Multiplying
(3+4i) × (1−2i)
= 11 − 2i
Modulus & argument
Find |3+4i| and Arg(3+4i).
θ = atan2(4,3) ≈ 53.13°
De Moivre power
(1+i)⁸
(1+i)⁸ = 16
Key formulas
- Multiplication: (a+bi)(c+di) = (ac−bd) + (ad+bc)i.
- Division: multiply by the conjugate: z₁/z₂ = z₁z̄₂ / |z₂|².
- Modulus: |z| = √(a²+b²). Argument: θ = atan2(b, a), in (−π, π].
- De Moivre's theorem: [r(cos θ + i sin θ)]ⁿ = rⁿ(cos nθ + i sin nθ).
- nth roots: ⁿ√z has n values, at angles (θ+2πk)/n for k = 0, …, n−1, each with modulus |z|^(1/n).
By Suhaib Hassan · Reviewed 29 September 2026 · How CalculatePilot verifies formulas →
Sources
- Churchill, R. V., & Brown, J. W. (2013). Complex Variables and Applications (9th ed.). McGraw-Hill, chapters 1–2.
- ISO 80000-2:2019, clause 8 (complex numbers).
Frequently asked questions
Why does dividing complex numbers use the conjugate?
Multiplying the denominator by its conjugate z̄ turns it into the real number |z|², eliminating i from the denominator — the complex equivalent of rationalizing a square root.
What is the geometric meaning of multiplying complex numbers?
Multiplication multiplies the moduli and adds the arguments — geometrically, it scales and rotates. This is exactly what De Moivre's theorem generalizes to powers and roots.
Why does a complex number have n different nth roots?
Because angles that differ by a full turn (2π) describe the same point, but raising to the nth power multiplies the angle by n — so n different starting angles, spaced 2π/n apart, all land on the same result.
What is the complex conjugate used for?
Besides division, z·z̄ = |z|² always gives a real number, which is essential for finding magnitudes, normalizing vectors, and in signal processing for computing power from a complex amplitude.
How is Euler's formula related to polar form?
Euler's formula e^(iθ) = cos θ + i sin θ lets polar form r(cos θ + i sin θ) be written compactly as r·e^(iθ) — the same number, and the form that makes multiplication, powers and roots most natural.
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