Complex Number Calculator

About the Complex Number Calculator

Add, subtract, multiply and divide complex numbers written in any form — rectangular a + bi, polar r∠θ (or r cis θ) and exponential re^(iθ) — and see each answer in all three forms with its modulus, argument and conjugate. Every operation is worked step by step: multiplying out with i² = −1, dividing by the conjugate, and the same result by the polar route as a check.

Two more modes raise a number to any whole power with De Moivre’s theorem and find all n-th roots, laid out on an Argand diagram. Answers stay exact wherever the numbers allow — 1 + i, i√3, √2/2, π/4, the cube roots of 8 — and are given to the decimal places you choose otherwise. Everything is worked out in your browser.

Common uses

• Checking complex number homework in algebra, trigonometry and further mathematics.
• Converting phasors between rectangular and polar form for AC circuit work.
• Finding all roots of a complex number, such as the n-th roots of unity, with a picture of where they lie.
• Seeing why multiplying complex numbers rotates and scales them on the Argand diagram.

The three forms and how to convert

A complex number z = a + bi is the point (a, b) on the Argand diagram. Its modulus r = |z| = √(a² + b²) is the distance from 0, and its argument θ is the angle from the positive real axis. This calculator gives the principal argument, −180° < θ ≤ 180° (−π < θ ≤ π), the single-valued choice described in NIST’s Digital Library of Mathematical Functions, §1.9. Because arctan only returns angles between −90° and 90°, the quadrant decides: θ = arctan(b/a) when a > 0, θ = π − arctan|b/a| in the second quadrant and θ = −π + arctan|b/a| in the third.

Then z = r(cos θ + i sin θ) = r∠θ, and by Euler’s formula e^(iθ) = cos θ + i sin θ the same number is re^(iθ). Going back, a = r cos θ and b = r sin θ.

Multiplying, dividing and De Moivre’s theorem

In rectangular form, multiply out and use i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. To divide, multiply the top and bottom by the conjugate of the bottom, which turns it into the real number c² + d². In polar form both are simpler: multiply the moduli and add the arguments, or divide the moduli and subtract the arguments.

Repeating the multiplication gives De Moivre’s theorem (DLMF 1.9.22): (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ for every whole number n, so zⁿ = rⁿ(cos nθ + i sin nθ). The angle nθ is then brought back into −180° < nθ ≤ 180°.

All n-th roots

Every non-zero complex number has exactly n different n-th roots: w_k = r^(1/n) ∠ (θ + 360°·k)/n for k = 0, 1, …, n − 1. They lie on a circle of radius r^(1/n), spaced 360°/n apart, so for n ≥ 3 they are the corners of a regular polygon, and they always add up to 0. The principal root is w₀, the one a calculator’s z^(1/n) gives.

Limitations

• Exact answers use fractions, square roots and π. Values that need nested radicals or other numbers (sin 72° = √(10 + 2√5)/4, e, ln 2) are given as decimals, to up to 10 decimal places.
• Powers are whole numbers from −1,000 to 1,000 and roots from the 2nd to the 64th. Powers above 64 are worked with De Moivre’s theorem in floating point, about 15 significant digits, and a power whose modulus is above about 10³⁰⁸ (or closer to 0 than about 10⁻³²³) cannot be shown.
• sqrt, ln, z^w for a non-whole w and arg give principal values, with −π < arg z ≤ π.
• A polar angle such as 10∠45°/2 is read as 10∠(45°/2); write (10∠45°)/2 to divide the number.

Privacy

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