Complex Numbers Calculator
Add, subtract, multiply and divide complex numbers exactly, in a + bi or polar r∠θ form. For one number, get its modulus, argument, polar and exponential forms, conjugate, powers and n-th roots, drawn on the Argand diagram.
How to Use
- Choose two numbers to add, subtract, multiply or divide, or one number to study it.
- Type numbers as 3 + 4i, −2i, 1/2 − 3i, 5j (engineering) or in polar form as 2∠30°.
- Read the result in rectangular and polar form, its modulus and argument.
- For one number, choose n to see zⁿ and its n n-th roots; the table gives the exponential form and reciprocal.
- The Argand diagram draws each number as an arrow from the origin; the roots form a regular polygon.
Worked Example
Dividing. (1 + 2i) ÷ (3 − i): multiply top and bottom by 3 + i. The top becomes 3 + i + 6i + 2i² = 1 + 7i and the bottom 9 − i² = 10, so the answer is 1/10 + 7/10 i.
Cube roots of 8. 8 is 8∠0°. Its cube roots have length ∛8 = 2 and angles 0°, 120° and 240°: 2, −1 + i√3 and −1 − i√3, the corners of an equilateral triangle.
The common mistake: i² = +1. In (3 + 4i)(1 − 2i) the term 4i × (−2i) = −8i² = +8, because i² = −1. Getting this sign wrong gives −5 − 2i instead of 11 − 2i.
Show Work
Formulas
Imaginary, but Useful
Complex numbers forced their way in through cubic equations: in 1545 Gerolamo Cardano’s formula needed square roots of negative numbers even when the final answers were real, and Rafael Bombelli worked out the rules for calculating with them in 1572. Descartes called them “imaginary”, meaning it as an insult, and the name stuck.
Caspar Wessel (1799) and Jean-Robert Argand (1806) drew them as points in a plane, Euler gave e^(iπ) + 1 = 0, and Gauss proved that every polynomial has a complex root. Charles Steinmetz made them the standard language of AC circuits in the 1890s.
About This Calculator
This calculator does complex arithmetic exactly with fractions, so divisions give answers such as 1/10 + 7/10 i rather than rounded decimals. It reads rectangular, engineering (j) and polar input, finds the modulus as an exact surd where possible, the argument in degrees and radians, the exponential form, powers by De Moivre’s theorem and all n-th roots, and draws them on the Argand diagram.
Everything runs in your browser; nothing is sent anywhere.
Related calculators: Quadratic Formula Calculator, Trigonometry Calculator, and Fractal Explorer.
Frequently Asked Questions
How do I multiply complex numbers?
Multiply out the brackets and use i² = −1. (3 + 4i)(1 − 2i) = 3 − 6i + 4i − 8i² = 3 − 2i + 8 = 11 − 2i. In polar form it is easier still: multiply the lengths and add the angles. |3 + 4i| = 5 and |1 − 2i| = √5, so the product has length 5√5 = 11.18.
How do I divide complex numbers?
Multiply top and bottom by the conjugate of the bottom, which makes the bottom real. (1 + 2i) ÷ (3 − i) = (1 + 2i)(3 + i) ÷ ((3 − i)(3 + i)) = (1 + 7i) ÷ 10 = 1/10 + 7/10 i.
What are the modulus and argument?
The modulus |z| = √(a² + b²) is the distance from 0 on the Argand diagram; the argument is the angle from the positive real axis, atan2(b, a). For 3 + 4i: |z| = 5 and arg z = 53.13°, so it is 5∠53.13° in polar form, or 5e^(0.9273i).
How do I find powers and roots?
De Moivre’s theorem: (r∠θ)ⁿ = rⁿ∠nθ. (1 + i) is √2∠45°, so (1 + i)⁸ = 16∠360° = 16. The n-th roots are r^(1/n)∠(θ + 360°k)/n for k = 0 to n − 1, evenly spaced on a circle: the cube roots of 8 are 2, −1 + i√3 and −1 − i√3.
Where are complex numbers used?
Electrical engineering writes AC impedance as R + jX (with j because i means current): a 30 Ω resistor in series with a 40 Ω reactance is 30 + 40j = 50∠53.13° Ω. They also describe waves, rotations, signal processing (the Fourier transform), quantum mechanics and fractals such as the Mandelbrot set.
How do I use the Complex Numbers Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
AC circuits
30 + 40j Ω is 50 Ω at 53.13°.
Homework
(1 + 2i) ÷ (3 − i) = 1/10 + 7/10 i, step by step.
Roots of unity
The 6 sixth roots of 1 form a hexagon.
Powers
(1 + i)⁸ = 16 by De Moivre.
Rotations
Multiplying by i turns any point 90° counter-clockwise.
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