Complex Number Calculator
Add, subtract, multiply, divide and raise complex numbers, find nth roots, modulus, argument and conjugate, in rectangular, polar and exponential form.
Arguments are principal values in (−180°, 180°]. Values are rounded to 10 significant figures. A real or imaginary part smaller than 10⁻¹² of the numbers involved is rounding noise and is shown as 0, so e^(iπ) + 1 gives exactly 0.
| Number | Rectangular | Polar | Modulus |
|---|---|---|---|
| z₁ | 3 + 4i | 5∠53.13010235° | 5 |
| z₂ | 1 − 2i | 2.236067977∠-63.43494882° | 2.236067977 |
A complex number a + bi has a real part a and an imaginary part b, where i² = −1. They are used for AC circuits, signal processing, control systems and roots of polynomials. This calculator reads numbers in rectangular, polar or exponential form, performs arithmetic, powers and nth roots, and shows every answer in all three forms with its modulus, argument and conjugate.
How to use this tool
- Type z₁ as 3+4i, in polar form such as 5∠30°, or in exponential form such as 2e^(iπ/3).
- Choose the operation; for roots, set n.
- Type z₂ when the operation needs it.
- Read the result in rectangular, polar and exponential form; roots are listed in a table.
Formula
(a + bi)(c + di) = (ac − bd) + (ad + bc)i · z₁ ÷ z₂ = z₁·z̄₂ ÷ |z₂|² · |z| = √(a² + b²) · nth roots: |z|^(1/n) ∠ (θ + 360°k)/n
Polar form r∠θ means r(cos θ + i sin θ) = re^(iθ). De Moivre's theorem gives powers and roots: raise the modulus to the power and multiply the angle. Non-integer powers use the principal logarithm, z^w = e^(w·ln z).
Example
Multiplying (3 + 4i)(1 − 2i) gives 3 − 6i + 4i − 8i² = 11 − 2i, with modulus √125 ≈ 11.18. Dividing instead gives −1 + 2i. The cube roots of 8 are 2 and −1 ± 1.732i, spaced 120° apart on a circle of radius 2.
Frequently asked questions
Is the angle in degrees or radians?
Polar results are shown in degrees and the exponential form in radians. For input, 5∠30° is always degrees; an angle without ° follows the setting under the inputs.
Why does i^i give a real number?
i = e^(iπ/2), so i^i = e^(i·iπ/2) = e^(−π/2) ≈ 0.2079. That is the principal value; other branches differ by factors of e^(2πk).
Can I use j instead of i?
Yes. Engineers often write j for the imaginary unit, so 3+4j is read the same as 3+4i.
Last updated:
Page address with these inputs: https://kakao.mobile/tools/complex-number-calculator/
Cite or embed this tool
kakaoCalc. (2026, November 9). Complex Number Calculator. https://kakao.mobile/tools/complex-number-calculator/
“Complex Number Calculator.” kakaoCalc, 9 Nov. 2026, kakao.mobile/tools/complex-number-calculator/.
Paste this code into your page. The widget runs in your visitors' browsers, shows no ads and keeps a link back to this page. Embedding terms