Complex number calculator
Complex arithmetic, modulus and argument, conjugates, conversion between rectangular, polar and exponential forms, integer and fractional powers, and all n roots of a complex number — with near-zero components cleaned up.
Runs in your browserEvery computation happens in your browser — your data never leaves this device.
Input
Accepts 1+2i, -3i, i, 1.5-0.5i and 3∠45; near-zero components are cleaned so you never see 1e-17 noise.
Result
11 + 2i
11.180339887510.304846468811 - 2i11.1803398875 ∠ 10.3048464688°11.1803398875·e^(i 10.3048464688°)What this tool does
- Work through complex analysis or circuit exercises: (1+2i)(3−4i) returns 11 + 2i without expanding and collecting terms by hand.
- See the same result in every notation at once: modulus, argument, conjugate, polar r∠θ and exponential r·e^(iθ) are all shown.
- Apply De Moivre’s formula: powers use fast exponentiation and n-th roots list all n solutions, e.g. the fourth roots of 1+i sit 90° apart.
- Check magnitude and phase before plotting: switch between degrees and radians so you never convert 45° to π/4 by hand.
Example
Input
z₁ = 1 + 2i, z₂ = 3 − 4i, operation “z₁ × z₂”, degree mode
Output
11 + 2i; modulus 11.1803398875, argument 10.3048464688°, conjugate 11 − 2i, polar 11.1803398875 ∠ 10.3048464688°, exponential 11.1803398875·e^(i 10.3048464688°)
Every component is rounded to 12 significant digits and near-zero noise is zeroed, so (1+i)/(1−i) gives a clean i instead of a 1e-17 tail.
Frequently asked questions
How can I enter a complex number?
All the usual forms work: 1+2i, -3i, i, -i, 2 and 1.5-0.5i, with j accepted as the imaginary unit; polar input such as 3∠45 is read in degrees. Spaces are ignored, so “1 - 2i” equals “1-2i”.
Why do n-th roots give n results?
Root extraction is multi-valued over the complex numbers: every n-th root of z has modulus |z|^(1/n) and argument (θ + 2πk)/n for k = 0…n−1, so exactly n points are spread evenly on one circle. All of them are listed in increasing angle, and the tests verify that each one raised to the n-th power returns the original number.
Can the exponent be fractional?
Yes. Integer exponents use fast exponentiation for extra accuracy, while fractional ones go through polar form rⁿ·e^(inθ), so 0.5 behaves like the principal square root. Zero to a non-positive-integer exponent is rejected because it is not well defined.
Why is there no 1e-17 in the output?
Each component is rounded to 12 significant digits and anything below 1e-12 is zeroed. Squaring 1+i therefore gives a clean 2i rather than 1.2246e-16 + 2i floating-point noise.
Which outputs does the angle unit affect?
Only the argument, the polar form and the exponential form: 45° in degree mode becomes 0.7853981634 in radian mode. The arithmetic itself (addition, multiplication, powers, roots) is unit-independent, while polar input like 3∠45 is always interpreted in degrees.
Keywords：complex number复数计算复数运算complex calculatorpolar form极坐标exponential form指数形式modulus argument模长辐角nth rootsn 次方根de moivre棣莫弗公式