About the Complex Number Calculator
A complex number has a real part and an imaginary part, written a + bi where i squared is -1. This calculator does the usual arithmetic on them: addition, subtraction, multiplication and division, plus whole-number powers and all the n-th roots of a number. For every result it also shows the real and imaginary parts, the modulus (distance from zero), the angle in degrees and radians, and the conjugate. It is aimed at students of algebra, electronics and physics who need quick, checkable answers.
Updated October 2026
How to use the Complex Number Calculator
- 1Choose the operation from the list.
- 2Enter the real and imaginary parts of number A.
- 3For add, subtract, multiply and divide, enter the real and imaginary parts of number B.
- 4For powers and roots, enter the whole number n instead.
- 5Read the result in a + bi form together with its modulus, angle and conjugate.
Working with a + bi
Addition and subtraction work on the parts separately: add the real parts together and the imaginary parts together. Multiplication follows the ordinary distributive rule, with the extra step that i squared becomes -1. Division uses the conjugate to clear the imaginary part from the bottom. The calculator follows exactly these steps, so you can use it to check handwritten work.
Powers and roots with polar form
- To raise a number to the power n, raise its modulus to n and multiply its angle by n (De Moivre's theorem).
- To find n-th roots, take the n-th root of the modulus and divide the angle by n, then add a full turn divided by n for each further root.
- The roots of any non-zero number form a regular polygon around zero.
- Zero raised to a negative power is undefined, and the tool says so.
Where complex numbers are used
In alternating-current circuits, impedance is a complex number: the real part is resistance and the imaginary part is reactance. Multiplying and dividing complex impedances gives the same results as engineers work out by hand. In signal processing the Fourier transform produces complex values, and in geometry multiplying by a complex number rotates and scales points. For related work try the Trigonometry Calculator for sine and cosine, the Quadratic Equation Solver when a quadratic has negative discriminant, and the Matrix Calculator.
Features
- Add, subtract, multiply and divide
- Whole-number powers from -1000 to 1000
- All n-th roots from 1 to 12, listed one by one
- Modulus and angle in degrees and radians
- Conjugate of every result
- Division by zero reported clearly
- Runs in your browser
Frequently asked questions
What is a complex number?
A complex number is a number of the form a + bi, where a and b are ordinary real numbers and i is the imaginary unit with i squared equal to -1. The number a is the real part and b is the imaginary part. Complex numbers let you take square roots of negative numbers and are used in electrical engineering, signal processing and physics.
How do you multiply two complex numbers?
Multiply each part of the first by each part of the second and use i squared equals -1. For (3 + 4i) times (1 - 2i), you get 3 - 6i + 4i - 8i squared, which is 3 - 2i + 8, so the answer is 11 - 2i. The calculator does this for you and displays the result.
How do you divide complex numbers?
Multiply the top and bottom by the conjugate of the bottom number, which flips the sign of its imaginary part. That makes the denominator a real number, and the answer can be split into real and imaginary parts. For example (3 + 4i) divided by (1 - 2i) equals -1 + 2i.
What are the modulus and argument?
The modulus is the distance of the number from zero on the complex plane, the square root of a squared plus b squared. The argument is the angle from the positive real axis. Together they give the polar form: the number equals modulus times cosine of the angle plus i times sine of the angle.
How many roots does a complex number have?
A non-zero complex number has exactly n different n-th roots, spaced evenly around a circle on the complex plane. The square roots of -1 are i and -i. The calculator lists every root for degrees 2 to 12, and the first one shown is the principal root.
Can I use decimals and negative numbers?
Yes. Both parts accept decimals and negative values, for example 2.5 and -3. The parts are shown rounded to six decimal places, and tiny rounding noise such as 0.0000000001 is cleaned to zero.