Complex Number Calculator
Add, multiply, divide and exponentiate complex numbers, convert between rectangular and polar form, and find all n-th roots with the principal value clearly marked.
Last reviewed by the Radiatus Cloud team
Need this done properly for your business?
Radiatus delivers secure cloud, DevOps & compliance engineering.
Two representations of the same number
The rectangular form, a plus bi, makes addition and subtraction trivial: add the parts separately. The polar form, a magnitude and an angle, makes multiplication and powers trivial: multiply the magnitudes and add the angles. Neither is more correct, and choosing the one that matches the operation is what makes complex arithmetic straightforward rather than tedious. Multiplication in rectangular form requires four real multiplications and a sign change that is easy to get wrong; in polar form it is one multiplication and one addition.
Every non-zero number has n distinct n-th roots
The number 1 has three cube roots, not one: the familiar 1 and two complex numbers spaced 120 degrees apart around the unit circle. This follows immediately from the polar form, since adding a full turn to the angle gives the same number but a different root when the angle is divided by n. Programming languages return only the principal root, which is why an equation solved by taking a root usually loses solutions unless the others are found deliberately.
The argument has a branch cut
The angle of a complex number is only defined up to multiples of a full turn, and the convention is to return a value between minus pi and pi. That makes the argument discontinuous across the negative real axis, where it jumps from just under pi to just over minus pi. Any calculation that crosses that line, such as accumulating a phase or taking a logarithm, has to handle the jump explicitly or it produces a discontinuity that looks like a bug in the physics.
Related tools
- Percentage Calculator — Calculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
- Statistical Calculator — Calculate mean, median, mode, standard deviation, and more.
- Probability Calculator — Calculate probabilities for single and combined events, including conditional probability and Bayes' theorem.
- Scientific Calculator — Perform advanced mathematical operations including trig, log, and exponents.
Frequently Asked Questions
When should I use polar form?
For multiplication, division, powers and roots, where it reduces the work to multiplying magnitudes and adding angles. Rectangular form is better for addition and subtraction, where the parts simply add.
Why does a number have several roots?
Because the angle of a complex number is defined only up to a full turn. Adding a turn before dividing by n gives a different angle and therefore a different root, and there are exactly n of them equally spaced around a circle.
What is the principal value?
The single root or angle a convention picks out of the many possible ones, normally the one with an angle between minus pi and pi. Programming languages return it, which is why an equation solved by taking a root silently loses the other solutions.
What is the branch cut?
The line where the argument jumps, conventionally the negative real axis. Just above it the angle is close to pi and just below it is close to minus pi, so any calculation crossing that line sees a discontinuity that must be handled deliberately.
How do I write a complex number here?
Forms such as 3+4i, -2i, 5, 1-i and 2.5e-3+1.2i are all accepted. The imaginary unit can be written i or j, since engineering convention uses j to avoid a clash with current.
Privacy & Security
Everything runs in your browser; nothing is uploaded.
How to Use
Enter two complex numbers to see every operation and both representations.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.
Related Tools
Percentage Calculator
MathCalculate percentages, percentage change, increase and decrease, and reverse percentages, with the working shown for each result.
Statistical Calculator
MathCalculate mean, median, mode, standard deviation, and more.
Probability Calculator
MathCalculate probabilities for single and combined events, including conditional probability and Bayes' theorem.