Complex Number Calculator
Complex Number Calculator works out Real part of the result and Imaginary part of the result from Operation, Enter the numbers as and a: real part. The default figures on this page give result (rectangular form) of 11 - 2i; change any entry and the result, the working and the sources update together.
Your figures
Updates as you typeHow this is worked out
5 steps
- a = 3 + 4i (modulus 5, argument 53.130102 deg)
- b = 1 - 2i (modulus 2.236068, argument -63.434949 deg)
- (x + yi)(u + vi) = (xu - yv) + (xv + yu)i
- Result = 11 - 2i
- |result| = sqrt(11^2 + -2^2) = 11.18034
Its 5 worked examples are re-run on every release, last on 15 September 2026. What changed
Sources and references
2 sources
What the Complex Number Calculator calculates
Adds, subtracts, multiplies, divides, powers and roots complex numbers and converts between rectangular, polar and exponential form with modulus and principal argument.
How to use this Complex Number Calculator
Enter Operation, Enter the numbers as, a: real part and a: imaginary part. Real part of the result updates as you type, and “Use a worked example” loads checked figures you can adjust. Change one figure at a time to see what drives the result.
How the result is worked out
Open “How this is worked out” beside the result for the step-by-step calculation behind Real part of the result, Imaginary part of the result and Result (rectangular form).
The method follows NIST Digital Library of Mathematical Functions, Chapter 4: Elementary Functions and OpenStax, Precalculus 2e, Chapter 8.5: Polar Form of Complex Numbers.
(3 + 4i)(1 - 2i) example
With Operation: multiply, a: real part: 3, a: imaginary part: 4, b: real part: 1, the calculator returns Real part of the result: 11, Imaginary part of the result: -2, Modulus of the result: 11.1803399. This example is checked against an independent source in the test suite.
What you can compare
Change one input at a time to see how it affects Real part of the result, Imaginary part of the result and Result (rectangular form). The inputs you can vary are Operation, Enter the numbers as, a: real part, a: imaginary part, b: real part and b: imaginary part.
Results are general information, not financial, tax, legal, medical or engineering advice. Check any decision that matters with a qualified professional.
See something wrong? Report a problem with this calculator
Frequently asked questions
How to do complex in calculator?
Adds, subtracts, multiplies, divides, powers and roots complex numbers and converts between rectangular, polar and exponential form with modulus and principal argument. Open “How this is worked out” beside the result to see each step for your own figures.
What is the most complex calculator?
Adds, subtracts, multiplies, divides, powers and roots complex numbers and converts between rectangular, polar and exponential form with modulus and principal argument. On this page it is computed from Operation, Enter the numbers as and a: real part and shown as Real part of the result, Imaginary part of the result and Result (rectangular form).
What information do I need for the Complex Number Calculator?
You need Operation, Enter the numbers as, a: real part, a: imaginary part, b: real part, b: imaginary part, a: modulus and a: argument. The page opens with default figures that give result (rectangular form) of 11 - 2i, so you can see the scale before entering your own.
What does this Complex Number Calculator show?
It shows Real part of the result, Imaginary part of the result, Result (rectangular form), Modulus of the result, Argument of the result (radians), Argument of the result (degrees), Polar form and Exponential form, with the working and any warnings for your entries.
Can I check how the Complex Number Calculator result was calculated?
Yes. The “How this is worked out” section shows the calculation steps, and the sources section identifies the references used for the method.