Primary formula
(a+bi)(c+di) = (ac-bd) + (ad+bc)i, and |a+bi| = sqrt(a^2+b^2).
Math calculator
Complex Number Calculator
Calculate complex-number arithmetic in rectangular form, plus conjugates, integer powers, magnitude and principal polar angle.
Last reviewed: July 1, 2026Math algebra and base engine v1.0.0
Math calculator
Complex Number Calculator
Calculations run locally from the values entered. Exact values and decimal approximations are labelled separately.
Exact math engine
Mode
Add, subtract, multiply or divide two complex numbers.
Enter values and choose Calculate to show the result below the calculator.
Formula and assumptions
Input assumptions
Real and imaginary parts are parsed as rational values. Argument is reported as the principal atan2 approximation. Integer powers are limited to nonnegative exponents through 20.
Precision note
Exact integer, rational, base, matrix and equation outputs are labelled. Decimal and polar values are shown as approximations when exact symbolic output is not practical.
Algebra and base-number flow
Parse
Read integer, rational, base, matrix, complex, or equation inputs from labelled fields.
Validate
Apply domain, bit-width, matrix-dimension and expression-scope checks before calculating.
Calculate
Use deterministic TypeScript, BigInt, exact rational arithmetic and Decimal.js where appropriate.
Label
Separate exact values, approximations, table rows, exports and warnings.
What the Complex Number Calculator does
Complex Number Calculator is built for complex arithmetic with exact rectangular components and labelled polar approximations. The calculator appears first, then the result card, detailed method, educational content, FAQs, references and disclaimer follow in the approved NexaCalc layout.
How to use the Complex Number Calculator
Choose the mode that matches your calculation, enter values using the visible labels and helper text, then select Calculate. Reset restores the default example.
Use Copy, Print, Share, Export text or Export CSV after a result appears. Exports are generated locally in the browser.
Mathematical definition
(a+bi)(c+di) = (ac-bd) + (ad+bc)i, and |a+bi| = sqrt(a^2+b^2).
Formula or algorithm
The engine reuses NexaCalc exact rational and BigInt helpers, then applies the selected operation with explicit domain and performance limits.
Accepted input
Inputs are parsed from text. Integer, rational, matrix, base, complex and equation fields use specific validators so unsupported tokens produce validation errors instead of hidden coercion.
- Real and imaginary parts are parsed as rational values.
- Argument is reported as the principal atan2 approximation.
- Integer powers are limited to nonnegative exponents through 20.
Domain and restrictions
The calculator rejects operations outside its stated mathematical scope, such as division by zero, invalid base digits, nonsquare inverse matrices, singular inverses and unsupported equation forms.
Exact versus approximate results
Exact result labels are used for integer, rational, matrix and equation outputs where exact arithmetic is available. Decimal approximation labels are used for polar angles, irrational roots and rounded display values.
Step-by-step calculation
The result card lists the main algorithm steps and a detailed table where useful. Matrix and base tools also expose exportable rows for independent checking.
Worked example
(3 + 4i)(1 - 2i) = 11 - 2i.
Common mistakes
Most mistakes come from mixing conventions, entering a value outside the mathematical domain, or treating a rounded approximation as exact.
- Forgetting that i^2 = -1.
- Dividing by zero complex number.
- Treating one argument angle as the only possible angle.
Limitations
This page is a practical calculator rather than a full computer algebra system or arbitrary-precision programming environment.
- Polar angle is approximate and multi-valued by nature.
- General complex functions are out of scope.
Privacy and performance
Calculations are local to the browser session. Inputs are not uploaded, stored, or logged by NexaCalc, and limits are applied to keep pages fast on shared hosting.
Educational disclaimer
This calculator provides mathematical results from the values, conventions and methods you enter. Verify important academic, engineering or professional work independently.
Frequently asked questions
What does the Complex Number Calculator do?
It supports complex arithmetic with exact rectangular components and labelled polar approximations. Results are calculated locally and show method notes, exact values, approximations and warnings where relevant.
How does the Complex Number Calculator calculate results?
(a+bi)(c+di) = (ac-bd) + (ad+bc)i, and |a+bi| = sqrt(a^2+b^2).
Are results exact?
Integer, rational, base, modular, matrix and equation coefficients are exact where labelled. Polar angles, irrational roots and long fractional base expansions are labelled as approximations or repeating expansions.
Does this calculator use eval or run expressions as code?
No. Equation inputs are tokenized and parsed by a limited mathematical parser. Unsupported tokens, division by a variable and expressions outside the stated scope are rejected.
Are calculations stored or uploaded?
No. Calculations run in the browser session from manual inputs. NexaCalc does not require accounts, databases or paid APIs for these tools.
Why are some operations limited?
Limits keep browser sessions responsive and avoid unbounded integer, matrix, base-cycle or expression parsing work.
Can I copy or export the result?
Yes. Result actions include copy, print, share, text export and CSV export where table data is available.
What should I verify independently?
Verify high-stakes academic, engineering, coding or professional work independently, especially when a result uses a convention, bit width, approximation or parser restriction.
Does the page create duplicate calculator routes?
No. Each tool has one canonical /math/ route. Common aliases redirect to the canonical page.
What are the main limitations of the Complex Number Calculator?
Polar angle is approximate and multi-valued by nature. General complex functions are out of scope.
References
- OpenStax Prealgebra 2e, arithmetic foundations for integers, factors, multiples and absolute value. Source.
- OpenStax College Algebra 2e, algebra prerequisites, polynomial expressions and quadratic equations. Source.
- OpenStax Precalculus 2e, functions, absolute value, equations and graph interpretation. Source.
- OpenStax Algebra and Trigonometry 2e, matrices, logarithms, exponents and algebraic methods. Source.
- MDN Web Docs, BigInt reference for arbitrary-size integer arithmetic in JavaScript. Source.
- decimal.js API documentation by MikeMcl, arbitrary-precision decimal arithmetic used for labelled approximations. Source.
Math Phase 3 references and calculation conventions reviewed on July 1, 2026.
Educational disclaimer
This calculator provides mathematical results from the values, conventions and methods you enter. Verify important academic, engineering or professional work independently.