Quick answer
Complex numbers add and subtract component-wise, multiply as (ac − bd) + (ad + bc)i and divide as ((ac + bd) + (bc − ad)i) ÷ (c² + d²). (1 + 2i) + (3 + 4i) = 4 + 6i, with modulus 7.2111 and argument 56.31°.
Enter the real and imaginary parts of two numbers and choose an operation. The calculator returns the result, modulus and argument.
Related tools: Prime Number Calculator: Check whether a number is prime and find the next prime. Random Number Generator: Random integers in a range, optionally repeatable with a seed. Scientific Calculator: Evaluate expressions with functions, constants, powers and factorials.
How to use the Complex Number Calculator
- Enter real part of z₁.
- Enter imaginary part of z₁.
- Choose operation: +, −, × and ÷.
- Enter real part of z₂.
- Enter imaginary part of z₂.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Real part of z₁ | A number | — |
| Imaginary part of z₁ | A number | — |
| Operation | Choose one: +; −; ×; ÷ | Starts at + |
| Real part of z₂ | A number | — |
| Imaginary part of z₂ | A number | — |
Formula and method
Formula
(a+bi)(c+di) = (ac−bd) + (ad+bc)i; (a+bi)/(c+di) = ((ac+bd) + (bc−ad)i) ÷ (c²+d²)
IEEE-754 double precision; shown to at most 6 decimal places.
Standard rectangular-form arithmetic.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Real part of z₁ = 1; Imaginary part of z₁ = 2; Operation = +; Real part of z₂ = 3; Imaginary part of z₂ = 4.
| Result | Value |
|---|---|
| Result | 4 + 6i |
| Modulus | 7.211103 |
| Argument | 56.309932° |
Example 2
Inputs: Real part of z₁ = 1; Imaginary part of z₁ = 2; Operation = ×; Real part of z₂ = 3; Imaginary part of z₂ = 4.
| Result | Value |
|---|---|
| Result | -5 + 10i |
| Modulus | 11.18034 |
| Argument | 116.565051° |
Example 3
Inputs: Real part of z₁ = 1; Imaginary part of z₁ = 2; Operation = ÷; Real part of z₂ = 3; Imaginary part of z₂ = 4.
| Result | Value |
|---|---|
| Result | 0.44 + 0.08i |
| Modulus | 0.447214 |
| Argument | 10.304846° |
The first line is the headline result (result). Below it the complex Number Calculator lists modulus and argument, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Modulus and argument
The modulus is √(a² + b²) = √(16 + 36) = √52 = 7.2111 for 4 + 6i. The argument is atan2(6, 4) = 56.31°, the angle from the positive real axis. Together they give polar form.
Division
Division multiplies top and bottom by the conjugate of the divisor. Dividing by 0 + 0i is invalid and is rejected.
How the result changes with real part of z₁
Holding every other input at the first worked example, this table changes only real part of z₁. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Real part of z₁ | Result | Modulus | Argument |
|---|---|---|---|
| 0.5 | 3.5 + 6i | 6.946222 | 59.743563° |
| 1 | 4 + 6i | 7.211103 | 56.309932° |
| 2 | 5 + 6i | 7.81025 | 50.194429° |
| 4 | 7 + 6i | 9.219544 | 40.601295° |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Real part of z₂: entering "0" shows "Cannot divide by 0 + 0i." beside that field and no result.
Choosing between the Complex Number Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Prime Number Calculator](/math/prime-number-calculator/) | number | Result |
| [Random Number Generator](/math/random-number-generator/) | minimum, maximum and how many | Random numbers |
| [Scientific Calculator](/math/scientific-calculator/) | expression and angle mode | Result |
| [Roman Numeral Converter](/math/roman-numeral-converter/) | convert and value | Roman numeral |
| [Fibonacci Sequence Calculator](/math/fibonacci-sequence-calculator/) | position n | F(10) |
Assumptions and limitations
- Dividing by 0 + 0i is invalid
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for real part of z₂ to confirm this.
Frequently asked questions
What is i squared?
i squared is −1, which is the defining property of the imaginary unit.
What is a conjugate?
The same number with the sign of the imaginary part flipped.
Is the argument in degrees?
Yes, shown in degrees.
What does the Complex Number Calculator calculate?
Add, subtract, multiply and divide complex numbers. It returns Result, Modulus and Argument.
What formula does the Complex Number Calculator use?
(a+bi)(c+di) = (ac−bd) + (ad+bc)i; (a+bi)/(c+di) = ((ac+bd) + (bc−ad)i) ÷ (c²+d²). Standard rectangular-form arithmetic.
What do I need to enter in the complex Number Calculator?
You enter Real part of z₁, Imaginary part of z₁, Operation, Real part of z₂ and Imaginary part of z₂.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Dividing by 0 + 0i is invalid.
What does the “Operation” option change?
It sets how the complex Number is worked out. The choices are +, −, × and ÷.
What is an example of a complex Number result?
With real part of z₁ 1, imaginary part of z₁ 2, operation +, real part of z₂ 3 and imaginary part of z₂ 4, the result is 4 + 6i.
Related calculators
- Prime Number Calculator — Check whether a number is prime and find the next prime.
- Random Number Generator — Random integers in a range, optionally repeatable with a seed.
- Scientific Calculator — Evaluate expressions with functions, constants, powers and factorials.
- Roman Numeral Converter — Convert between numbers and Roman numerals (1–3999).
- Fibonacci Sequence Calculator — The nth Fibonacci number and the sequence leading to it.
- Factorial Calculator — n! computed exactly.
- Big Number Calculator — Exact arithmetic on integers of any size.
- Integral Calculator — Antiderivative of a function of x, with optional definite bounds.
- Limit Calculator — Two-sided limit of a function of x, with a numeric check.
Sources
- OpenStax, Algebra and Trigonometry 2e (Rice University): exponents, radicals, logarithms and inverse trigonometric functions
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.