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Complex Number Calculator

Performs arithmetic on two complex numbers written as strings such as '3+4i', or computes the modulus, conjugate or polar form of one, and reports the real and imaginary parts, modulus and argument of the result.

When to use

You need to combine complex numbers in rectangular form (a + bi), convert between rectangular and polar form, or find |z| and arg(z).

Do not use when: You want the complex roots of a polynomial (use quadratic-equation or cubic-equation) or trigonometric values of a real angle (use trigonometry).

Formula

(a+bi) ± (c+di) = (a±c) + (b±d)i; (a+bi)(c+di) = (ac − bd) + (ad + bc)i; (a+bi)/(c+di) = ((ac + bd) + (bc − ad)i) / (c² + d²); |z| = √(a² + b²); arg z = atan2(b, a); conj(z) = a − bi

Inputs

Parameter	Type	Required	Description
`z1`	string	yes	First complex number in rectangular form, e.g. '3+4i', '-2.5i', '7' or '1-i' (j is accepted for i).
`z2`	string	no	Second complex number; required for add, subtract, multiply and divide, ignored otherwise.
`operation`	enum: add \| subtract \| multiply \| divide \| modulus \| conjugate \| polar	default add	Binary operations use z₁ and z₂; modulus, conjugate and polar use z₁ only.

Outputs

Output	Type	Unit	Description
`result`	string		The result in rectangular form, e.g. '-1+2i' (for polar: 'r∠θ°').
`real`	number		Re(result) (for polar: Re(z₁)).
`imaginary`	number		Im(result) (for polar: Im(z₁)).
`modulus`	number		√(real² + imaginary²) of the result.
`argument_degrees`	number	°	atan2(imaginary, real) in degrees, in (−180°, 180°].
`argument_radians`	number	rad	The principal argument in radians.
`polar_form`	string		The result as 'r∠θ°' and 'r·e^(iθ)' with θ in radians (6 significant digits).
`conjugate`	string		The conjugate of the result (imaginary part negated).
`expression`	string		The evaluated operation as text, e.g. '(3+4i) ÷ (1-2i) = -1+2i'.

Example

(3+4i) ÷ (1−2i): {"z1":"3+4i","z2":"1-2i","operation":"divide"} → {"result":"-1+2i","real":-1,"imaginary":2,"modulus":2.236068,"argument_degrees":116.5651,"argument_radians":2.034444,"conjugate":"-1-2i","expression":"(3+4i) ÷ (1-2i) = -1+2i"}

Polar form of 3+4i: {"z1":"3+4i","operation":"polar"} → {"result":"5∠53.1301°","real":3,"imaginary":4,"modulus":5,"argument_degrees":53.1301,"argument_radians":0.927295,"polar_form":"5∠53.1301° = 5·e^(0.927295i)"}

GET https://tttkmbb.com/api/v1/calculate/complex-numbers?z1=3%2B4i&z2=1-2i&operation=divide

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/complex-numbers (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/complex-numbers · Markdown: https://tttkmbb.com/math/complex-numbers.md · JSON definition: https://tttkmbb.com/math/complex-numbers.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="complex-numbers"
OpenAPI operationId: calculate_complex_numbers
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

Wikipedia – Complex number (reference)
NIST Digital Library of Mathematical Functions §1.9 – Calculus of a complex variable: complex numbers (standard)
Wolfram MathWorld – Complex Number (reference)

FAQ

How is division computed?

Numerator and denominator are multiplied by the conjugate of the divisor: (3+4i)/(1−2i) = (3+4i)(1+2i)/(1² + 2²) = (−5 + 10i)/5 = −1 + 2i.

Which range is the argument in?

The principal value from atan2, −180° < θ ≤ 180° (−π < θ ≤ π), measured counter-clockwise from the positive real axis.

Related calculators

Quadratic Equation Solver — Source of complex conjugate roots.
Cubic Equation Solver — Cubics with a complex root pair.
Trigonometry Calculator — cos θ and sin θ used in the polar form.