{"success":true,"version":"v1","request":{"tool":"get_calculator_schema","calculator_id":"complex-numbers"},"result":{"entity_type":"calculator","id":"complex-numbers","calculator_id":"complex-numbers","canonical_url":"https://tttkmbb.com/math/complex-numbers","name":"Complex Number Calculator","title":"Complex Number Calculator – Add, Subtract, Multiply, Divide, Modulus, Conjugate and Polar Form","category":"math","category_name":"Math","tool_name":"calculate_complex_numbers","featured_mcp_tool":false,"description":"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.","use_when":"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).","inputs":[{"name":"z1","label":"z₁","type":"string","required":true,"description":"First complex number in rectangular form, e.g. '3+4i', '-2.5i', '7' or '1-i' (j is accepted for i).","example":"3+4i"},{"name":"z2","label":"z₂","type":"string","required":false,"description":"Second complex number; required for add, subtract, multiply and divide, ignored otherwise.","example":"1-2i"},{"name":"operation","label":"Operation","type":"enum","required":false,"default":"add","values":[{"value":"add","label":"z₁ + z₂","aliases":["+","plus","sum"]},{"value":"subtract","label":"z₁ − z₂","aliases":["-","minus","difference"]},{"value":"multiply","label":"z₁ × z₂","aliases":["*","times","product"]},{"value":"divide","label":"z₁ ÷ z₂","aliases":["/","over","quotient"]},{"value":"modulus","label":"|z₁|","aliases":["abs","absolute value","magnitude"]},{"value":"conjugate","label":"Conjugate of z₁","aliases":["conj"]},{"value":"polar","label":"Polar form of z₁","aliases":["polar form","to polar"]}],"description":"Binary operations use z₁ and z₂; modulus, conjugate and polar use z₁ only.","example":"divide"}],"outputs":[{"name":"result","label":"Result","type":"string","decimals":4,"description":"The result in rectangular form, e.g. '-1+2i' (for polar: 'r∠θ°')."},{"name":"real","label":"Real part","type":"number","decimals":6,"description":"Re(result) (for polar: Re(z₁))."},{"name":"imaginary","label":"Imaginary part","type":"number","decimals":6,"description":"Im(result) (for polar: Im(z₁))."},{"name":"modulus","label":"Modulus |z|","type":"number","decimals":6,"description":"√(real² + imaginary²) of the result."},{"name":"argument_degrees","label":"Argument","type":"number","unit":"°","decimals":4,"description":"atan2(imaginary, real) in degrees, in (−180°, 180°]."},{"name":"argument_radians","label":"Argument (radians)","type":"number","unit":"rad","decimals":6,"description":"The principal argument in radians."},{"name":"polar_form","label":"Polar form","type":"string","decimals":4,"description":"The result as 'r∠θ°' and 'r·e^(iθ)' with θ in radians (6 significant digits)."},{"name":"conjugate","label":"Conjugate","type":"string","decimals":4,"description":"The conjugate of the result (imaginary part negated)."},{"name":"expression","label":"Expression","type":"string","decimals":4,"description":"The evaluated operation as text, e.g. '(3+4i) ÷ (1-2i) = -1+2i'."}],"input_schema":{"type":"object","properties":{"z1":{"description":"First complex number in rectangular form, e.g. '3+4i', '-2.5i', '7' or '1-i' (j is accepted for i).","type":"string","examples":["3+4i"]},"z2":{"description":"Second complex number; required for add, subtract, multiply and divide, ignored otherwise.","type":"string","examples":["1-2i"]},"operation":{"description":"Binary operations use z₁ and z₂; modulus, conjugate and polar use z₁ only.","type":"string","enum":["add","subtract","multiply","divide","modulus","conjugate","polar"],"default":"add","examples":["divide"]}},"additionalProperties":false,"required":["z1"]},"output_schema":{"type":"object","properties":{"result":{"description":"The result in rectangular form, e.g. '-1+2i' (for polar: 'r∠θ°').","type":"string"},"real":{"description":"Re(result) (for polar: Re(z₁)).","type":"number"},"imaginary":{"description":"Im(result) (for polar: Im(z₁)).","type":"number"},"modulus":{"description":"√(real² + imaginary²) of the result.","type":"number"},"argument_degrees":{"description":"atan2(imaginary, real) in degrees, in (−180°, 180°]. Unit: °.","type":"number","x-unit":"°"},"argument_radians":{"description":"The principal argument in radians. Unit: rad.","type":"number","x-unit":"rad"},"polar_form":{"description":"The result as 'r∠θ°' and 'r·e^(iθ)' with θ in radians (6 significant digits).","type":"string"},"conjugate":{"description":"The conjugate of the result (imaginary part negated).","type":"string"},"expression":{"description":"The evaluated operation as text, e.g. '(3+4i) ÷ (1-2i) = -1+2i'.","type":"string"}}},"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","sources":[{"name":"Wikipedia – Complex number","url":"https://en.wikipedia.org/wiki/Complex_number","type":"reference","retrieved_at":"2026-09-24"},{"name":"NIST Digital Library of Mathematical Functions §1.9 – Calculus of a complex variable: complex numbers","url":"https://dlmf.nist.gov/1.9","type":"standard","retrieved_at":"2026-09-24"},{"name":"Wolfram MathWorld – Complex Number","url":"https://mathworld.wolfram.com/ComplexNumber.html","type":"reference","retrieved_at":"2026-09-24"}],"freshness":{"type":"static","max_age_seconds":null,"note":"Deterministic formula with fixed constants; results never go stale. Inputs supplied by the caller determine the output."},"examples":[{"name":"(3+4i) ÷ (1−2i)","inputs":{"z1":"3+4i","z2":"1-2i","operation":"divide"},"expected":{"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"},"url":"https://tttkmbb.com/api/v1/calculate/complex-numbers?z1=3%2B4i&z2=1-2i&operation=divide"},{"name":"Polar form of 3+4i","inputs":{"z1":"3+4i","operation":"polar"},"expected":{"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)"},"url":"https://tttkmbb.com/api/v1/calculate/complex-numbers?z1=3%2B4i&operation=polar"}],"faq":[{"q":"How is division computed?","a":"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."},{"q":"Which range is the argument in?","a":"The principal value from atan2, −180° < θ ≤ 180° (−π < θ ≤ π), measured counter-clockwise from the positive real axis."}],"tags":["complex numbers","complex number calculator","imaginary numbers","polar form","modulus argument","complex division"],"related":[{"calculator_id":"quadratic-equation","reason":"Source of complex conjugate roots."},{"calculator_id":"cubic-equation","reason":"Cubics with a complex root pair."},{"calculator_id":"trigonometry","reason":"cos θ and sin θ used in the polar form."}],"links":{"html":"https://tttkmbb.com/math/complex-numbers","markdown":"https://tttkmbb.com/math/complex-numbers.md","json":"https://tttkmbb.com/math/complex-numbers.json","api":"https://tttkmbb.com/api/v1/calculate/complex-numbers","schema":"https://tttkmbb.com/api/v1/calculators/complex-numbers","openapi":"https://tttkmbb.com/openapi.json","mcp":"https://tttkmbb.com/mcp"},"version":"v1","updated_at":"2026-09-24"},"timestamp":"2026-09-24T01:51:38Z","next_actions":[{"tool":"run_calculator","calculator_id":"complex-numbers","reason":"Run Complex Number Calculator with the inputs above."}],"links":{"markdown":"https://tttkmbb.com/math/complex-numbers.md"}}