Complex Number Calculator
Add, subtract, multiply and divide complex numbers written as a + bi, and find the conjugate, the modulus and the polar form.
Calculator
- Add, subtract, multiply, divide
- Conjugate and modulus
- Rectangular and polar form
- Division-by-zero guard
Result & calculation receipt
- Method
- Rectangular-form complex arithmetic (multiply)
- Formula
(a+bi)(c+di) = (ac−bd)+(ad+bc)i- Substitution
- 3 + 2i × 1 + 7i = -11 + 23i
- Steps
- 3 + 2i × 1 + 7i = -11 + 23i
- Modulus = 25.495098, argument = 2.016902 rad = 115.56°.
- Verification
- Rectangular form -11 + 23i matches polar form 25.495098(cos 2.016902 + i·sin 2.016902).
- Exact result
- -11 + 23i
- Decimal result
- real: -11 · imaginary: 23 · modulus: 25.49509757 · argumentRad: 2.01690188
- Engine
- math-solvers v1.4.0
How this calculator works
Complex numbers combine a real part and an imaginary part (a multiple of i, where i² = −1). Addition and subtraction act on the parts separately; multiplication uses the FOIL rule with i² = −1; division multiplies numerator and denominator by the conjugate of the denominator to clear the i from the bottom. The calculator also gives the conjugate, the modulus √(a² + b²) and the polar form, and refuses division by the complex zero.
Worked example
(3 + 2i)(1 + 7i) = 3 + 21i + 2i + 14i² = 3 + 23i − 14 = −11 + 23i. Dividing (1 + i) by (1 − i) multiplies top and bottom by the conjugate 1 + i to get i. The modulus of 3 + 4i is √(3² + 4²) = 5. Division results are checked by multiplying the quotient back by the divisor.
Key insight
Complex numbers are points in a plane: the real part is the x-coordinate and the imaginary part the y-coordinate. The modulus is the distance from the origin and the argument is the angle — which is why the polar form makes multiplication (rotate and scale) so natural.
Frequently asked questions
How do I multiply complex numbers?
Use the distributive (FOIL) rule and replace i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i. The calculator shows the expansion.
How do I divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator. That makes the denominator the real number c² + d², clearing the i from the bottom.
What is the conjugate of a complex number?
The conjugate of a + bi is a − bi — the same real part with the imaginary part negated. Multiplying a number by its conjugate gives the real value a² + b².
What is the modulus?
The modulus |a + bi| = √(a² + b²) is the distance of the number from the origin in the complex plane — its size. For 3 + 4i it is 5.
Why is division by 0 + 0i rejected?
The complex zero has modulus zero, so dividing by it is undefined, just like dividing by zero for real numbers. The calculator returns a structured error.
Methodology & verification
Complex arithmetic is performed in rectangular form: addition and subtraction act componentwise, multiplication uses (a + bi)(c + di) = (ac − bd) + (ad + bc)i, and division multiplies by the conjugate of the denominator (denominator c² + d²), with the complex zero rejected as a division-by-zero error. The conjugate, modulus √(a² + b²) and polar form are reported. Division results are verified by multiplying the quotient by the divisor and confirming it recovers the dividend.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.