Complex Number Calculator
Add, subtract, multiply or divide two complex numbers a + bi and c + di, with the result in a + bi form and its modulus.
How it works
Add/Sub: (a ± c) + (b ± d)i · Multiply: (ac − bd) + (ad + bc)i · Divide: [(ac + bd) + (bc − ad)i] ÷ (c² + d²) · |z| = √(re² + im²)
Complex numbers combine a real part and an imaginary part (a multiple of i, where i² = −1). Addition and subtraction act part by part. Multiplication uses the distributive law and the rule i² = −1, which folds the product of the two imaginary parts back into the real part. Division multiplies top and bottom by the conjugate of the divisor, turning the denominator into the real number c² + d². The modulus is the number’s distance from the origin in the complex plane, √(re² + im²).
Worked example
Add (3 + 4i) and (1 + 2i): sum the real parts 3 + 1 = 4 and the imaginary parts 4 + 2 = 6, giving 4 + 6i. Its modulus is √(4² + 6²) = √52 ≈ 7.2111. Multiplying the same two would instead give (3·1 − 4·2) + (3·2 + 4·1)i = −5 + 10i.
Sources: Complex number — Wikipedia
Frequently asked questions
How do you divide complex numbers?
Multiply the numerator and denominator by the conjugate of the denominator (c − di). That makes the denominator the real number c² + d², after which you split into real and imaginary parts. Dividing by 0 + 0i is undefined, so it is blocked.
What is the modulus of a complex number?
It is the number’s distance from the origin in the complex plane, |a + bi| = √(a² + b²). For 3 + 4i it is √(9 + 16) = 5. The modulus measures magnitude regardless of direction.
Why does multiplying imaginary parts change the real part?
Because i² = −1. When you expand (a + bi)(c + di), the bi × di term becomes bd·i² = −bd, a real quantity, which is why the real part of the product is ac − bd rather than just ac.
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This calculator is for educational purposes. Double-check important results before acting on them.