Complex Number Calculator
The complex number calculator helps you to find the outcome of basic operations of complex numbers. It is possible to choose between complex numbers' algebraic and polar forms.
a × b11 - 2i
a + b4 + 2i
a ÷ b-1 + 2i
Modulus |a|5
Argument of a53.130102°0.92729522 rad
Polar form of a5(cos 53.13° + i sin 53.13°)
Every operation
| Operation | Result |
|---|---|
| a + b | 4 + 2i |
| a − b | 2 + 6i |
| a × b | 11 - 2i |
| a ÷ b | -1 + 2i |
| conjugate of a | 3 - 4i |
| a² | -7 + 24i |
The method
i² = −1; (a+bi)(c+di) = (ac−bd) + (ad+bc)i
Dividing uses the conjugate
To divide, multiply top and bottom by the conjugate of the denominator. That makes the denominator real, because (c+di)(c−di) = c² + d² — the same trick as rationalising a surd.
Modulus and argument
Every complex number is a point in the plane, so it also has a distance from the origin (the modulus) and an angle (the argument). Multiplying complex numbers multiplies their moduli and adds their arguments — which is why complex multiplication is a rotation combined with a scaling.