Complex Conjugate Calculator
The complex conjugate calculator is here to become your favorite tool to find the complex conjugate of a number.
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.