Complex Number Calculator
Enter two complex numbers a + bi and choose an operation. Calvo gives the result with modulus, argument, conjugate and polar form.
Complex number calculator
+i
+i(only needed for + − × ÷)
n (for power)
The result will appear here.
Formulas
| Operation | Formula |
|---|---|
| Add / subtract | (a + bi) ± (c + di) = (a ± c) + (b ± d)i |
| Multiply | (a + bi)(c + di) = (ac − bd) + (ad + bc)i |
| Divide | (a + bi)/(c + di) = [(ac + bd) + (bc − ad)i] / (c² + d²) |
| Modulus | |z| = √(a² + b²) |
| Argument | θ = atan2(b, a) |
| Conjugate | z̄ = a − bi |
| Polar form | z = r(cos θ + i sin θ) |
| De Moivre | zn = rn(cos nθ + i sin nθ) |
Worked example
(3 + 4i)(1 − 2i) = (3·1 − 4·(−2)) + (3·(−2) + 4·1)i = 11 − 2i. The modulus of 3 + 4i is √(9 + 16) = 5.
FAQ
How do I multiply complex numbers?
Use (a + bi)(c + di) = (ac - bd) + (ad + bc)i, because i squared is -1.
How do I divide complex numbers?
Multiply the top and bottom by the conjugate of the denominator, then divide by c squared plus d squared.
What is the modulus of a complex number?
The distance from the origin: square root of a squared plus b squared.
What angle does Calvo give?
The principal argument between -180 and 180 degrees, also shown in radians.
Does the square root show both roots?
Yes. A non-zero complex number has two square roots, the principal one and its negative.