Complex Number Calculator (Add, Multiply, Divide, Polar Form)
Enter two complex numbers a + bi and c + di and choose the operation.
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Complex Number Calculator (Add, Multiply, Divide, Polar Form) by TheCalcTools
How the Complex Number Calculator (Add, Multiply, Divide, Polar Form) works
Add and subtract the real and imaginary parts separately. Multiply with i² = −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.
To divide, multiply top and bottom by the conjugate of the divisor: (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²). The modulus is √(x² + y²) and the argument is atan2(y, x).
The formula
Worked example
Using the default values shown in the calculator:
| a (real part of the first number) | 3 |
|---|---|
| b (imaginary part of the first number) | 4 |
| Operation | Divide |
| c (real part of the second number) | 1 |
| d (imaginary part of the second number) | -2 |
- (3 + 4i) ÷ (1 − 2i) = -1 + 2i.
- |z| = 2.236068, argument 116.5651°.
Result:
| Result | -1 + 2i |
|---|---|
| Modulus |z| | 2.2361 |
| Argument (degrees) | 116.5651 |
| Polar form | 2.236068 ∠ 116.565051° |
| Conjugate | -1 − 2i |
Frequently asked questions
What is (3 + 4i) ÷ (1 − 2i)?
−1 + 2i.
Where are complex numbers used?
AC circuits, signal processing, control systems, quantum mechanics and fractals.
Sources and further reading
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