Complex Number Calculator
Add, subtract, multiply and divide complex numbers, convert to polar form, and find all the nth roots — with the conjugate and De Moivre steps shown, not just answers.
Example — (3 + 4i) ÷ (1 − 2i)−1 + 2i
How we got it — the example, step by step
- Multiply top and bottom by the conjugate of the bottom, 1 + 2i. The bottom becomes real: 1² + 2² = 5.
- The top becomes (3 + 4i)(1 + 2i) = 3 + 6i + 4i + 8i² = −5 + 10i.
- Divide both parts by 5: −1 + 2i.
Check it backwards
Multiply the answer by the original divisor: (−1 + 2i)(1 − 2i) = −1 + 2i + 2i − 4i² = 3 + 4i — the number we divided.
Questions people also ask
Why does (3 + 4i) have modulus exactly 5?
The modulus is the distance from the origin: √(3² + 4²) = √25 = 5 — the 3-4-5 triangle, sitting in the complex plane. Its argument is atan2(4, 3) ≈ 53.13°.