Complex Number Calculator
Add, subtract, multiply, or divide complex numbers, and convert the result between rectangular a + bi and polar r(cos θ + i sin θ) forms.
Result (a + bi)
11 − 10i
Breakdown
- Real part
- 11
- Imaginary part
- -10
- Modulus |z|
- 14.87
- Argument θ (radians)
- -0.74
- Argument θ (degrees)
- -42.27
- Conjugate
- 11 + 10i
Last updated:
FAQ
How do you multiply complex numbers?
Expand like binomials and use i² = −1: (3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 11 − 10i. In polar form, moduli multiply and angles add — often faster by hand.
Why multiply by the conjugate when dividing?
Multiplying top and bottom by c − di turns the denominator into c² + d², a plain real number. (3 + 2i)/(1 − 4i) = −0.29 − 0.71i once the dust settles.
When is polar form useful?
Polar form shines for powers and roots: De Moivre's theorem gives (r∠θ)ⁿ = rⁿ∠(nθ). Computing (1 + i)¹⁰ is far easier as (√2∠45°)¹⁰ = 32∠450° = 32i than by repeated binomial expansion.
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