📐 Complex Root Calculator
Find square roots of complex numbers
z = a + bi
How to Use This Calculator
Enter Real Part
Type the real part (a) of the complex number
Enter Imaginary Part
Type the imaginary part (b) of the complex number
Calculate
See both square roots in a+bi form
Complex Root Formula
√(a + bi) using polar form
Method:
- Convert to polar: r = √(a² + b²), θ = arctan(b/a)
- Take square root: √r · e^(i·θ/2)
- Convert back: Real = √r·cos(θ/2), Imag = √r·sin(θ/2)
Example: √(4 + 3i)
r = √(16 + 9) = 5
θ = arctan(3/4) ≈ 36.87°
√5 · e^(i·18.44°) ≈ 2 + 1i (approximately)
About Complex Root Calculator
The Complex Root Calculator finds the square roots of complex numbers using polar form representation. Every complex number has exactly two square roots (except 0, which has one).
Key Concepts
- Polar Form: z = r·e^(iθ) = r(cos θ + i sin θ)
- Square Root: √z = √r · e^(i·θ/2)
- Two Roots: Second root is negative of first
Applications
- Electrical Engineering: AC circuit analysis
- Signal Processing: Frequency domain analysis
- Mathematics: Polynomial roots, complex analysis
Frequently Asked Questions
How many square roots does a complex number have?
Every nonzero complex number has exactly two square roots. The second root is the negative of the first.
What's the principal square root?
The principal square root is the one with argument in [-π/2, π/2]. It's the one typically calculated.
Can I calculate roots of other powers?
Yes! Use r^(1/n) · e^(i·θ/n) for the nth root. There are n distinct nth roots.