Math

Complex Root Calculator

Find the principal square root of a complex number a + bi.

Calculated Output

Real part of √z 2

Imaginary part of √z (i) 1

Formula: √(a+bi) = ±(√((|z|+a)/2) + sgn(b)·√((|z|−a)/2)·i)

Step-by-Step Breakdown

1. Values used:

2. Real part a = 3

3. Imaginary part b = 4 i

5. Real part of √z = 2

6. Imaginary part of √z = 1i

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The math behind it

With modulus |z| = √(a² + b²), the principal root is √((|z|+a)/2) + sgn(b)·√((|z|−a)/2) i. The second root is its negative.

√(3 + 4i) = 2 + i, since (2 + i)² = 3 + 4i.

How do I get all n-th roots?

Use De Moivre's theorem in polar form; there are exactly n distinct n-th roots evenly spaced around a circle.