Square Root Calculator with Complex Numbers

This square root calculator handles complex numbers, providing both principal and secondary roots. Enter a complex number in the form a + bi, and the calculator will compute its square roots using the standard complex number square root formula.

How to Use This Calculator

To calculate the square root of a complex number:

Enter the real part (a) of your complex number in the first input field.
Enter the imaginary part (b) of your complex number in the second input field.
Click the "Calculate" button to see the results.
Use the "Reset" button to clear all inputs and results.

The calculator will display both the principal square root and the secondary square root of your complex number.

The Square Root Formula

The square root of a complex number \( z = a + bi \) is given by:

\( \sqrt{z} = \pm \left( \sqrt{\frac{|z| + a}{2}} + i \cdot \text{sgn}(b) \cdot \sqrt{\frac{|z| - a}{2}} \right) \) where: \( |z| = \sqrt{a^2 + b^2} \) and sgn(b) is the sign of b

This formula provides both roots of the complex number, with the principal root having a non-negative real part.

Worked Examples

Example 1: Simple Complex Number

Calculate the square roots of \( 3 + 4i \).

First, compute \( |z| = \sqrt{3^2 + 4^2} = 5 \).
Then, calculate the real part of the root: \( \sqrt{\frac{5 + 3}{2}} = \sqrt{4} = 2 \).
Calculate the imaginary part: \( \text{sgn}(4) \cdot \sqrt{\frac{5 - 3}{2}} = 1 \cdot \sqrt{1} = 1 \).
The principal root is \( 2 + i \), and the secondary root is \( -2 - i \).

Example 2: Purely Imaginary Number

Calculate the square roots of \( 0 + 5i \).

Compute \( |z| = \sqrt{0^2 + 5^2} = 5 \).
Real part: \( \sqrt{\frac{5 + 0}{2}} = \sqrt{2.5} \approx 1.581 \).
Imaginary part: \( \text{sgn}(5) \cdot \sqrt{\frac{5 - 0}{2}} = 1 \cdot \sqrt{2.5} \approx 1.581 \).
The principal root is approximately \( 1.581 + 1.581i \), and the secondary root is \( -1.581 - 1.581i \).

Frequently Asked Questions

What is the difference between principal and secondary roots?

The principal square root is the one with a non-negative real part, while the secondary root is its negative. Both are valid solutions to the square root equation.

Can I calculate square roots of negative real numbers?

Yes, this calculator handles negative real numbers by treating them as complex numbers with an imaginary part of 0.

What if I enter a zero for both parts?

The square root of zero is zero, and the calculator will display this result.