Square Roots of Complex Number Calculator

Introduction

A complex number is typically written in the form a + bi, where a and b are real numbers, and i is the imaginary unit with the property that i² = -1. The square roots of a complex number can be found using algebraic methods, and the result will always be another complex number.

This calculator implements the standard formula for finding square roots of complex numbers, providing both the principal and secondary roots. The results are presented in both rectangular (a + bi) and polar forms for complete understanding.

How to Use the Calculator

1. Enter the real part (a) of your complex number in the first input field.
2. Enter the imaginary part (b) of your complex number in the second input field.
3. Click the "Calculate" button to compute the square roots.
4. Review the results displayed in both rectangular and polar forms.
5. Use the "Reset" button to clear the inputs and start over.

Formula

The square roots of a complex number z = a + bi can be found using the following formula:

Where:

• a is the real part of the complex number
• b is the imaginary part of the complex number
• sgn(b) is the sign function of b (1 if b ≥ 0, -1 if b < 0)

This formula provides both the principal (positive) and secondary (negative) square roots of the complex number.

Examples

Example 1: Simple Complex Number

Let's find the square roots of 3 + 4i.

1. First, calculate the magnitude: √(3² + 4²) = √(9 + 16) = √25 = 5
2. Compute the real part of the roots: √((3 + 5)/2) = √(4) = 2
3. Compute the imaginary part of the roots: sgn(4)·√((-3 + 5)/2) = 1·√(1) = 1
4. The square roots are: 2 + i and -2 - i

Example 2: Purely Real Number

Find the square roots of 9 (which is 9 + 0i).

1. Calculate the magnitude: √(9² + 0²) = √81 = 9
2. Compute the real part of the roots: √((9 + 9)/2) = √(9) = 3
3. Compute the imaginary part of the roots: sgn(0)·√((-9 + 9)/2) = 0·√(0) = 0
4. The square roots are: 3 and -3

Example 3: Purely Imaginary Number

Find the square roots of 4i (which is 0 + 4i).

1. Calculate the magnitude: √(0² + 4²) = √16 = 4
2. Compute the real part of the roots: √((0 + 4)/2) = √(2) ≈ 1.414
3. Compute the imaginary part of the roots: sgn(4)·√((-0 + 4)/2) = 1·√(2) ≈ 1.414
4. The square roots are: 1.414 + 1.414i and -1.414 - 1.414i

Interpreting Results

The calculator provides results in two forms:

• Rectangular form: a + bi, where a is the real part and b is the imaginary part
• Polar form: r(cosθ + i·sinθ), where r is the magnitude and θ is the angle in radians

For each square root, the rectangular form shows the exact values, while the polar form provides the magnitude and angle, which can be useful for further calculations or visualizations.