Complex Number Square Root Calculator
Verified Calculation Engine
The online Complex Number Square Root Calculator helps you calculate instantly and solve problems related to Complex Numbers. This tool provides accurate results using standard formulas and step-by-step calculation; you can view the formula with example in the calculator where available. Whether you are a student, teacher, or professional, this calculator simplifies complex calculations and saves time. Enter the required values below and get instant results. Results are shown clearly, with optional step-by-step explanation where applicable. The tool is free to use and works in any modern browser—no download or installation required. Bookmark this page for quick access whenever you need reliable Math Numbers calculations.
Finds both square roots of a complex number: √(a + bi).
√(a + bi) = ±[√((|z| + a)/2) + i sgn(b)√((|z| - a)/2)]
where |z| = √(a² + b²) and sgn(b) is sign of b
Or using polar: √z = √r[cos(θ/2) + i sin(θ/2)] and √r[cos((θ+2π)/2) + i sin((θ+2π)/2)]
A complex number has exactly two square roots
Inputs
Results
Worked Examples
Example 1: Basic Square Root
Find square roots of 3 + 4i
- real: 3
- imaginary: 4
- root1: 2 + i
- root2: -2 - i
- rootsRectangular: 2 + i, -2 - i
Example 2: Board Level
Find square roots of -1
- real: -1
- imaginary: 0
- root1: i
- root2: -i
- rootsRectangular: i, -i
About this calculator
Overview
Complex Number Square Root Calculator evaluates Complex Number Square Root for the math / complex-numbers library. Primary input cue: Real Part [required]: Real part a. Method anchor: √(a + bi) = ±[√((|z| + a)/2) + i sgn(b)√((|z| - a)/2)] where |z| = √(a² + b²) and sgn(b) is sign of b Or using polar: √z = √r[cos(θ/2) + i sin(θ/2)] and √r[cos(.
When to use
Use Complex Number Square Root Calculator when you already know the inputs and need complex number square root without setting up a spreadsheet.
Inputs explained
- Real Part [required]: Real part a
- Imaginary Part [required]: Imaginary part b
Formula / method
√(a + bi) = ±[√((|z| + a)/2) + i sgn(b)√((|z| - a)/2)] where |z| = √(a² + b²) and sgn(b) is sign of b Or using polar: √z = √r[cos(θ/2) + i sin(θ/2)] and √r[cos((θ+2π)/2) + i sin((θ+2π)/2)]. A complex number has exactly two square roots (calculator id: complex-number-square-root-calculator).
Worked example
Worked values: real = 3, imaginary = 4 → root1 = 2 + i, root2 = -2 - i, rootsRectangular = ['2 + i', '-2 - i']
Interpreting results
Read the primary output as complex number square root under the method on this page.
Assumptions
- Empty required inputs are not silently replaced with zeros for complex number square root. [complex-number-square-root-calculator]
- Complex Number Square Root is computed only from the fields you enter on this calculator.
- Real-number arithmetic is used unless complex number square root explicitly needs integers.
Limitations
- If multiple conventions exist for complex number square root, this calculator uses the convention implemented in its engine path.
- This page does not provide a full textbook proof for every identity related to complex number square root.
- Complex/branch cuts and undefined points are not always interactively annotated for complex number square root.
Important note
Educational mathematics tool. Check edge cases and domain restrictions when results look unexpected.
How to Use This Calculator
- Enter the required values in the input fields.
- Click the Calculate button.
- View the computed result instantly.
Formula Used
√(a + bi) = ±[√((|z| + a)/2) + i sgn(b)√((|z| - a)/2)] where |z| = √(a² + b²) and sgn(b) is sign of b Or using polar: √z = √r[cos(θ/2) + i sin(θ/2)] and √r[cos((θ+2π)/2) + i sin((θ+2π)/2)]. A complex number has exactly two square roots (calculator id: complex-number-square-root-calculator).
Example Calculation
Worked values: real = 3, imaginary = 4 → root1 = 2 + i, root2 = -2 - i, rootsRectangular = ['2 + i', '-2 - i']
Frequently Asked Questions
What is Complex Number Square Root Calculator?
- Complex Number Square Root Calculator evaluates Complex Number Square Root for the math / complex-numbers library. Primary input cue: Real Part [required]: Real part a. Method anchor: √(a + bi) = ±[√((|z| + a)/2) + i sgn(b)√((|z| - a)/2)] where |z| = √(a² + b²) and sgn(b) is sign of b Or using polar: √z = √r[cos(θ/2) + i sin(θ/2)] and √r[cos(.
How does Complex Number Square Root Calculator work?
- √(a + bi) = ±[√((|z| + a)/2) + i sgn(b)√((|z| - a)/2)] where |z| = √(a² + b²) and sgn(b) is sign of b Or using polar: √z = √r[cos(θ/2) + i sin(θ/2)] and √r[cos((θ+2π)/2) + i sin((θ+2π)/2)]. A complex number has exactly two square roots (calculator id: complex-number-square-root-calculator).
Why use this Math Numbers calculator?
- Use Complex Number Square Root Calculator when you already know the inputs and need complex number square root without setting up a spreadsheet.