Online Square Root Calculator Tool

Square Root Calculator

Professional-grade calculator with step-by-step explanations

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Quick Guide:

Enter your number in the input field on the left, then click the "Calculate √" button. Your result will appear on the right with detailed step-by-step calculations below. Use the copy button to copy results, as they may be longer than displayed.

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Result Information

Mathematical Properties

Step-by-Step Calculation Methods

Newton's Method (Newton-Raphson)

x_n+1 = ½(x_n + a/x_n)

Convergence Visualization:

Iteration	Approximation	Error	Accuracy

Babylonian Method (Heron's Method)

x_n+1 = ½(x_n + S/x_n)

Historical Note:

This method was used by ancient Babylonians around 2000 BCE and later refined by Heron of Alexandria. It's essentially the same as Newton's method applied to square roots.

Prime Factorization Method

Method Explanation:

This method breaks down the number into prime factors, then extracts perfect squares. It's particularly useful for finding exact values when possible.

Long Division Method

Traditional Approach:

This is the classic pencil-and-paper method taught in schools, useful for understanding the step-by-step process of finding square roots digit by digit.

Calculation History

No calculations yet. Start by entering a number above!

Square Root Tutorial

What is a Square Root?

A square root of a number x is a value that, when multiplied by itself, gives x. For example, √9 = 3 because 3 × 3 = 9.

Perfect Squares

Numbers like 1, 4, 9, 16, 25, 36... are perfect squares because their square roots are whole numbers (1, 2, 3, 4, 5, 6...).

Irrational Numbers

Most square roots (like √2, √3, √5) are irrational numbers with infinite, non-repeating decimal expansions.

Historical Facts

The Radical Symbol (√)

The radical symbol √ was first used in print by German mathematician Christoph Rudolff in 1525 in his book "Coss".

Ancient Origins

The concept of square roots dates back to ancient Babylonians around 2000 BCE. They used geometric methods and approximation techniques.

Greek Mathematics

Ancient Greeks discovered that √2 is irrational, leading to a crisis in their mathematical philosophy that only accepted rational numbers.

Advanced Features

Copy Functionality

Click any copy button to copy results to clipboard. Useful for long decimal results.

Multiple Methods

View different calculation approaches including Newton's method and prime factorization.

Detailed Analysis

Get comprehensive information about mathematical properties and convergence analysis.

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Feature	Details
Price	Free
Rendering	Client-Side Rendering
Language	JavaScript
Paywall	No

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About This Tool

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