Square Root Calculator (Newton's Method)

Newton's Method for Square Roots:

\[ x_{n+1} = \frac{1}{2}\left(x_n + \frac{S}{x_n}\right) \]

Number to find square root:

(positive number)

Iterations:

(1-20)

Initial guess (optional):

(positive number)

Square Root Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Newton's Method for Square Roots?

Newton's Method (also called the Newton-Raphson method) is an iterative algorithm for finding successively better approximations to the roots (or zeroes) of a real-valued function. For square roots, it provides a fast-converging method to calculate √S without a calculator.

2. How Does the Calculator Work?

The calculator uses Newton's Method equation:

\[ x_{n+1} = \frac{1}{2}\left(x_n + \frac{S}{x_n}\right) \]

Where:

\( S \) — The number you want the square root of
\( x_n \) — Current approximation
\( x_{n+1} \) — Next, better approximation

Explanation: Starting with an initial guess (usually S/2), each iteration applies the formula to get a better approximation. The method converges quadratically, meaning the number of correct digits roughly doubles with each step.

3. Importance of Square Root Calculation

Details: Square roots are fundamental in mathematics, physics, engineering, and statistics. Understanding manual calculation methods builds numerical intuition and is valuable when calculators aren't available.

4. Using the Calculator

Tips:

Enter any positive number to find its square root
More iterations (5-10) give more accurate results
For numbers less than 1, a better initial guess might be 1
The method works best for positive numbers

5. Frequently Asked Questions (FAQ)

Q1: Why use Newton's method instead of other approaches?
A: Newton's method converges much faster than simple bisection or guess-and-check methods, often reaching high precision in just a few iterations.

Q2: How accurate is this method?
A: With 5 iterations, you typically get 6-8 decimal places of accuracy for most numbers. Each iteration doubles the number of accurate digits.

Q3: Does the initial guess matter?
A: The method will converge regardless of initial guess (as long as it's positive), but better guesses lead to faster convergence.

Q4: Can this calculate cube roots or other roots?
A: Yes, with a modified formula. The general Newton's method can find roots of any differentiable function.

Q5: What's the history of this method?
A: While often attributed to Isaac Newton, the method was known to ancient mathematicians and was refined by Joseph Raphson in 1690.