In Newton-Raphson’s method, the formula for finding the square root of the number N is:

• x = (x + N / x) / 2

Newton-Raphson's method is a powerful technique for finding roots of equations, including calculating square roots. To find the square root of a number \( N \), you can use the iterative formula:

Formula Breakdown

The formula is:

x = (x + N / x) / 2

In this equation:

• x is your current estimate of the square root.
• N is the number for which you want to find the square root.

How It Works

1. Start with an initial guess for \( x \). A common choice is \( N/2 \) or any positive number.

2. Plug this value into the formula to get a new estimate.

3. Repeat the process using the new estimate until the value stabilizes (the change is very small).

Example Calculation

For instance, to find the square root of 9:

• Start with \( x = 4 \).
• Calculate: \( x = (4 + 9 / 4) / 2 = (4 + 2.25) / 2 = 3.125 \).
• Repeat with \( x = 3.125 \) to refine the estimate.

This method converges quickly, making it efficient for finding square roots.