Approximating the Square Root of 2: Methods and Techniques

Approximating the square root of 2 is a classic problem used to illustrate various mathematical concepts and methods. There are several ways to approach this, and here we will explore a few of them. We'll start by approximating the square root of 2 using fractions and then delve into more sophisticated methods like the Babylonian method and Newton's method.

Fractional Approximation

In a previous example, the square root of 2 was approximated using the fraction 10/7. This method involves starting with an initial guess and making successive refinements.

Let's start with a 10 and b 7. The initial approximation is:

(frac{10}{7}^2 100/49) or approximately 2.041.

We then use the formula (b(n/a a/b)). For our first iteration:

a 10, b 7, n 2

Transforming 10/7 to (27/10 * 7 24/17) and squaring it gives approximately 1.993. Thus, a 24, b 17 and n 2.

Repeating the process:

a 24, b 17, n 2

24 * (217/24 * 17) 58/41

58/41^2 3364/1681 approx 2.0012)

This process can be repeated arbitrarily to get closer approximations.

Guess and Check Method

A simple and intuitive method is the guess and check method, particularly useful for manual calculations. Start with a value and refine it by trying values in between.

For example, knowing that 1^2 1 and 2^2 4, we can see that sqrt{2} lies between 1 and 2. A better guess can be 1.4, since 1.4^2 1.96. A higher guess of 1.5, where 1.5^2 2.25, can help us narrow it down to between 1.4 and 1.5.

Babylonian Method (Herons Method)

The Babylonian method (also known as Heron's method) is an iterative method widely used for calculating square roots accurately. It is a more refined approach that converges quickly to the correct value.

Steps:

Choose an initial guess: x_0 1.5 Calculate x_{n 1} frac{1}{2}(x_n frac{2}{x_n}) Repeat until convergence

Applying the method:

x_0 1.5 x_1 frac{1}{2}(1.5 frac{2}{1.5}) frac{1.5 1.3333}{2} 1.4167) x_2 frac{1}{2}(1.4167 frac{2}{1.4167}) approx 1.4142)

Continuing the iterations, the value will converge to the square root of 2, which is approximately 1.4142.

Using a Calculator

For high precision and ease, the simplest and most effective method is to use a calculator. Simply input sqrt{2} to get the precise value of approximately 1.41421356.

Newton's Method

Newton's method, or Newton-Raphson method, is an iterative method that uses calculus to converge to the square root. It is particularly useful for refinement and provides a more advanced approach.

For the function f(x) x^2 - 2, the iterative formula is:

x_{n 1} x_n - frac{f(x_n)}{f'(x_n)} x_n - frac{x_n^2 - 2}{2x_n} frac{1}{2}(x_n frac{2}{x_n}))

Starting with an initial guess such as x_0 1.5 and applying the update iteratively:

x_1 frac{1.5 frac{2}{1.5}}{2} 1.4167

x_2 approx 1.4142)

Continuing this process, we can achieve a highly precise result.

Conclusion

The Babylonian method is often the simplest and most effective for hand calculations due to its straightforward and iterative nature. For high accuracy and convenience, using a calculator is the most efficient method. Newton's method provides an advanced and calculus-based approach for refining approximations.