Problem 48 - Algebra and Trigonometry with Analytic Geometry

Question

Rational approximations to square roots can be found using a formula discovered by the ancient Babylonians. Let \(x_{1}\) be the first rational approximation for \(\sqrt{n}\). If we let $$ x_{2}=\frac{1}{2}\left(x_{1}+\frac{n}{x_{1}}\right) $$ then \(x_{2}\) will be a better approximation for \(\sqrt{n}\), and we can repeat the computation with \(x_{2}\) replacing \(x_{1}\). Starting with \(x_{1}=\frac{3}{2}\), find the next two rational approximations for \(\sqrt{2}\).

Step-by-Step Solution

Answer

The next two approximations are \( \frac{17}{12} \) and \( \frac{577}{408} \).

Step 1: Understand the Formula

The Babylonian method finds successive approximations. Starting with an initial guess \( x_1 \), the formula for the next approximation \( x_2 \) is \( x_2 = \frac{1}{2} \left(x_1 + \frac{n}{x_1}\right) \). This process refines the guess for \( \sqrt{n} \).

Step 2: Apply the Formula for the First Approximation

Given \( x_1 = \frac{3}{2} \) and \( n = 2 \), substitute these values into the formula: \( x_2 = \frac{1}{2} \left( \frac{3}{2} + \frac{2}{\frac{3}{2}} \right) \). Calculate \( \frac{2}{\frac{3}{2}} = \frac{4}{3} \).

Step 3: Complete the First Approximation Calculation

Now substitute the calculation from Step 2: \( x_2 = \frac{1}{2} \left( \frac{3}{2} + \frac{4}{3} \right) = \frac{1}{2} \left( \frac{9}{6} + \frac{8}{6} \right) = \frac{1}{2} \left( \frac{17}{6} \right) = \frac{17}{12} \). Thus, \( x_2 = \frac{17}{12} \) is the second approximation.

Step 4: Use the Second Approximation for Further Refinement

Now use \( x_2 = \frac{17}{12} \) to find \( x_3 \): \( x_3 = \frac{1}{2} \left( \frac{17}{12} + \frac{2}{\frac{17}{12}} \right) \). First, calculate \( \frac{2}{\frac{17}{12}} \) by multiplying by the reciprocal: \( \frac{24}{17} \).

Step 5: Complete the Calculation for the Second Approximation

Now substitute in the calculated values: \( x_3 = \frac{1}{2} \left( \frac{17}{12} + \frac{24}{17} \right) \). Find a common denominator: \( \frac{17 \times 17}{204} = \frac{289}{204} \) and \( \frac{24 \times 12}{204} = \frac{288}{204} \). Thus, \( x_3 = \frac{1}{2} \left( \frac{577}{204} \right) = \frac{577}{408} \) after simplification.

Key Concepts

Rational Approximation

Rational approximation is a technique used to find a close estimate of a number, often when an exact value is difficult to determine. This approach is especially useful for non-perfect square roots, like \( \sqrt{2} \).

• We start with an initial guess, also known as an approximation.
• The goal is to refine this guess so it gets closer to the true value.
• Rational approximations are expressed as fractions, making calculations manageable.

In our exercise, we're using the Babylonian method to achieve a rational approximation for \( \sqrt{2} \). It's a process of iteration and refinement, allowing our starting point to evolve into a more accurate estimate.

Square Roots

A square root is a value that, when multiplied by itself, gives the original number. For example, \( \sqrt{4} = 2 \) because \( 2 \times 2 = 4 \).

• Square roots can be whole numbers or irrational.
• Finding the exact square root of non-perfect squares like 2 results in an irrational number.
• The decimal form of \( \sqrt{2} \) goes on without repeating.

These reasons make it convenient to use rational approximations to find a usable estimate of \( \sqrt{2} \) rather than its exact irrational value.

Iterative Calculation

Iterative calculation is a method where calculations are repeated over and over. Each iteration refines the result from the last.

• The Babylonian method is inherently iterative.
• We start with an initial approximation and use a formula to find a better one.
• This improved value becomes the starting point for the next iteration.

In our example, after calculating \( x_2 \) using the initial guess \( x_1 = \frac{3}{2} \), we get an improved approximation \( x_3 \) by repeating the process. This iterative nature is key to achieving greater accuracy over successive steps.

Numerical Methods

Numerical methods are mathematical techniques used to approximate solutions. These methods, like the Babylonian method, are essential in situations where exact answers are difficult or impossible to find.

• They are particularly useful in engineering, physics, and computer science.
• Allow for calculations that would otherwise be too complex.
• Empower us to solve problems that involve irrational numbers and complex equations practically.

In the context of our task, a numerical method helps find closer estimates to \( \sqrt{2} \) by quickly iterating our calculations until we reach a satisfactory level of precision.