Problem 48 - Precalculus: Functions and Graphs

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 \( x_2 = \frac{17}{12} \) and \( x_3 = \frac{577}{408} \).

Step 1: Understand the Problem

We need to find two more rational approximations for \( \sqrt{2} \) using the given formula starting from \( x_1 = \frac{3}{2} \). The formula for the next approximation is \( x_{2} = \frac{1}{2} \left( x_{1} + \frac{n}{x_{1}} \right) \) where \( n = 2 \).

Step 2: Substitute into the Formula for the First Iteration

Substitute \( x_1 = \frac{3}{2} \) into the formula to find \( x_2 \): \[x_2 = \frac{1}{2} \left( \frac{3}{2} + \frac{2}{\frac{3}{2}} \right)\] Simplify the expression inside the parentheses first.

Step 3: Simplify the Expression Inside the Parentheses

Calculate \( \frac{2}{\frac{3}{2}} = \frac{2 \times 2}{3} = \frac{4}{3} \). So, the expression becomes: \[\frac{3}{2} + \frac{4}{3}\] Find a common denominator to add the fractions.

Step 4: Add the Fractions

The common denominator for \( \frac{3}{2} \) and \( \frac{4}{3} \) is 6. Convert and add the fractions: \[\frac{3}{2} = \frac{9}{6}, \quad \frac{4}{3} = \frac{8}{6}\]\[\frac{9}{6} + \frac{8}{6} = \frac{17}{6}\]

Step 5: Calculate x2

Substitute the sum back into the formula for \( x_2 \): \[x_2 = \frac{1}{2} \left( \frac{17}{6} \right) = \frac{17}{12}\]

Step 6: Use x2 to Find x3

Now let \( x_1 = \frac{17}{12} \) and find \( x_3 \) using the formula:\[x_3 = \frac{1}{2} \left( \frac{17}{12} + \frac{2}{\frac{17}{12}} \right)\]Simplify \( \frac{2}{\frac{17}{12}} \) first.

Step 7: Simplify the Division

Calculate \( \frac{2}{\frac{17}{12}} = \frac{2 \times 12}{17} = \frac{24}{17} \). So, the expression becomes \[\frac{17}{12} + \frac{24}{17}\]Find a common denominator to add the fractions.

Step 8: Add the Fractions

The common denominator for \( \frac{17}{12} \) and \( \frac{24}{17} \) is 204. Convert and add the fractions:\[\frac{17}{12} = \frac{289}{204}, \quad \frac{24}{17} = \frac{288}{204}\]\[\frac{289}{204} + \frac{288}{204} = \frac{577}{204}\]

Step 9: Calculate x3

Substitute the sum back into the formula for \( x_3 \):\[x_3 = \frac{1}{2} \left( \frac{577}{204} \right) = \frac{577}{408}\] Thus, \( x_3 = \frac{577}{408} \).

Key Concepts

Rational Approximation

The concept of rational approximation involves finding values that are "good enough" estimates for numbers that are hard to compute exactly, such as square roots. The Babylonians developed a technique for estimating square roots with a rational approximation. This gave a rough estimate that was easy to improve with repeated calculations.

In our example, we're dealing with the square root of 2 (\(\sqrt{2}\)), which is a number that cannot be exactly represented as a fraction. Therefore, we use rational approximation to get as close as possible. In practical terms, this means that we start with an initial estimate (in this case \(x_1 = \frac{3}{2}\)) and refine it to make it more accurate. As you go through each step, the results "approximate" the true value better than the previous iteration, which is the beauty of rational approximation.

Square Roots

Square roots are simply numbers that produce a given number when multiplied by themselves. For instance, the square root of 16 is 4 because 4 times 4 is 16. However, not all numbers have neat, whole-number square roots. Consider \(\sqrt{2}\): there isn't a precise fraction or simple number that results in multiplying back to exactly 2. This is why we turn to methods like the Babylonian method for approximation.

Understanding the nature of square roots helps us see why rational approximations are useful. We try to balance between the guesswork and precision since exact answers aren't feasible for certain numbers like root 2. This is why when computing approximations for \(\sqrt{n}\), it is important to start with a reasonable guess and then refine it further.

Mathematical Formula

The Babylonian method, famous for its simplicity, uses a mathematical formula to refine approximations. The formula \(x_2 = \frac{1}{2} \left( x_1 + \frac{n}{x_1} \right)\) helps us find a better approximation of \(\sqrt{n}\). This step-by-step and iterative process allows one to gradually approach a more accurate result.

With each iteration, the formula does two things: it averages the current estimate and the quotient of the desired number divided by the estimate. This balance ensures that the next estimate is not too high or low, centering it closer to the true square root. By plugging numbers into this formula and simplifying, we get progressively better approximations.

Iteration Method

Iteration involves repeating steps in a process to get closer to a desired result. The iteration method is crucial in refining rational approximations. We start with an initial guess and apply the Babylonian formula repeatedly. Each iteration uses the output of the previous step as its starting point.

The brilliance of this method is in its simplicity and efficiency. With each cycle, or iteration, the method adjusts and corrects previous inaccuracies, honing in on the right answer. Inviting you to be patient, each iteration builds upon the previous, making the final approximation increasingly accurate. For students, understanding iteration as a process of continual improvement is not only vital for problems like finding square roots but also an essential skill in solving diverse mathematical challenges.