Problem 56 - Calculus: Early Transcendentals

Question

Express the number as a ratio of intergers. \( 5. \overline {71358} \)

Step-by-Step Solution

Answer

The number expressed as a ratio of integers is \( \frac{190451}{33333} \).

Step 1: Understanding the Repeating Decimal

The number given is \( 5. \overline{71358} \), which means that 71358 is the repeating part. Thus, the number can be expressed as \( 5.713587135871358... \)

Step 2: Express as a Series of Decimals

Let \( x = 5.7135871358... \). This number can be seen as \( x = 5 + 0.7135871358... \). Focus on the decimal part \( 71358... \).

Step 3: Form an Equation with the Repeating Part

To eliminate the repeating part, let's set \( y = 0.\overline{71358} \). Therefore, \( x = 5 + y \). Next, find a way to express \( y \) as a fraction.

Step 4: Establish a Multiplier to Clear the Repetition

Multiply \( y \) by \( 100000 \) (because 71358 has 5 digits) to move the decimal point over the full repeating block: \( 100000y = 71358.71358... \), which can be rewritten as \( 100000y = 71358 + y \).

Step 5: Solve for the Non-Terminating Decimal Fraction

By subtracting \( y = 0.71358... \) from \( 100000y = 71358.71358... \), we get \( 100000y - y = 71358 \). Simplifying gives \( 99999y = 71358 \). Solve for \( y \): \( y = \frac{71358}{99999} \).

Step 6: Simplifying the Fraction

To simplify \( \frac{71358}{99999} \), find the greatest common divisor (GCD) and reduce. The GCD here is 3, so \( \frac{71358 \div 3}{99999 \div 3} = \frac{23786}{33333} \).

Step 7: Combine integer and Decimal to form the Final Ratio

We have \( x = 5 + \frac{23786}{33333} \). Convert this to a single fraction: \( x = \frac{5 \times 33333 + 23786}{33333} = \frac{166665 + 23786}{33333} = \frac{190451}{33333} \).

Key Concepts

Ratio of Integers

When dealing with repeating decimals, we often need to express them as a ratio of two integers. This process transforms the repeating, endless series of numbers into a fraction, which can be more easily analyzed and manipulated.
To start, recognize what the repeating decimal looks like. For example, the number \(5.\overline{71358}\) represents a repeating block '71358'. Here, this is the portion of the decimal that repeats forever.

• Integer Component: The integer part, in this case, is \(5\).
• Repeating Decimal: The repeating block is \(71358\).

We represent this repeating decimal using a combination of multiplication and subtraction. By knowing how to separate the integer and decimal components, we can create a coherent mathematical expression. This expression captures both components of the repeating decimal in one single fraction.

Fraction Conversion

Fraction conversion involves changing a repeating decimal into a fraction. This helps in better understanding and utilizing the number.
Let's break down this conversion using our repeating decimal example \(0.\overline{71358}\), which can be noted as \(y = 0.7135871358...\). To find the fraction form:

• Set up an Equation: Let \(y\) equal the repeating decimal, such that \(y = 0.\overline{71358}\).
• Clear the Repeating Part: Multiply \(y \) by \(100000\), because the block '71358' has 5 digits. This results in: \(100000y = 71358.71358...\).
• Create Integer Equation: Simplify by recognizing the repeating decimal by subtracting: \(100000y - y = 71358\).

This transforms into a simple ratio: \(99999y = 71358\), leading to the fraction \(y = \frac{71358}{99999}\). At this point, the decimal has been successfully converted to a fraction.

Simplifying Fractions

Once you have your fraction from a repeating decimal, simplifying it makes it easier to work with. Simplification involves reducing the fraction to its smallest equivalent form.

• Identify the Greatest Common Divisor (GCD): In our example, the fraction is \(\frac{71358}{99999}\). First, find the GCD of these numbers. Here, it is 3.
• Divide Numerator and Denominator by GCD: Reduce the fraction by dividing both the numerator and the denominator by their GCD: \(\frac{71358 \div 3}{99999 \div 3} = \frac{23786}{33333}\).

Achieving this reduced form of the fraction makes any future calculations much simpler. Furthermore, expressing the whole number together with the simplification provides the final expression: \(x = 5 + \frac{23786}{33333}\).
This can also be reformulated as a single fraction \(\frac{190451}{33333}\), integrating the integer part and the reduced decimal fraction efficiently.