Problem 23 - Thomas Calculus

Question

Express each of the numbers in Exercises \(23-30\) as the ratio of two integers. $$0 . \overline{23}=0.232323 \ldots$$

Step-by-Step Solution

Answer

The repeating decimal \( 0.232323... \) can be expressed as \( \frac{23}{99} \).

Step 1: Represent the Repeating Decimal

Let the repeating decimal be represented as \( x = 0.232323... \). This is a geometric progression.

Step 2: Remove Repetition by Multiplication

Multiply both sides of the equation by 100 (since the repeating part has two digits) to shift the decimal: \( 100x = 23.232323... \).

Step 3: Subtract to Eliminate Repetition

Subtract the initial equation from this equation to eliminate the repeating part: \( 100x - x = 23.232323... - 0.232323... \). This results in: \( 99x = 23 \).

Step 4: Solve for x

Divide both sides by 99 to solve for \( x \): \( x = \frac{23}{99} \). Thus, \( 0.232323... \) can be expressed as the fraction \( \frac{23}{99} \).

Step 5: Simplify the Fraction

Check if \( \frac{23}{99} \) can be simplified. Since 23 is a prime number, and it doesn’t divide 99, the fraction is already in its simplest form.

Key Concepts

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In the world of repeating decimals, this concept comes into play when we express the repeating sequence as a sum of infinite terms, each an element of the progression. This helps us solve the problem by understanding that the repeating pattern can be captured using algebraic methods.

For example, consider the decimal 0.232323... Here, the repeated section '23' is a type of mini geometric series. We take advantage of the pattern (which repeats every two digits) by multiplying the decimal by 100, shifting all the digits over, creating the equation. The essence of this step is to strip the number of its repetition so we can more easily handle it analytically.

Ratio of Two Integers

A repeating decimal represents an infinite sequence that can be neatly expressed as a ratio of two integers, also known as a fraction. This conversion is possible because the repeating section implies a predictable and steady pattern. In our example of 0.232323..., the trick is to set the repeating decimal equal to a variable, such as \( x \), and use algebraic manipulation to boil it down to a ratio.

Once you establish an equation through multiplication, subtraction helps eliminate the repeating elements. After subtraction, you are left with a simple algebraic equation. Solving this introduces the concept of expressing infinite decimals in a much more manageable fractional form, \( \frac{23}{99} \) in this case. This shows the connection between repeating patterns and their fractional counterparts.

Fraction Simplification

Simplifying fractions involves reducing the numerator and the denominator to their smallest whole number values while maintaining the same value of the fraction. Simplification helps us interpret the ratio more intuitively and makes mathematical operations more manageable.

In our example, the fraction \( \frac{23}{99} \) is checked for simplification. We consider whether the numerator (23) and the denominator (99) share any common factors other than 1. Since 23 is a prime number and does not divide evenly into 99, the fraction is already in its simplest form. Understanding simplification allows students to express numbers in their most concise and efficient ways, which is a critical skill in mathematics.