Problem 40 - Algebra and Trigonometry with Analytic Geometry

Question

Find the rational number represented by the repeating decimal. $$0.0 \overline{71}$$

Step-by-Step Solution

Answer

The repeating decimal 0.0\overline{71} is equivalent to \( \frac{71}{990} \).

Step 1: Assign a Variable

Let \( x \) represent the repeating decimal. So, we have \( x = 0.07171717\ldots \).

Step 2: Multiply to Isolate the Repeating Part

Multiply \( x \) by 100 (since there are two repeating digits) to shift the decimal 2 places to the right: \( 100x = 7.171717\ldots \).

Step 3: Subtract to Eliminate the Repeating Decimal

Subtract the original \( x = 0.07171717\ldots \) from \( 100x = 7.171717\ldots \). This gives: \( 100x - x = 7.171717\ldots - 0.07171717\ldots \). Simplifying, we get \( 99x = 7.1 \).

Step 4: Simplify the Equation

Express 7.1 as a fraction: \( 7.1 = \frac{71}{10} \). So, \( 99x = \frac{71}{10} \).

Step 5: Solve for x

Solve for \( x \) by dividing both sides by 99: \( x = \frac{71}{990} \).

Step 6: Verify the Solution

Convert \( \frac{71}{990} \) back to a decimal to check: \( \frac{71}{990} = 0.07171717\ldots \), which matches the original repeating decimal.

Key Concepts

Rational Numbers

Rational numbers are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. For example, numbers like 1/2, -3/4, and 5 are all rational numbers. Their defining characteristic is the ability to be written as a fraction.

• Any number that can be expressed in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b eq 0 \), is considered a rational number.
• This includes fractions, but it also includes whole numbers and some decimals.
• Repeating decimals, such as 0.07171717..., are also rational because they can be converted into fractions.

Understanding rational numbers is crucial when dealing with repeating decimals, as the goal is to convert these decimals into a rational form that can be expressed as a fraction.

Fraction Conversion

Fraction conversion involves transforming a decimal or a repeating decimal into a fraction. This process is essential when working with repeating decimals like 0.0\( \overline{71} \). The steps entail using algebra to eliminate the repeating part and solve for a fraction form:

• Assign the repeating decimal to a variable, often denoted as \( x \).
• Multiply \( x \) by a power of 10 that matches the length of the repeating sequence. For 0.07171717..., multiply by 100.
• Subtract the original equation from the multiplied version to eliminate the repeat.
• The result gives a simple equation, which can be solved to find the fraction.

By converting repeating decimals to fractions, you can express these numbers precisely, avoiding any approximation that comes with keeping the decimal form.

Decimal Representation

Decimal representation expresses numbers using the base-ten system, and it may include finite decimals, infinite repeating decimals, or infinite non-repeating decimals. Exploring decimal types helps in understanding how numbers are structured.

• Finite decimals, such as 0.5 or 1.25, have limited digits after the decimal point and represent rational numbers.
• Infinite repeating decimals, like 0.07171717..., show a subset of digits that repeat indefinitely. These are always rational numbers because they can convert to fractions.
• Infinite non-repeating decimals, such as \( \pi \), are typically irrational, meaning they cannot be expressed as a fraction.

Understanding the structure of decimal representation allows us to see why repeating decimals can always conform to a rational form. It also highlights the differences between repeating and non-repeating decimals from a mathematical perspective.

Algebraic Manipulation

Algebraic manipulation is used to solve equations and convert repeating decimals into fractions through a strategic process of isolation and simplification. This method relies on understanding how to handle variables and coefficients systematically:

• Start by assigning the repeating decimal to a variable, followed by manipulating this variable through multiplication.
• By moving the decimal point via multiplication, you create an equation that helps isolate the repeating part.
• Subtracting these equations eliminates the repeating section, leading to a clean equation with variables.
• The final step involves solving for \( x \) by simplifying the equation, usually resulting in a simple, rational fraction.

Using algebraic manipulation allows us to systematically transform complex repeating decimals into simplified fractions easily. This technique is fundamental not only in recognizing patterns but also in showcasing how decimals relate back to their fraction counterparts.