How to Convert Repeating Decimals to Fractions (Rational Numbers)

📚 Understanding Repeating Decimals and Fractions

A repeating decimal, also known as a recurring decimal, is a decimal number that has a digit or group of digits that repeat infinitely. Converting these decimals into fractions, also known as rational numbers, is a fundamental skill in mathematics. Let's explore this concept in detail.

📜 A Brief History

The concept of rational numbers and their decimal representations has been around for centuries. Ancient civilizations, including the Egyptians and Babylonians, used fractions extensively. The modern decimal system, however, was developed later, with mathematicians like Simon Stevin contributing significantly to its standardization in the 16th century. Understanding how to represent repeating decimals as fractions allows us to appreciate the link between different number systems.

🔑 Key Principles for Conversion

• 🧮 Identifying the Repeating Block: The first step is to identify the repeating digit or sequence of digits. For example, in $0.\overline{3}$, the digit $3$ repeats. In $0.\overline{142857}$, the sequence $142857$ repeats.
• ✍️ Setting up the Equation: Let $x$ equal the repeating decimal. Multiply $x$ by a power of 10 such that the repeating block starts immediately after the decimal point.
• ➖ Subtracting to Eliminate the Repeating Part: Subtract the original equation from the new equation. This eliminates the repeating decimal portion.
• ➗ Solving for x: Solve the resulting equation for $x$. This will give you the fraction representation of the repeating decimal.
• 🌱 Simplifying the Fraction: Simplify the fraction to its lowest terms. This ensures the fraction is in its simplest form.

➗ Converting Simple Repeating Decimals

Let's look at how to convert decimals with a single repeating digit.

1. Example 1: Convert $0.\overline{3}$ to a fraction.
   • Let $x = 0.\overline{3}$.
   • Multiply by 10: $10x = 3.\overline{3}$.
   • Subtract the original equation: $10x - x = 3.\overline{3} - 0.\overline{3}$, which simplifies to $9x = 3$.
   • Solve for $x$: $x = \frac{3}{9}$.
   • Simplify: $x = \frac{1}{3}$.
2. Example 2: Convert $0.\overline{7}$ to a fraction.
   • Let $x = 0.\overline{7}$.
   • Multiply by 10: $10x = 7.\overline{7}$.
   • Subtract the original equation: $10x - x = 7.\overline{7} - 0.\overline{7}$, which simplifies to $9x = 7$.
   • Solve for $x$: $x = \frac{7}{9}$.

➕ Converting Repeating Decimals with Multiple Repeating Digits

Now, let's tackle decimals with a repeating block of digits.

1. Example 1: Convert $0.\overline{142857}$ to a fraction.
   • Let $x = 0.\overline{142857}$.
   • Multiply by $10^6$ (since there are 6 repeating digits): $1000000x = 142857.\overline{142857}$.
   • Subtract the original equation: $1000000x - x = 142857.\overline{142857} - 0.\overline{142857}$, which simplifies to $999999x = 142857$.
   • Solve for $x$: $x = \frac{142857}{999999}$.
   • Simplify: $x = \frac{1}{7}$.
2. Example 2: Convert $0.\overline{23}$ to a fraction.
   • Let $x = 0.\overline{23}$.
   • Multiply by $10^2$ (since there are 2 repeating digits): $100x = 23.\overline{23}$.
   • Subtract the original equation: $100x - x = 23.\overline{23} - 0.\overline{23}$, which simplifies to $99x = 23$.
   • Solve for $x$: $x = \frac{23}{99}$.

📝 Converting Mixed Repeating Decimals

Mixed repeating decimals have a non-repeating part followed by a repeating part.

1. Example: Convert $0.1\overline{6}$ to a fraction.
   • Let $x = 0.1\overline{6}$.
   • Multiply by 10: $10x = 1.\overline{6}$.
   • Multiply by 10 again: $100x = 16.\overline{6}$.
   • Subtract the two equations: $100x - 10x = 16.\overline{6} - 1.\overline{6}$, which simplifies to $90x = 15$.
   • Solve for $x$: $x = \frac{15}{90}$.
   • Simplify: $x = \frac{1}{6}$.

🌍 Real-World Applications

Converting repeating decimals to fractions has practical applications in various fields, including:

• 💰 Finance: Calculating interest rates or loan payments.
• 📏 Engineering: Ensuring precision in measurements and calculations.
• 💻 Computer Science: Converting numbers between different formats for accurate data processing.

💡 Conclusion

Converting repeating decimals to fractions is a valuable skill with applications in various fields. By understanding the underlying principles and practicing with examples, you can master this concept and enhance your mathematical proficiency. Remember to always identify the repeating block, set up the equation, eliminate the repeating part through subtraction, solve for the variable, and simplify the fraction. With practice, you'll find converting repeating decimals to fractions much easier. 😉

✍️ Practice Quiz

Convert the following repeating decimals to fractions:

1. $0.\overline{5}$
2. $0.\overline{12}$
3. $0.\overline{789}$
4. $0.2\overline{3}$
5. $0.1\overline{45}$
6. $1.\overline{3}$
7. $2.\overline{01}$