Write As A Fraction In Lowest Terms:$\begin{array}{l} .126126126 \ldots=\frac{a}{b} \\ A=\square \\ B=\square \end{array}$

Feb 28, 2025 by ADMIN 123 views

**Repeating Decimals and Fractions: A Mathematical Exploration**

Introduction

In mathematics, repeating decimals are a type of decimal that has a block of digits that repeats indefinitely. These decimals can be represented as fractions, which are essential in various mathematical operations. In this article, we will explore how to write repeating decimals as fractions in lowest terms.

Understanding Repeating Decimals

A repeating decimal is a decimal that has a block of digits that repeats indefinitely. For example, 0.123123123... is a repeating decimal where the block of digits "123" repeats indefinitely. Repeating decimals can be represented as fractions using the concept of infinite geometric series.

Converting Repeating Decimals to Fractions

To convert a repeating decimal to a fraction, we can use the following steps:

Identify the repeating block of digits.
Let x be the repeating decimal.
Multiply x by 10^n, where n is the number of digits in the repeating block.
Subtract the original decimal from the product to eliminate the repeating block.
Solve for x to get the fraction.

Example 1: Converting 0.126126126... to a Fraction

Let's convert the repeating decimal 0.126126126... to a fraction using the steps above.

Identify the repeating block of digits: 126
Let x be the repeating decimal: x = 0.126126126...
Multiply x by 10^3 (since there are 3 digits in the repeating block): 10^3x = 126.126126...
Subtract the original decimal from the product: 10^3x - x = 126.126126... - 0.126126...
Simplify the equation: 999x = 126
Solve for x: x = 126/999

Simplifying the Fraction

The fraction 126/999 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 126 and 999 is 9.

126 ÷ 9 = 14 999 ÷ 9 = 111

So, the simplified fraction is 14/111.

Example 2: Converting 0.090909... to a Fraction

Let's convert the repeating decimal 0.090909... to a fraction using the steps above.

Identify the repeating block of digits: 09
Let x be the repeating decimal: x = 0.090909...
Multiply x by 10^2 (since there are 2 digits in the repeating block): 10^2x = 9.090909...
Subtract the original decimal from the product: 10^2x - x = 9.090909... - 0.090909...
Simplify the equation: 99x = 9
Solve for x: x = 9/99

Simplifying the Fraction

The fraction 9/99 can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 9 and 99 is 9.

9 ÷ 9 = 1 99 ÷ 9 = 11

So, the simplified fraction is 1/11.

Conclusion

Repeating decimals can be represented as fractions using the concept of infinite geometric series. By following the steps outlined above, we can convert repeating decimals to fractions in lowest terms. This is essential in various mathematical operations, such as algebra and calculus. In this article, we explored how to convert two repeating decimals, 0.126126126... and 0.090909..., to fractions in lowest terms.

Future Directions

In future articles, we can explore more advanced topics in mathematics, such as:

Infinite Series: We can delve deeper into the concept of infinite series and explore how they are used to represent repeating decimals.
Calculus: We can explore how repeating decimals are used in calculus, particularly in the study of limits and infinite series.
Number Theory: We can explore the properties of repeating decimals and how they relate to number theory, particularly in the study of prime numbers and modular arithmetic.

References

Wikipedia: Repeating Decimal
Math Is Fun: Repeating Decimals
Khan Academy: Repeating Decimals

Glossary

Repeating Decimal: A decimal that has a block of digits that repeats indefinitely.
Fraction: A way of representing a number as the ratio of two integers.
Greatest Common Divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.
Infinite Geometric Series: A series of numbers that has a common ratio and is infinite in length.
Repeating Decimals and Fractions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored how to convert repeating decimals to fractions in lowest terms. In this article, we will answer some frequently asked questions about repeating decimals and fractions.

Q&A

Q: What is a repeating decimal?

A: A repeating decimal is a decimal that has a block of digits that repeats indefinitely. For example, 0.123123123... is a repeating decimal where the block of digits "123" repeats indefinitely.

Q: How do I identify the repeating block of digits in a repeating decimal?

A: To identify the repeating block of digits, look for a pattern in the decimal that repeats indefinitely. For example, in the decimal 0.126126126..., the block of digits "126" repeats indefinitely.

Q: How do I convert a repeating decimal to a fraction?

A: To convert a repeating decimal to a fraction, follow these steps:

Identify the repeating block of digits.
Let x be the repeating decimal.
Multiply x by 10^n, where n is the number of digits in the repeating block.
Subtract the original decimal from the product to eliminate the repeating block.
Solve for x to get the fraction.

Q: What is the greatest common divisor (GCD) and how do I use it to simplify fractions?

A: The greatest common divisor (GCD) is the largest number that divides two or more numbers without leaving a remainder. To simplify a fraction, divide both the numerator and the denominator by their GCD.

Q: Can I use a calculator to convert repeating decimals to fractions?

A: Yes, you can use a calculator to convert repeating decimals to fractions. However, it's essential to understand the concept behind the conversion to ensure accuracy.

Q: Are there any limitations to converting repeating decimals to fractions?

A: Yes, there are limitations to converting repeating decimals to fractions. For example, some repeating decimals may not have a finite decimal representation, and therefore, cannot be converted to a fraction.

Q: Can I use repeating decimals in real-world applications?

A: Yes, repeating decimals have many real-world applications, such as:

Finance: Repeating decimals are used to calculate interest rates and investments.
Science: Repeating decimals are used to represent physical constants, such as the speed of light.
Engineering: Repeating decimals are used to calculate dimensions and measurements.

Q: Can I use fractions in real-world applications?

A: Yes, fractions have many real-world applications, such as:

Cooking: Fractions are used to measure ingredients and calculate recipes.
Building: Fractions are used to calculate dimensions and measurements.
Science: Fractions are used to represent physical constants and calculate scientific formulas.