0.47 Recurring As A Fraction

Unveiling the Mystery: 0.47 Recurring as a Fraction

Understanding recurring decimals and their fractional equivalents can seem daunting, but with a systematic approach, it becomes surprisingly straightforward. This article delves into the process of converting the recurring decimal 0.47 recurring (denoted as 0.474747...) into a fraction. We'll explore the underlying mathematical principles, provide step-by-step instructions, and address common questions, making this complex topic accessible to everyone. Mastering this skill will enhance your understanding of decimal-fraction relationships and improve your problem-solving abilities in mathematics.

Understanding Recurring Decimals

A recurring decimal, also known as a repeating decimal, is a decimal representation of a number where one or more digits repeat infinitely. The repeating digits are indicated by placing a bar over them. For example, 0.47 recurring is written as 0.$\overline{47}$. This notation signifies that the digits "47" repeat endlessly: 0.47474747...

Recurring decimals represent rational numbers – numbers that can be expressed as a fraction of two integers. This means that every recurring decimal has an equivalent fraction, and finding that fraction is the focus of this article. Conversely, every rational number has a decimal representation that is either terminating or recurring. Irrational numbers, like π (pi) or √2, have non-repeating, non-terminating decimal representations.

Method 1: Using Algebra to Solve for the Fraction

This algebraic method is a powerful and general approach for converting any recurring decimal to a fraction. Let's apply it to 0.$\overline{47}$:

Step 1: Assign a Variable

Let x = 0.$\overline{47}$. This assigns a variable to the recurring decimal.

Step 2: Multiply to Shift the Decimal Point

Multiply both sides of the equation by 100 (since there are two recurring digits). This shifts the repeating block to the left of the decimal point:

100x = 47.$\overline{47}$

Step 3: Subtract the Original Equation

Subtracting the original equation (x = 0.$\overline{47}$) from the equation in Step 2 eliminates the repeating decimal part:

100x - x = 47.$\overline{47}$ - 0.$\overline{47}$

This simplifies to:

99x = 47

Step 4: Solve for x

Divide both sides by 99 to solve for x:

x = 47/99

Therefore, the fraction equivalent of 0.$\overline{47}$ is 47/99. This fraction is in its simplest form because 47 is a prime number and does not share any common factors with 99 (other than 1).

Method 2: Understanding the Place Value System

This method helps build an intuitive understanding of why the algebraic approach works. Let's break down 0.$\overline{47}$ using the place value system:

0.$\overline{47}$ = 47/100 + 47/10000 + 47/1000000 + ...

This is an infinite geometric series where the first term (a) is 47/100 and the common ratio (r) is 1/100. The formula for the sum of an infinite geometric series is:

Sum = a / (1 - r)

Substituting our values:

Sum = (47/100) / (1 - 1/100) = (47/100) / (99/100) = 47/99

This confirms our result from the algebraic method. This approach demonstrates the underlying mathematical structure of recurring decimals as infinite sums.

Simplifying Fractions: A Quick Review

While 47/99 is already in its simplest form, understanding how to simplify fractions is crucial in mathematics. A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. To simplify a fraction, you find the greatest common divisor (GCD) of the numerator and denominator and divide both by it. For example, to simplify 12/18:

1. Find the GCD: The GCD of 12 and 18 is 6.
2. Divide: Divide both the numerator and denominator by 6: 12/6 = 2 and 18/6 = 3.
3. Simplified Fraction: The simplified fraction is 2/3.

The prime factorization method is often helpful in finding the GCD. For example, the prime factorization of 12 is 2 x 2 x 3, and the prime factorization of 18 is 2 x 3 x 3. The common factors are 2 and 3, so the GCD is 2 x 3 = 6.

Dealing with Longer Recurring Decimals

The algebraic method described above easily extends to recurring decimals with more repeating digits. For example, let's consider 0.123$\overline{456}$:

1. Assign a variable: x = 0.123$\overline{456}$
2. Multiply to align repeating digits: Multiply by 1000 to get 1000x = 123.$\overline{456}$
3. Multiply again to shift decimal: Multiply by 1000000 to get 1000000x = 123456.$\overline{456}$
4. Subtract: 1000000x - 1000x = 123456.$\overline{456}$ - 123.$\overline{456}$ which simplifies to 999000x = 123333
5. Solve: x = 123333 / 999000. This fraction can then be simplified by finding the GCD of 123333 and 999000.

This method demonstrates the adaptability of the algebraic technique for handling more complex recurring decimals. The key is to multiply by the appropriate power of 10 to align the repeating digits before subtraction.

Terminating Decimals vs. Recurring Decimals

It's important to distinguish between terminating and recurring decimals. Terminating decimals have a finite number of digits after the decimal point. For example, 0.5, 0.75, and 0.125 are terminating decimals. They can also be expressed as fractions: 1/2, 3/4, and 1/8 respectively.

Recurring decimals, on the other hand, have an infinite number of repeating digits. The process of converting them to fractions involves the algebraic method or the geometric series approach discussed earlier. Understanding this distinction helps in selecting the appropriate method for converting decimals to fractions.

Frequently Asked Questions (FAQ)

Q1: Can all decimals be expressed as fractions?

A1: No. Only rational numbers can be expressed as fractions. Irrational numbers, such as π and √2, have non-terminating and non-repeating decimal representations.

Q2: What if the recurring decimal starts after some non-recurring digits?

A2: For example, consider 0.12$\overline{34}$. You would first separate the non-recurring part: 0.12 + 0.00$\overline{34}$. Convert the recurring part to a fraction using the algebraic method. Then, add the non-recurring part expressed as a fraction (12/100).

Q3: Is there a calculator or software that can convert recurring decimals to fractions?

A3: While some calculators might handle simple cases, most will not be able to directly convert all recurring decimals to fractions, particularly those with long repeating blocks. The algebraic method remains the most reliable approach.

Q4: Why is understanding this conversion important?

A4: The ability to convert recurring decimals to fractions is fundamental in various mathematical fields. It underpins concepts in algebra, calculus, and number theory. It also strengthens your understanding of the relationship between decimal and fractional representations of numbers.

Conclusion: Mastering the Conversion

Converting 0.47 recurring (0.$\overline{47}$) to a fraction, 47/99, illustrates the power and elegance of mathematical methods. The algebraic approach, coupled with an understanding of infinite geometric series, offers a robust and reliable technique for converting any recurring decimal to its fractional equivalent. This process is not just about finding an answer; it's about grasping the fundamental connection between different number systems and developing a deeper appreciation for the beauty and logic inherent in mathematics. Remember to practice these methods with various examples to build confidence and proficiency. The more you practice, the easier it will become to handle even the most complex recurring decimals and unravel the mysteries of their fractional identities.