Rewrite .17777 As A Fraction

Rewriting 0.17777... as a Fraction: A Deep Dive into Repeating Decimals

Understanding how to convert repeating decimals, like 0.17777..., into fractions is a fundamental skill in mathematics. This seemingly simple task unlocks a deeper understanding of number systems and provides a solid foundation for more advanced mathematical concepts. This comprehensive guide will not only show you how to convert 0.17777... into a fraction but also why the method works, exploring the underlying principles and addressing common questions.

Introduction: Decimals and Fractions – Two Sides of the Same Coin

Before we delve into the specifics of converting 0.17777..., let's establish a clear understanding of the relationship between decimals and fractions. Both represent parts of a whole. Fractions express this relationship using a numerator (top number) and a denominator (bottom number), while decimals use a base-ten system, with digits to the right of the decimal point representing tenths, hundredths, thousandths, and so on. The ability to convert between these two representations is crucial for various mathematical operations and problem-solving.

The number 0.17777... is a repeating decimal. The digit 7 repeats infinitely. This is often denoted as 0.17̅ or 0.17 recurring. Converting this type of decimal to a fraction requires a specific approach that leverages algebraic manipulation.

Step-by-Step Conversion of 0.17777... to a Fraction

Here's a step-by-step guide to converting the repeating decimal 0.17777... into a fraction:

1. Assign a Variable:

Let's represent the repeating decimal with a variable, say 'x':

x = 0.17777...

2. Multiply to Shift the Repeating Part:

We need to manipulate the equation to isolate the repeating part. Since the repeating part starts after the first two digits, we'll multiply both sides of the equation by 10:

10x = 1.77777...

3. Subtract to Eliminate the Repeating Part:

Now, we subtract the original equation (x = 0.17777...) from the equation we just created (10x = 1.77777...). Notice that this clever subtraction eliminates the repeating part:

10x - x = 1.77777... - 0.17777...

This simplifies to:

9x = 1.6

4. Solve for x:

Now, we can easily solve for x by dividing both sides of the equation by 9:

x = 1.6 / 9

5. Convert to a Proper Fraction:

The result is a decimal fraction (1.6/9). To convert this to a proper fraction, we multiply both the numerator and the denominator by 10 to remove the decimal point in the numerator:

x = (1.6 * 10) / (9 * 10) = 16/90

6. Simplify the Fraction:

Finally, we simplify the fraction by finding the greatest common divisor (GCD) of the numerator and denominator. The GCD of 16 and 90 is 2. Dividing both the numerator and the denominator by 2, we get the simplified fraction:

x = 8/45

Therefore, 0.17777... is equal to 8/45.

The Mathematical Explanation Behind the Method

The method we used relies on the properties of infinite geometric series. A repeating decimal can be expressed as the sum of an infinite geometric series. Let's break down why the subtraction step works:

0.17777... = 0.1 + 0.07 + 0.007 + 0.0007 + ...

This is a geometric series with the first term (a) = 0.07 and the common ratio (r) = 0.1. The sum of an infinite geometric series is given by the formula:

S = a / (1 - r), where |r| < 1

In our case:

S = 0.07 / (1 - 0.1) = 0.07 / 0.9 = 7/90

Adding the non-repeating part (0.1):

0.1 + 7/90 = 9/90 + 7/90 = 16/90 = 8/45

This confirms our result obtained through the step-by-step method. The subtraction technique effectively isolates the infinite geometric series, allowing us to use the summation formula to find the fractional equivalent.

Different Approaches for Converting Repeating Decimals

While the method outlined above is straightforward and effective, there are other approaches to convert repeating decimals to fractions. These alternative methods often involve slightly different algebraic manipulations but ultimately achieve the same result. One such approach could involve multiplying by powers of 10 until the repeating portion lines up perfectly for subtraction.

Frequently Asked Questions (FAQ)

Q1: What if the repeating part starts immediately after the decimal point?

A1: If the repeating part starts immediately after the decimal, the process is slightly simpler. For example, to convert 0.777... to a fraction, you would follow these steps:

1. Let x = 0.777...
2. Multiply by 10: 10x = 7.777...
3. Subtract: 10x - x = 7.777... - 0.777... = 7
4. Solve for x: 9x = 7 => x = 7/9

Q2: What if there's a non-repeating part before the repeating part?

A2: For decimals with a non-repeating part followed by a repeating part, you need to separate the non-repeating part and handle the repeating part using the method described above. Then, add the fractional equivalents of the non-repeating and repeating parts together.

Q3: Can all repeating decimals be converted to fractions?

A3: Yes, all repeating decimals can be expressed as fractions. This is a fundamental property of rational numbers (numbers that can be expressed as a fraction of two integers).

Q4: Are there any limitations to this method?

A4: The method is generally applicable to all repeating decimals. However, it's important to ensure accurate calculations during the subtraction and simplification steps. Errors in these steps can lead to an incorrect fractional representation.

Conclusion: Mastering the Conversion of Repeating Decimals

Converting repeating decimals to fractions is a valuable skill that reinforces your understanding of number systems and mathematical operations. The step-by-step approach outlined above, along with the underlying mathematical principles explained, provides a clear and comprehensive guide to successfully performing this conversion. Remember the key steps: assign a variable, multiply to shift the repeating part, subtract to eliminate the repeating part, solve for the variable, convert to a proper fraction, and finally simplify. By mastering this technique, you'll be well-equipped to tackle more complex mathematical problems and develop a deeper appreciation for the interconnectedness of different numerical representations. The ability to seamlessly move between decimal and fractional forms lays the foundation for future learning in algebra, calculus, and other advanced mathematical fields.