72 Repeating As A Fraction

Unraveling the Mystery of 72 Repeating as a Fraction: A Deep Dive into Decimal Conversions

Have you ever encountered the repeating decimal 0.727272...? This seemingly simple number holds a fascinating mathematical puzzle: how do we express this repeating decimal as a fraction? Understanding this process not only solves this specific problem but also unlocks a broader understanding of decimal-to-fraction conversions, particularly those involving repeating decimals. This article will guide you through the steps, explaining the underlying principles, and answering frequently asked questions. We'll explore various methods and demonstrate why this seemingly simple problem offers a surprisingly rich mathematical journey.

Understanding Repeating Decimals

Before diving into the conversion, it's crucial to understand what a repeating decimal is. A repeating decimal, also known as a recurring decimal, is a decimal representation of a number where one or more digits repeat infinitely. In our case, the digits "72" repeat endlessly. This signifies that the number is a rational number—a number that can be expressed as a fraction of two integers. This is a key concept; all repeating decimals are rational numbers. Irrational numbers, like pi (π) or the square root of 2, have decimal representations that neither terminate nor repeat.

Method 1: The Algebraic Approach

This method elegantly uses algebra to solve for the fraction. Let's represent our repeating decimal, 0.727272..., as the variable 'x':

x = 0.727272...

To eliminate the repeating part, we multiply both sides of the equation by 100 (because two digits repeat):

100x = 72.727272...

Now, subtract the original equation (x) from this new equation (100x):

100x - x = 72.727272... - 0.727272...

This simplifies to:

99x = 72

Now, solve for x by dividing both sides by 99:

x = 72/99

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 9:

x = 8/11

Therefore, the repeating decimal 0.727272... is equivalent to the fraction 8/11.

Method 2: The Geometric Series Approach

This method leverages the concept of geometric series. A geometric series is a series where each term is a constant multiple of the previous term. We can express 0.727272... as an infinite sum:

0.72 + 0.0072 + 0.000072 + ...

This is a geometric series with the first term (a) = 0.72 and the common ratio (r) = 0.01. The sum of an infinite geometric series is given by the formula:

Sum = a / (1 - r)

Substituting our values:

Sum = 0.72 / (1 - 0.01) = 0.72 / 0.99

To express this as a fraction, we can multiply both the numerator and denominator by 100 to remove the decimals:

Sum = 72/99

Again, simplifying this fraction by dividing by 9 gives us:

Sum = 8/11

This confirms our previous result: 0.727272... = 8/11.

Understanding the Underlying Principle

Both methods demonstrate the same fundamental principle: manipulating the repeating decimal to create an equation where the repeating part cancels out, leaving a simple algebraic equation to solve for the fractional representation. The key is to multiply the original equation by a power of 10 that shifts the repeating digits to align perfectly for subtraction. The power of 10 used depends on the number of digits that repeat. If three digits repeat, you'd multiply by 1000, and so on.

Dealing with More Complex Repeating Decimals

The techniques described above can be applied to more complex repeating decimals. For instance, consider the decimal 0.142857142857... where the sequence "142857" repeats. You would multiply the equation by 1,000,000 (10 to the power of 6, since six digits repeat) and then follow the same subtraction and simplification process.

Non-Repeating Decimals and Irrational Numbers

It's important to note that the methods described here only work for repeating decimals. Non-repeating decimals, which continue infinitely without a repeating pattern, cannot be expressed as simple fractions. These are often irrational numbers, such as π or √2. Their decimal representations go on forever without any repeating sequence.

Frequently Asked Questions (FAQ)

• Q: What if the repeating decimal starts after a non-repeating part?

A: For decimals like 0.272727..., you handle the non-repeating part separately. Let x = 0.272727... . Then, 100x = 27.272727..., and 100x - x = 27, giving 99x = 27, x = 27/99 = 3/11. Then add the non-repeating part: 3/11 + 2/10 = 3/11 + 1/5 = (15 + 11)/55 = 26/55

• Q: Can I use a calculator to convert a repeating decimal to a fraction?

A: Most calculators cannot directly handle infinitely repeating decimals. They provide approximations. The algebraic and geometric series methods are the most reliable ways to achieve the exact fractional representation.

• Q: Are there any limitations to these methods?

A: While these methods are generally effective, exceptionally long repeating sequences might lead to cumbersome calculations. However, the underlying principles remain the same.

• Q: Why is understanding this important?

A: Converting repeating decimals to fractions is a fundamental skill in mathematics, crucial for understanding rational numbers and their properties. It’s essential in various fields, from algebra and calculus to engineering and computer science.

Conclusion

Converting a repeating decimal like 0.727272... into a fraction might seem like a simple task, but it reveals the elegant interplay between decimal representation and rational numbers. Both the algebraic and geometric series methods provide powerful tools to solve this type of problem and demonstrate the underlying mathematical principles. Mastering these techniques allows you to tackle more complex repeating decimal conversions with confidence, expanding your understanding of fractions and their relationship to the seemingly endless world of decimal numbers. This seemingly simple exercise lays a strong foundation for more advanced mathematical concepts and problem-solving skills. Remember, practice is key to mastering this skill. So, grab a pen and paper, and try converting some other repeating decimals—you'll be surprised by how quickly you grasp the process!