.65 Repeating As A Fraction

Decoding the Mystery: 0.666... as a Fraction

Have you ever encountered the seemingly endless decimal 0.666...? It's a fascinating number that often leaves people wondering how to represent it as a fraction. This article delves into the mystery of repeating decimals, specifically 0.666..., explaining how to convert it to its fractional equivalent and exploring the underlying mathematical principles. We'll also tackle some common misconceptions and address frequently asked questions. Understanding this seemingly simple conversion unlocks a deeper appreciation for the elegant relationship between decimals and fractions.

Understanding Repeating Decimals

Before diving into the specifics of 0.666..., it's crucial to grasp the concept of repeating decimals. A repeating decimal is a decimal number where one or more digits repeat infinitely. These repeating digits are often indicated with a bar above them. For instance, 0.666... is often written as 0.$\overline{6}$, signifying that the digit 6 repeats indefinitely. Other examples include 0.333... (0.$\overline{3}$) and 0.142857142857... (0.$\overline{142857}$).

Converting 0.666... to a Fraction: The Algebraic Approach

The most straightforward method for converting 0.666... to a fraction involves using algebra. Let's follow these steps:

1. Assign a Variable: Let's represent the repeating decimal with a variable, say 'x':

   x = 0.666...

2. Multiply to Shift the Decimal: Multiply both sides of the equation by 10 to shift the decimal point one place to the right:

   10x = 6.666...

3. Subtract the Original Equation: Subtract the original equation (x = 0.666...) from the equation obtained in step 2:

   10x - x = 6.666... - 0.666...

4. Simplify: The repeating decimals cancel out, leaving:

   9x = 6

5. Solve for x: Divide both sides by 9:

   x = 6/9

6. Simplify the Fraction: Reduce the fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 3:

   x = 2/3

Therefore, 0.666... is equal to 2/3.

Geometric Series Approach: A Deeper Dive

The conversion of 0.666... to 2/3 can also be understood through the concept of an infinite geometric series. The decimal 0.666... can be written as:

0.6 + 0.06 + 0.006 + 0.0006 + ...

This is an infinite geometric series with the first term (a) = 0.6 and the common ratio (r) = 0.1. The formula for the sum of an infinite geometric series is:

Sum = a / (1 - r) (This formula is valid only when |r| < 1)

Substituting the values, we get:

Sum = 0.6 / (1 - 0.1) = 0.6 / 0.9 = 6/9 = 2/3

This confirms our previous result, demonstrating the elegant connection between repeating decimals and geometric series.

Addressing Common Misconceptions

Several misconceptions often surround repeating decimals. Let's clarify some of them:

• Rounding Error: It's incorrect to say that 0.666... is approximately 2/3. It's exactly 2/3. The repeating nature of the decimal implies infinite precision, eliminating any rounding error.

• Terminating Decimal: 0.666... is not a terminating decimal. A terminating decimal has a finite number of digits after the decimal point. Repeating decimals continue infinitely.

• Approximation: While you might use approximations like 0.67 or 0.667 in practical calculations, these are only approximations. They don't accurately represent the true value of 0.666..., which is precisely 2/3.

Generalizing the Conversion Process

The algebraic method we used for 0.666... can be generalized to convert any repeating decimal to a fraction. The key is to identify the repeating block of digits and manipulate the equation accordingly. For example, let's convert 0.121212... (0.$\overline{12}$) to a fraction:

1. Assign a variable: x = 0.121212...
2. Multiply to shift: 100x = 12.121212... (We multiply by 100 because there are two repeating digits)
3. Subtract: 100x - x = 12.121212... - 0.121212...
4. Simplify: 99x = 12
5. Solve: x = 12/99 = 4/33

Therefore, 0.121212... is equal to 4/33.

Beyond the Basics: Exploring Other Repeating Decimals

The techniques described above can be applied to a wide range of repeating decimals. The complexity might increase with longer repeating blocks or mixed repeating and non-repeating parts, but the underlying principles remain consistent. The key is careful manipulation of equations and a thorough understanding of the properties of infinite geometric series.

Frequently Asked Questions (FAQ)

Q: Why is 0.999... equal to 1?

A: This is a classic example of a repeating decimal. Using the same algebraic approach:

x = 0.999... 10x = 9.999... 10x - x = 9.999... - 0.999... 9x = 9 x = 1

Therefore, 0.999... is exactly equal to 1.

Q: Can all repeating decimals be converted to fractions?

A: Yes, all repeating decimals can be expressed as fractions. This is a fundamental property of the real number system.

Q: What if the repeating decimal has a non-repeating part?

A: For decimals with a non-repeating part followed by a repeating part, we can handle them similarly. For example, to convert 0.12$\overline{3}$ to a fraction:

1. x = 0.12333...
2. 100x = 12.333...
3. 1000x = 123.333...
4. 1000x - 100x = 111
5. 900x = 111
6. x = 111/900 = 37/300

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

A: You can convert a fraction to a decimal by performing long division. If the division results in a repeating pattern, you have a repeating decimal.

Q: Are there any limitations to this method?

A: While this method works effectively for most repeating decimals, the complexity might increase for exceptionally long repeating blocks. However, the underlying principles remain consistent.

Conclusion: Embracing the Elegance of Mathematics

Understanding the conversion of repeating decimals like 0.666... to fractions reveals the inherent elegance and interconnectedness within mathematics. The algebraic and geometric series approaches not only provide practical methods for conversion but also deepen our appreciation for the fundamental principles governing numbers. This understanding extends beyond simple conversions, providing a solid foundation for further exploration of mathematical concepts such as infinite series and number systems. The seemingly simple question of "what is 0.666... as a fraction?" opens a door to a deeper understanding of the fascinating world of mathematics.