0.666666666666667 as a fraction

Daniel Parker

2 min read

·

26-02-2025

1

0

Share

The decimal 0.666666666666667 might look complicated, but it represents a simple fraction. Understanding how to convert this repeating decimal into its fractional equivalent is a fundamental skill in mathematics. This article will guide you through the process, exploring the concept of repeating decimals and providing a clear, step-by-step solution. We'll also delve into why the slight variation from a perfectly repeating 6 (the extra 7 at the end) exists.

What are Repeating Decimals?

Repeating decimals, also known as recurring decimals, are decimals with a digit or group of digits that repeat infinitely. The number 0.666666... (where the 6s go on forever) is a perfect example. We often represent this using a bar over the repeating digits, like this: 0. $\overline{6}$. This notation clearly shows that the 6 repeats indefinitely.

Converting 0.666666666666667 to a Fraction

The number 0.666666666666667 isn't quite a perfectly repeating decimal due to rounding. Computers and calculators often round off repeating decimals, leading to the slight variation. To find the fractional equivalent, let's initially ignore the final 7 and work with 0.$\overline{6}$.

Here's how to convert 0.$\overline{6}$ to a fraction:

Step 1: Set up an equation.

Let x = 0.666666...

Step 2: Multiply by 10.

10x = 6.666666...

Step 3: Subtract the original equation from the multiplied equation.

10x - x = 6.666666... - 0.666666...

This simplifies to:

9x = 6

Step 4: Solve for x.

Divide both sides by 9:

x = 6/9

Step 5: Simplify the fraction.

Both 6 and 9 are divisible by 3:

x = 2/3

Therefore, 0.$\overline{6}$ is equal to 2/3.

The Significance of the Extra 7

The extra 7 in 0.666666666666667 is a result of rounding. The true value is infinitely close to 2/3 but is not exactly 2/3. This rounding occurs because computers and calculators have limited precision. They can’t store an infinite number of 6s. The 7 is an approximation to represent the remainder that would continue infinitely if exact precision were possible.

Practical Applications and Further Exploration

Understanding the conversion of repeating decimals to fractions is crucial in various fields, including:

• Engineering: Precise calculations often require fractional representations.
• Computer Science: Understanding floating-point arithmetic and its limitations.
• Finance: Calculating interest and other financial computations.

Exploring further into the realm of irrational numbers and their decimal representations provides a deeper understanding of mathematical concepts.

Conclusion

While 0.666666666666667 is a rounded representation, its fractional equivalent is best understood by focusing on the repeating decimal 0.$\overline{6}$, which simplifies neatly to the fraction 2/3. Remember that the slight deviation arises from the inherent limitations of representing an infinitely repeating decimal in a finite space. By understanding this concept, you gain a clearer understanding of the relationship between decimals and fractions.