0.666666666666667 as a fraction

Charlotte

2 min read

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01-03-2025

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The decimal 0.666666666666667 is a fascinating number, especially when considering its fractional representation. While it appears to be endlessly repeating, the trailing '7' suggests a slight rounding. Let's explore how to convert this decimal into a fraction, addressing the subtle implication of that final digit.

Understanding Repeating Decimals

Before diving into the conversion, let's understand repeating decimals. A repeating decimal is a decimal number with a digit or a group of digits that repeat infinitely. The number 0.6666... (often written as 0.6) is a classic example of a repeating decimal. It's a rational number, meaning it can be expressed as a simple fraction.

Converting 0.666666666666667 to a Fraction

The presence of the '7' at the end complicates things slightly. This likely indicates rounding from a longer decimal representation or a calculation result. We have two approaches to consider:

Approach 1: Ignoring the trailing 7 (for a purely repeating decimal)

If we ignore the trailing 7 and assume the decimal is truly 0.6, the conversion is straightforward:

1. Let x = 0.6666... We're assigning a variable to the repeating decimal.

2. Multiply by 10: 10x = 6.6666...

3. Subtract the original equation: 10x - x = 6.6666... - 0.6666... This simplifies to 9x = 6.

4. Solve for x: x = 6/9.

5. Simplify the fraction: 6/9 simplifies to 2/3.

Therefore, if the decimal were truly 0.6, the fraction would be 2/3.

Approach 2: Accounting for the trailing 7 (the realistic approach)

The trailing 7 signifies that 0.666666666666667 is an approximation. To convert this approximation to a fraction, we can use the following method:

1. Consider the decimal as 0.666666666666667

2. Write it as a fraction: 666666666666667/1000000000000000

3. Simplify (This part requires a calculator or software): This fraction can be simplified, but doing so by hand is impractical. A calculator or specialized software will find the greatest common divisor and simplify it. The result will likely be a very large fraction which, when expressed as a decimal, will give you back the 0.666666666666667.

Conclusion

The most accurate representation of 0.666666666666667 as a fraction depends on the source of the decimal. If it’s a rounded version of the repeating decimal 0.6, then 2/3 is the best representation. However, if it’s a result of a calculation, the second approach (treating the 7 as significant) provides a more precise, albeit more complex, fractional representation. Remember that in the real world, many decimals are approximations, and their fractional equivalents may involve less elegant simplification.