8.571428571428571 As A Fraction

Repeating Decimal to Fraction Conversion:

\[ 8.\overline{571428} = \frac{60}{7} \]

Fraction Result:

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1. What is 8.571428... as a Fraction?

The repeating decimal 8.571428571428... is exactly equal to the fraction 60/7. This is a repeating decimal where "571428" repeats infinitely.

2. Conversion Method

To convert 8.571428... to a fraction:

\[ x = 8.\overline{571428} \] \[ 1000000x = 8571428.\overline{571428} \] \[ 999999x = 8571428 - 8 = 8571420 \] \[ x = \frac{8571420}{999999} = \frac{60}{7} \]

The fraction simplifies to 60/7 after dividing numerator and denominator by 142857.

3. Mathematical Proof

Verification: When you divide 60 by 7, you get exactly 8.571428571428..., confirming the conversion.

4. Using the Calculator

Tips: The calculator is pre-set for this specific conversion. For other repeating decimals, additional logic would be needed.

5. Frequently Asked Questions (FAQ)

Q1: Why does 60/7 produce this repeating decimal?
A: Because 7 doesn't divide evenly into any power of 10, creating a repeating pattern in base 10.

Q2: How long is the repeating sequence?
A: For 1/7, the sequence is 6 digits long (142857). For 60/7, it's the same sequence shifted (571428).

Q3: Can all repeating decimals be converted to fractions?
A: Yes, all repeating decimals represent exact fractions of integers.

Q4: What's special about 1/7 and its multiples?
A: They produce cyclic numbers where the digits rotate in a specific order.

Q5: How accurate is this conversion?
A: Mathematically exact - the fraction and repeating decimal are two representations of the same number.