1. What is 0.142857 Repeating?

The repeating decimal 0.142857142857... (often written as 0.\overline{142857}) is exactly equal to the fraction 1/7. This is one of the classic examples of a repeating decimal with a cyclic pattern.

2. Conversion Method

To convert 0.\overline{142857} to a fraction:

The repeating block has 6 digits, so we multiply by 10^6 (1,000,000) to shift the decimal point.

3. Mathematical Proof

Verification: We can verify this by performing the division 1 ÷ 7:

7 goes into 1.000000... as 0.142857142857..., which exactly matches our repeating decimal.

4. Interesting Properties

Cyclic Number: 142857 is a cyclic number - its digits rotate when multiplied by numbers from 1 to 6:

• 1 × 142857 = 142857
• 2 × 142857 = 285714
• 3 × 142857 = 428571
• 4 × 142857 = 571428
• 5 × 142857 = 714285
• 6 × 142857 = 857142

5. Frequently Asked Questions (FAQ)

Q1: Why does 1/7 produce such a long repeating sequence?
A: Because 7 is a prime number that doesn't divide evenly into 10, 100, or any power of 10, resulting in a maximum length repeating decimal (6 digits for 1/7).

Q2: Are there other fractions with similar properties?
A: Yes, other fractions with denominator 7 produce cyclic permutations: 2/7 = 0.\overline{285714}, 3/7 = 0.\overline{428571}, etc.

Q3: How can I recognize such repeating decimals?
A: For prime denominators p (except 2 and 5), the decimal expansion of 1/p has a repeating cycle of length dividing p-1.

Q4: What's special about 142857?
A: It's the smallest cyclic number (excluding trivial cases) and appears in many interesting mathematical contexts.

Q5: Can this pattern be extended to larger numbers?
A: Yes, similar patterns exist for other primes, like 1/17 = 0.\overline{0588235294117647} with a 16-digit cycle.