You know that 1/7 = 0.142857. Can you predict what the decimal expansions of 2/7, 3/7, 4/7, 5/7, 6/7 are, without actually doing the long division? If so, how? [Hint: Study the remainders while finding the value of 1/7 carefully.]
Question
You know that $\frac{1}{7}=0 . \overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}$, $\frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how?
[Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
Answer
Yes, we can predict the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ without doing long division, by observing the pattern that emerges from $\frac{1}{7} = 0.\overline{142857}$. The key is in observing the repeating cycle of digits and how they are expected to shift based on multiplication.
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Understanding $\frac{1}{7}=0.\overline{142857}$:
- The repeating cycle is 142857.
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Predicting Decimal Expansions:
- For $\frac{1}{7}$, the cycle starts with 142857.
- For $\frac{2}{7}$, multiply the pattern by 2: Since $2 \times 7 = 14$, and knowing $7 \times 2$ doesn’t exceed the base 10 significantly, we expect the pattern to shift by one position but start with the second digit of the original cycle and cycle through: $0.\overline{285714}$.
- For $\frac{3}{7}$, similar logic applies: we start with the third digit of the original cycle: $0.\overline{428571}$.
- For $\frac{4}{7}$, we start with the fourth digit of the original cycle: $0.\overline{571428}$.
- For $\frac{5}{7}$, we start with the fifth digit of the original cycle: $0.\overline{714285}$.
- For $\frac{6}{7}$, we start with the sixth digit of the original cycle: $0.\overline{857142}$.
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Explanation:
- The pattern shift occurs due to how the multiples relate to the base 10 and the multiplicative nature of the pattern within the fraction. By understanding the pattern in $\frac{1}{7}$, we can predict the patterns for subsequent fractions up to $\frac{6}{7}$ by cyclically shifting the digits based on the numerator.
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