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Q2:
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.]
Solution :
Given Variables & Initial Setup
We are given the decimal expansion of the rational number $\frac{1}{7}$:
$\frac{1}{7} = 0.\overline{142857}$
The bar over the digits $142857$ indicates that this block of six digits repeats infinitely. We are tasked with predicting the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7},$ and $\frac{6}{7}$ without performing long division, utilizing the properties of the remainders generated during the division of $1$ by $7$.
Step 1: Understanding the Cyclic Property of $\frac{1}{7}$
The number $142857$ is a well-known cyclic number. [Per number theory, a cyclic number is an integer in which cyclic permutations of the digits are produced when multiplied by successive integers]. Because $7$ is a prime number and the length of the repeating decimal block is $7 - 1 = 6$, the decimal expansions for any fraction $\frac{k}{7}$ (where $1 \le k \le 6$) will consist of the exact same six digits in the exact same cyclic order. The only difference is the starting digit.
Step 2: Analyzing the Remainders (The Hint)
When we perform the long division of $1 \div 7$, we append a zero to the remainder at each step to continue the division. The sequence of dividends and their corresponding quotients dictates the repeating block:
- $10 \div 7 = 1$ (Remainder $3$) → First digit is $1$
- $30 \div 7 = 4$ (Remainder $2$) → Second digit is $4$
- $20 \div 7 = 2$ (Remainder $6$) → Third digit is $2$
- $60 \div 7 = 8$ (Remainder $4$) → Fourth digit is $8$
- $40 \div 7 = 5$ (Remainder $5$) → Fifth digit is $5$
- $50 \div 7 = 7$ (Remainder $1$) → Sixth digit is $7$
[By the Division Algorithm], to find the decimal expansion of $\frac{k}{7}$, we simply look at the first step of dividing $k$ by $7$ (i.e., $10k \div 7$). The quotient gives us the starting digit of the cycle, and the rest of the digits follow the established cyclic order: $1 \rightarrow 4 \rightarrow 2 \rightarrow 8 \rightarrow 5 \rightarrow 7 \rightarrow 1$.
Step 3: Predicting the Decimal Expansions
Using the logic established in Step 2, we can predict the expansions through simple scalar multiplication or by identifying the starting digit:
| Fraction | Initial Division ($10k \div 7$) | Starting Digit | Cyclic Sequence Prediction |
|---|---|---|---|
| $\frac{2}{7}$ | $20 \div 7 = 2$ (Remainder $6$) | $2$ | $0.\overline{285714}$ |
| $\frac{3}{7}$ | $30 \div 7 = 4$ (Remainder $2$) | $4$ | $0.\overline{428571}$ |
| $\frac{4}{7}$ | $40 \div 7 = 5$ (Remainder $5$) | $5$ | $0.\overline{571428}$ |
| $\frac{5}{7}$ | $50 \div 7 = 7$ (Remainder $1$) | $7$ | $0.\overline{714285}$ |
| $\frac{6}{7}$ | $60 \div 7 = 8$ (Remainder $4$) | $8$ | $0.\overline{857142}$ |
Alternatively, this can be verified algebraically by multiplying the original repeating decimal by the respective numerators. For example:
$\frac{2}{7} = 2 \times \frac{1}{7} = 2 \times 0.\overline{142857} = 0.\overline{285714}$.
Because there are no carry-over digits that disrupt the cycle, the permutation holds perfectly true for all scalar multiples up to $6$.
Final Solution: Yes, the decimal expansions can be predicted without long division by utilizing the cyclic permutation of the digits $142857$. The predicted expansions are:
$\frac{2}{7} = 0.\overline{285714}$
$\frac{3}{7} = 0.\overline{428571}$
$\frac{4}{7} = 0.\overline{571428}$
$\frac{5}{7} = 0.\overline{714285}$
$\frac{6}{7} = 0.\overline{857142}$
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- Q1(i): Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$
- Q1(ii): Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- Q1(iv): Write the following in decimal form and say what kind of decimal expansion each has : (iv) $\frac{3}{13}$
- Q1(v): Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$
- Q1(vi): Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$
- Q3(i): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$
- Q3(ii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (ii) $0.4\overline{7}$
- Q3(iii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (iii) $0.\overline{001}$
- Q4: Express $0.99999 ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
- Q5: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.
- Q6: Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q \neq 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
- Q7: Write three numbers whose decimal expansions are non-terminating non-recurring.
- Q8: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
- Q9(i): Classify the following numbers as rational or irrational : (i) $\sqrt{23}$
- Q9(ii): Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$
- Q9(iii): Classify the following numbers as rational or irrational : (iii) $0.3796$
- Q9(iv): Classify the following numbers as rational or irrational : (iv) $7.478478...$
- Q9(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
