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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.
Solution :
Theoretical Foundation: Maximum Digits in a Repeating Block
To determine the maximum number of digits in the repeating block of the decimal expansion of a rational number $\frac{p}{q}$ (where $p$ and $q$ are coprime integers), we analyze the possible remainders generated during the long division process.
[Per the Division Algorithm], when dividing by a divisor $q$, the only possible non-zero remainders are the integers from $1$ to $q - 1$. If a remainder of $0$ is reached, the decimal terminates. If the decimal is non-terminating and repeating, the sequence of remainders must eventually repeat. Because there are exactly $q - 1$ possible non-zero remainders, the division process can generate at most $q - 1$ distinct remainders before a previous remainder reappears, forcing the sequence of quotient digits to cycle.
For the fraction $\frac{1}{17}$:
- The divisor is $q = 17$.
- The maximum possible number of distinct non-zero remainders is $17 - 1 = 16$.
Therefore, the maximum number of digits in the repeating block of $\frac{1}{17}$ is strictly bounded by 16.
Step-by-Step Long Division Execution
To verify this theoretical maximum, we perform the long division of $1 \div 17$. We append zeros to the dividend and track the quotient digits and remainders at each step [applying $a = bq + r$].
| Step | Current Dividend | Division Operation | Quotient Digit | Remainder |
|---|---|---|---|---|
| 1 | 10 | $10 = 17 \times 0 + 10$ | 0 | 10 |
| 2 | 100 | $100 = 17 \times 5 + 15$ | 5 | 15 |
| 3 | 150 | $150 = 17 \times 8 + 14$ | 8 | 14 |
| 4 | 140 | $140 = 17 \times 8 + 4$ | 8 | 4 |
| 5 | 40 | $40 = 17 \times 2 + 6$ | 2 | 6 |
| 6 | 60 | $60 = 17 \times 3 + 9$ | 3 | 9 |
| 7 | 90 | $90 = 17 \times 5 + 5$ | 5 | 5 |
| 8 | 50 | $50 = 17 \times 2 + 16$ | 2 | 16 |
| 9 | 160 | $160 = 17 \times 9 + 7$ | 9 | 7 |
| 10 | 70 | $70 = 17 \times 4 + 2$ | 4 | 2 |
| 11 | 20 | $20 = 17 \times 1 + 3$ | 1 | 3 |
| 12 | 30 | $30 = 17 \times 1 + 13$ | 1 | 13 |
| 13 | 130 | $130 = 17 \times 7 + 11$ | 7 | 11 |
| 14 | 110 | $110 = 17 \times 6 + 8$ | 6 | 8 |
| 15 | 80 | $80 = 17 \times 4 + 12$ | 4 | 12 |
| 16 | 120 | $120 = 17 \times 7 + 1$ | 7 | 1 |
At Step 16, we obtain a remainder of $1$. This is the exact same value we started with (the numerator of $\frac{1}{17}$). Because the remainder has repeated, the sequence of quotient digits will now repeat infinitely in the exact same order.
Visualizing the Cyclic Nature of Remainders
The following diagram illustrates the complete cycle of the 16 distinct remainders generated during the division process. The cycle proves that the repeating block has reached its maximum theoretical length.
Final Decimal Expansion
Collecting the quotient digits from Step 1 through Step 16, we construct the repeating decimal block:
$ \frac{1}{17} = 0.\overline{0588235294117647} $
Counting the digits under the vinculum (overline), we find exactly 16 digits.
Final Solution: The maximum number of digits in the repeating block of the decimal expansion of $\frac{1}{17}$ is 16. Performing the long division confirms that the repeating block is $0588235294117647$, which contains exactly 16 digits.
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) $\frac{329}{400}$
- 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.]
- 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.
- 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) $1.101001000100001...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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