NCERT Solutions for Class 9 Maths Exercise 1.3 Question 5
Understanding the Question 🧐
This question asks us to investigate the decimal expansion of the fraction &&\frac{1}{17}&&. It has two parts: first, we need to predict the maximum possible length of the repeating block of digits. Second, we have to perform the actual long division to find the repeating block and check if our prediction holds.
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.
Part 1: Predicting the Maximum Number of Digits 🤔
There’s a simple rule to determine the maximum length of the repeating block in a fraction’s decimal expansion.
For any fraction in the form &&\frac{p}{q}&& (where it’s in its simplest form), the maximum number of digits in the repeating block is &&q – 1&&.
Why this rule works: When you divide a number by &&q&&, the possible remainders you can get are &&0, 1, 2, …, q-1&&. If the remainder becomes 0, the decimal terminates. If it doesn’t, you are left with &&q-1&& possible non-zero remainders. The sequence of digits in the quotient will start repeating as soon as one of these remainders repeats. Since there are at most &&q-1&& different non-zero remainders, the repeating block can’t be longer than &&q-1&& digits.
For our fraction &&\frac{1}{17}&&, the value of &&q&& is 17.
Therefore, the maximum number of digits in the repeating block is:
&&17 – 1 = 16&&
Prediction: The repeating block of digits for &&\frac{1}{17}&& can have a maximum of 16 digits.
Part 2: Performing the Long Division to Verify 📝
Now, we must perform the long division of &&1 \div 17&& to find the actual length of the repeating block. This is a lengthy calculation that requires patience.
We start by dividing 1 by 17:
When we carry out the division, we get a sequence of digits in the quotient. We must continue until the remainder repeats. We started with a dividend of ‘1’. The division process continues until we get a remainder of ‘1’ again.
After a long and careful division, we find:
&&\frac{1}{17} = 0.0588235294117647…&&
At this point, after 16 division steps, the remainder becomes 1. Since we started with 1, the entire sequence of digits will now repeat.
So, the repeating block of digits is 0588235294117647.
Let’s count the number of digits in this block: There are 16 digits.
Conclusion and Key Points ✅
The maximum number of digits in the repeating block for &&\frac{1}{17}&& is 16. Our long division confirms that the actual length of the repeating block is indeed 16.
Therefore, &&\frac{1}{17} = 0.\overline{0588235294117647}&&.
- For a fraction &&\frac{1}{q}&&, the maximum length of the repeating decimal part is &&q – 1&&.
- The repetition in the decimal expansion begins as soon as a remainder from the division process is repeated.
- The set of possible non-zero remainders when dividing by &&q&& is &&\{1, 2, 3, …, q-1\}&&.
FAQ
Q: What is the rule for the maximum length of a repeating decimal block for a fraction 1/q?
A: For a fraction &&\frac{1}{q}&&, the maximum possible number of digits in the repeating block of its decimal expansion is &&q-1&&.
Q: Why is the maximum length q-1?
A: When dividing by a number &&q&&, there are &&q-1&& possible non-zero remainders (&&1, 2, …, q-1&&). If the remainder becomes 0, the decimal terminates. If not, the division process must repeat a remainder within &&q-1&& steps, which starts the repeating cycle of digits.
Q: What is the final decimal expansion of &&\frac{1}{17}&&?
A: The decimal expansion is &&0.\overline{0588235294117647}&&, where the entire block of 16 digits repeats itself.
Q: Is the length of the repeating block always the maximum possible (q-1)?
A: No, not always. For example, for the fraction &&\frac{1}{3}&&, the maximum possible length is &&3-1=2&&, but the actual length is just 1 (the digit ‘3’). The number &&\frac{1}{17}&& is a case where the length is the maximum possible.
Q: What would be the maximum number of repeating digits for the fraction &&\frac{1}{19}&&?
A: For the fraction &&\frac{1}{19}&&, the maximum number of digits in the repeating block would be &&19 – 1 = 18&&.
Further Reading
The study of repeating decimals is a fascinating part of number theory. To explore more, you can refer to the official NCERT textbooks and look into topics like “cyclic numbers” and “full reptend primes.”
