Find the best tutors and institutes for Class 10 Tuition
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
Top Tutors who teach Number Systems
Super Tutor
Choolissery, Thrissur
20 yrs of
Exp
12 students
392 hrs of classes completed
600
per hour
Online Classes
I was a teacher at a CBSE-affiliated school. I have six years of experience teaching on online platforms. I have been conducting Mathematics, Physics, and Chemistry classes for high school and higher secondary students for more than 20 years. I believe I can encourage and motivate students to achieve excellent academic results while inspiring them to develop good values and habits.
Sini ma'am is an excellent maths teacher. I joined her classes at exactly the right time when I needed clarity, and she explained every concept in a very simple and structured way. Her teaching style made even difficult topics easy to understand, and my confidence in maths improved a lot in a short period. She is very patient, supportive, and genuinely cares about her students’ understanding. I’m really grateful for the guidance and clarity she gave me.
Super Tutor
Pandri, Raipur
4 yrs of
Exp
10 students
2689 hrs of classes completed
575
per hour
Online Classes
Hello, my name is Atulya Kumar. I have completed my studies at IIT(ISM)_Dhanbad and I am currently pursuing a PhD at NIT-Rourkela. With a degree in engineering, I am dedicated to teaching. Since 2019, I have been teaching physics to JEE, NEET, and Class XI & XII board students. I take great pride in sharing that my students have achieved exceptional results, with some clearing NEET 2022 with an AIR of 2513 and NEET 2023 with an AIR of 1702. Additionally, many of my students have scored above the 98th percentile in JEE Mains 2023. Teaching physics is a personal challenge for me. I strive to create a comfortable learning environment where students can fully grasp the concepts and apply them effectively during examinations, resulting in excellent marks and ranks.
Top Tutor
Electronic City Electronic City Phase II, Bangalore
5 yrs of
Exp
4 students
38 hrs of classes completed
150
per hour
Online Classes
Tutor's home
I was completed my m.Tech in engineering. I was maintained tuition center from 7 years in my home to all subjects from 1st grade to 10th grades across different boards including CBSE,ICSE,STATE BOARD (expert in math and physics) and I have 2 years experience in engineering college and i took classes for engineering students . My teaching style is create every student in to think and learn and convert every student in to passionate about their subjects. Over the years i helped to so many students to grown their career and overcome their fear about maths ,physics.
Top Tutor
Virugambakkam, Chennai
20 yrs of
Exp
36 students
1399 hrs of classes completed
500
per hour
Online Classes
I am R.Vinotha ., M.Sc.,M.Phil.,B.Ed in mathematics.I am the trainer/tutor of this course .I was allotted 12 regular classes for a month.I am using through google meet app for online class and also i am taking mathematics tuition at my home.I will create seperate WhatApp group for my tuition students and communicates to them .I will send the class notes to them after completing the class .My class planner is us following: Regular classes : Any 3 days for a week Class duration : 1 hour Home works: Daily write two problems and send it to be in WhatsApp group Test: Daily test : Formulae test: Everyday i am asking 10 formulae through voice message in WhatsApp group after getting my message ,the students will answer the questions and send it tome via tuition group Problems test:Daily write two problems Regular test : Every saturday and sunday Monthly test : Every month fourth saturday doubts clarification time : Every day 7 pm -8 pm I will correct the test papers immedietly and send it to their personal windows and also resolve the students mistakes. Chapter's quiz : After completing every chapters ,i will conduct mathematical quiz for that chapters Revisions: I will revise previous years questio papers After this type of my coachings many of my students got 100 marks in theire board examination .I put lots of efforts to my students and they getting good marks in the board examination .My teaching type is friendly manner.I gave extra solved questions and answers to my students .
Taking class as well as gives coaching in different approaches that can be dealt with the topic thro variety in problems and helps to reach solution with clarity of the nuances to be careful with.
Top Tutor
Bavdhan Vidnyan Nagar, Pune
5 yrs of
Exp
1 student
54 hrs of classes completed
250
per hour
Online Classes
Tutor's home
I have taught in an institution to state ,ICSE , CBSE kids. Extra-ordinary Results. Ample of Study Material will be provided. We can have a demo class . You would enjoy the interactive and interesting sessions.
Find more Tutor for Number Systems in your City
- Bangalore Mathematics Tutors
- Delhi Mathematics Tutors
- Chennai Mathematics Tutors
- Gurgaon Mathematics Tutors
- Noida Mathematics Tutors
- Hyderabad Mathematics Tutors
- Mumbai Mathematics Tutors
- Ghaziabad Mathematics Tutors
- Chandigarh Mathematics Tutors
- Pune Mathematics Tutors
- Jaipur Mathematics Tutors
- Surat Mathematics Tutors
