Find the best tutors and institutes for Class 10 Tuition
Q7:
Write three numbers whose decimal expansions are non-terminating non-recurring.
Solution :
Theoretical Foundation: Decimal Expansions of Real Numbers
In the real number system, numbers are broadly classified into two categories based on their decimal expansions: rational numbers and irrational numbers. [Per the fundamental classification of Real Numbers], a rational number's decimal expansion either terminates (ends) or is non-terminating but recurring (repeats a specific block of digits infinitely). Conversely, an irrational number cannot be expressed as a simple fraction $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$). Consequently, the defining characteristic of an irrational number is that its decimal expansion is strictly non-terminating and non-recurring.
To satisfy the prompt, we must provide three distinct examples of irrational numbers. We will derive these using three different mathematical approaches: pattern construction, algebraic roots, and transcendental constants.
Step 1: Constructing a Pattern-Based Irrational Number
The most direct way to generate a non-terminating, non-recurring decimal is to construct a sequence of digits that follows a predictable, yet non-repeating pattern. We can achieve this by systematically increasing the number of zeros between a repeating non-zero digit.
- Let the first decimal place be $0$.
- Follow it with a $1$, then one $0$ ($0.01\dots$).
- Follow it with a $1$, then two $0$s ($0.01001\dots$).
- Follow it with a $1$, then three $0$s ($0.010010001\dots$).
Number 1: $0.01001000100001\dots$
[Justification: Because the block of zeros expands by one with each iteration, there is no fixed finite block of digits that will ever repeat infinitely. Thus, it is non-recurring. The ellipsis ($\dots$) indicates it is non-terminating.]
Step 2: Utilizing Square Roots of Non-Perfect Squares
In algebra, the square root of any positive integer that is not a perfect square (such as $2, 3, 5, 6, 7$) is inherently an irrational number. [Per the theorem of irrationality of roots: If $p$ is a prime number, then $\sqrt{p}$ is irrational].
Let us take the square root of the first prime number, $2$.
Number 2: $\sqrt{2} = 1.41421356237309504\dots$
[Justification: The decimal expansion of $\sqrt{2}$ has been mathematically proven to extend infinitely without any repeating sequence of digits.]
Step 3: Referencing Transcendental Mathematical Constants
Certain fundamental constants in mathematics cannot be expressed as the root of any non-zero polynomial equation with rational coefficients. These are known as transcendental numbers, which are a subset of irrational numbers. The most famous example is Pi ($\pi$), which represents the ratio of a circle's circumference to its diameter.
Number 3: $\pi = 3.141592653589793238\dots$
[Justification: Johann Lambert proved in 1761 that $\pi$ is irrational. Therefore, its decimal representation goes on forever without settling into a permanently repeating pattern.]
Final Solution: Three distinct numbers whose decimal expansions are non-terminating and non-recurring are:
1) $0.01001000100001\dots$ (A constructed non-repeating pattern)
2) $\sqrt{2} = 1.41421356\dots$ (The square root of a non-perfect square)
3) $\pi = 3.14159265\dots$ (A transcendental mathematical constant)
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.
- 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?
- 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
Top Tutor
Thane West, Thane
20 yrs of
Exp
37 students
5605 hrs of classes completed
350
per hour
Online Classes
This is Prita Chatterjee with 27 years of teaching in the Engineering College and Schools .I teach Maths for all the boards - IB AA(SL& HL) ,IB - AI(HL&SL),Integrated Maths ,MYP ,Cambridge A Level , Additional Maths, IGCSE O Level, ISC, ICSE , CBSE .Presently, I am tutoring students from U.S. France, Austrailia , Netherland , Japan, Singapore, Malaysia , Thailand , Phillipines , Dubai , Riyadh , Doha, and pan India . I would love to teach from basics . .If you would like me to teach your child , kindly let me know ,so we can schedule a demo ,as per suitability.
She is a good teacher with very well and skilled explanations and making sure that students do each and every sum from text book.Hoping that she teaches out of the book&concepts too.:) my only concern is I can’t expect the reply for my messages as quick.
Top Tutor
Karampura, Delhi
10 yrs of
Exp
2500
per hour
Online Classes
Student's Home
I am an IIM Kashipur alumnus, passionate about guiding students to achieve their immediate and long-term goals. As a personal tutor and founder of the online educational startup ‘MinDely,’ I have gained deep insights into child psychology and student needs, allowing me to tailor my teaching methods accordingly. I design lesson plans based on each student's grasp of concepts, ensuring a student-friendly and easy-to-understand approach. My weekly tests provide valuable insights for parents to track their child’s progress. I take pride in the success of my students, many of whom have graduated from IITs, DTU, and NSIT.
Super Tutor
Sector 5, Gurgaon
14 yrs of
Exp
29 students
2313 hrs of classes completed
450
per hour
Online Classes
Tutor's home
Mathematics is not just a subject—it is the backbone of science and the foundation for logical thinking and problem-solving. With over eleven years of teaching experience and a B.Tech degree, I am dedicated to nurturing students' mathematical abilities while instilling confidence and a love for learning. Recognitions & Achievements ✅ UrbanPro Excellency Awards – Top 10 in Class 10 Tuition (2024) & Top 100 Tutors for 2024 ✅ Teacher Innovation Award by ZIIEI (Sri Aurobindo Society) ✅ Certified by Leading Ed-Tech Platforms – UrbanPro Certification, GuruQ Certification These accolades reflect my commitment to providing high-quality education and delivering exceptional learning outcomes for my students. Why Choose Me as Your Child’s Tutor? As an educator, my primary goal is to empower students with a strong conceptual foundation while ensuring they enjoy the learning process. I believe that every child has unique strengths and learning needs, and my approach is tailored to bring out the best in them. 📌 Comprehensive Academic Support: I provide in-depth coaching in Mathematics, Science, and English for Grades V–X and specialize in Mathematics for Grades XI–XII and Undergraduate Courses. 📌 Expert in Competitive & Entrance Exam Preparation: I have successfully guided students in securing admissions to prestigious institutions and excelling in competitive exams, including: 🎯 School Entrance Exams – Sainik School, Military School, Navodaya Vidyalaya, Gurukul Entrance, Central Hindu School, Rai Sports School 🎯 Competitive & Government Job Exams – Olympiads, RIMC, NTSE, SSC MTS, CHSL, DSSSB 📌 Proven Track Record of Student Success: My students have consistently demonstrated remarkable improvement in grades and conceptual understanding. I maintain an individualized learning plan for each student, ensuring they receive personalized guidance and continuous support. Regular assessments, progress reports, and feedback sessions help track improvement and address learning gaps effectively. My Teaching Methodology & Approach I strongly believe that learning should be an engaging and interactive experience rather than rote memorization. My experiential learning approach is designed to: ✔ Encourage critical thinking and problem-solving skills ✔ Foster conceptual clarity through real-life applications ✔ Enhance creativity and independent learning To achieve this, I incorporate modern teaching tools and methodologies, such as: 🖥 Live Interactive Sessions – Engaging discussions to clarify doubts in real time 📊 PowerPoint Presentations & Animations – To visualize complex concepts easily 🖋 Whiteboard Teaching – Step-by-step breakdown of mathematical problems 📚 Comprehensive Study Materials, Worksheets & Assignments – Reinforcing learning through practice Online & Offline Classes: I conduct online classes via digital platforms and in-person sessions at my institute in Gurgaon. For students requiring additional support, I offer home tuition when feasible. Commitment to Student Success 🌟 Personalized Attention: I ensure that every student gets the individual focus they need to overcome challenges and master concepts. 🌟 Supportive Learning Environment: My classes are structured to be patient, encouraging, and motivating, allowing students to ask questions freely and develop confidence in their abilities. 🌟 Building Lifelong Skills: Beyond academics, I emphasize the development of problem-solving abilities, logical reasoning, and exam strategies that benefit students in their future careers. 🌟 Parental Involvement & Regular Feedback: I believe in open communication with parents to discuss their child’s progress, strengths, and areas for improvement. Regular performance reviews ensure that parents stay informed about their child's academic growth. Invest in Your Child’s Future Today! With a proven track record, structured teaching approach, and a passion for education, I am committed to helping students achieve excellence in Mathematics and beyond. If you are looking for a dedicated, award-winning tutor who can provide high-quality education, personalized guidance, and measurable results, I would be delighted to support your child’s learning journey. 📩 Contact me today to discuss how I can help your child excel!
Good techniques and enjoyed the class. Loved the way he explained things and gave tricks to save time as well.
Top Tutor
Tilak Nagar, Delhi
4 yrs of
Exp
2 students
56 hrs of classes completed
300
per hour
Online Classes
Student's Home
Tutor's home
I am a mathematics expert . I am teaching this subject since a decade . I ve handled thousands of students and bring numerous centum results in past . You will able to learn maths in the most pragmatic approach with me and develop a love for this subject .
Top Tutor
MSR nagar mathikere, Bangalore
20 yrs of
Exp
21 students
5675 hrs of classes completed
499
per hour
Online Classes
I enjoy teaching and making things easier and fun for the children. Didn't send my kids for any tuitions. Today the elder child is a CSE graduate , working for Amazon & the other is in her 3rd year MBBS !
I have been studying Maths and Science under her guidance since 8th grade. She explains every concept in a simple and clear way, making even difficult topics easy to understand. Her teaching style is patient and supportive, and she always makes sure all doubts are cleared. She focuses on strengthening basics and gives regular practice, which has really helped me improve. Because of her guidance, my confidence in both subjects has increased a lot. I am truly grateful for her dedication and support
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
