Find the best tutors and institutes for Class 10 Tuition
Q9(v):
Classify the following numbers as rational or irrational :
(v) $1.101001000100001...$
Solution :
Initial Setup & Theoretical Foundation
We are tasked with classifying the following real number:
$x = 1.101001000100001...$
To classify a real number as either rational ($\mathbb{Q}$) or irrational ($\mathbb{I}$) based on its decimal representation, we rely on the fundamental properties of real numbers:
- Rational Numbers ($\mathbb{Q}$): Possess decimal expansions that are either terminating (e.g., $0.25$) or non-terminating but repeating/recurring (e.g., $0.333...$). They can be expressed in the form $\frac{p}{q}$, where $p, q \in \mathbb{Z}$ and $q \neq 0$.
- Irrational Numbers ($\mathbb{I}$): Possess decimal expansions that are strictly non-terminating and non-repeating. They cannot be expressed as a ratio of two integers.
Step 1: Analyzing the Termination of the Decimal
Observe the given number $x = 1.101001000100001...$. The presence of the ellipsis ($...$) at the end of the number explicitly indicates that the decimal digits continue infinitely. [Per the standard mathematical notation for infinite sequences].
Therefore, the decimal expansion is definitively non-terminating.
Step 2: Analyzing the Recurrence (Repeating Pattern)
Next, we examine the sequence of digits after the decimal point to determine if there is a fixed, finite block of digits that repeats infinitely.
The sequence of fractional digits is: $101001000100001...$
Let us break down the structural pattern of zeros and ones:
- First segment: $10$ (contains exactly one zero)
- Second segment: $100$ (contains exactly two zeros)
- Third segment: $1000$ (contains exactly three zeros)
- Fourth segment: $10000$ (contains exactly four zeros)
Because the number of zeros between consecutive ones increases by exactly one each time, the sequence constantly changes its structure. [By the mathematical definition of periodicity, a sequence is periodic if $a_{n+p} = a_n$ for some constant period $p$. The strictly increasing run-length of zeros mathematically prevents any constant period $p$ from existing].
Therefore, the decimal expansion is non-repeating (non-recurring).
Step 3: Visualizing the Classification Framework
The following flowchart illustrates the logical path used to classify the given number based on its decimal expansion.
Step 4: Final Classification
Since the decimal expansion of $1.101001000100001...$ is proven to be both non-terminating and non-repeating, it cannot be expressed as a simple fraction $\frac{p}{q}$.
[Per the fundamental theorem of rational numbers, only numbers with terminating or repeating decimals belong to the set of rational numbers $\mathbb{Q}$. All other real numbers belong to the set of irrational numbers $\mathbb{I}$].
Final Solution: The number $1.101001000100001...$ is an irrational number.
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?
- 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) $7.478478...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
Top Tutors who teach Number Systems
Super Tutor
Kodambakkam, Chennai
15 yrs of
Exp
57 students
4041 hrs of classes completed
800
per hour
Online Classes
With a decade of experience in teaching mathematics, physics, and chemistry for students from grades 8 through 12 across CBSE, IB, and ICSE boards, I offer a comprehensive educational approach that caters to a diverse range of curricula and learning needs. My academic background includes a BE and an MBA, equipping me with both technical and managerial skills that enhance my teaching methodology. Throughout my career, I have specialized in home tutoring, where I have developed a personalized approach to education that focuses on each student's unique needs and strengths. My sessions are designed to be interactive and engaging, fostering a learning environment where students feel supported and motivated. I conduct both online and offline classes, each lasting one hour, and utilize a variety of teaching tools to facilitate learning. I employ a whiteboard to visually explain concepts and provide question papers to help students practice and prepare for their exams. This hands-on approach ensures that students not only understand theoretical concepts but also develop problem-solving skills and confidence in their subjects. My commitment to delivering high-quality education and my extensive experience make me well-suited to guide students through their academic journey, helping them achieve their full potential in mathematics, physics, and chemistry.
Shri Seshadri sir was able to break the jinx of my son not learning indefinitely, with his experience in smoothly sliding into a routine with such interactive, yet nonchalant transition for my son's education. Thankyou sir, thankyou UrbanPro.
Top Tutor
Talwandi Sector-B, Kota
10 yrs of
Exp
1 student
694 hrs of classes completed
1850
per hour
Online Classes
Tutor's home
My teaching exp.11 year in most reputed coaching in india. Mentor of jee top rank air-12 yatis agrawal, 19, 28 many more in top 100 in jee adavnced. Many of students got 100%tile in jee mains along with 100% marks in board exams. I also provide complete relevent study material like worksheet, practice sheet , dpps , assignment etc. Daily quizz based on assignment. Periodically test papers topic wise, minor and major as per shedule. I have helped thousands of students overcome their fear of maths.
Gaur City 2, Noida
5 yrs of
Exp
500
per hour
Online Classes
Student's Home
Tutor's home
I am a mathematics teacher I am giving home tution since 4 years I have done postgraduation in mathematics I am teaching in school last 4 years.
Top Tutor
Gharoli Gharoli, Delhi
16 yrs of
Exp
2 students
72 hrs of classes completed
400
per hour
Online Classes
I bring with me 17 years of diverse experience in the pharmaceutical industry along with academic expertise, combining strong practical knowledge with a passion for teaching. I am dedicated to simplifying complex concepts and delivering them in an engaging manner, ensuring that students not only gain subject knowledge but also develop the skills, confidence, and potential required to achieve academic excellence and become industry-ready professionals.
Explain very well and in detail friendly and honest teacher my overall experience is good recommend for tuition.
Top Tutor
Serlingampally, Hyderabad
20 yrs of
Exp
13 students
623 hrs of classes completed
400
per hour
Online Classes
Tutor's home
Teaching is my Passion. I have taught Mathematics, Physics and Chemistry to several students during 20+ years of my teaching. Many of my students who initially were scarred of Mathematics, have developed interest on the subject and opted MPC group in their Intermediate and are now earning Dollars as Software Engineers in USA, CANADA, UK and Australia.
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
