Find the best tutors and institutes for Class 10 Tuition
Q1(iii):
Classify the following numbers as rational or irrational:
(iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$
Solution :
Initial Setup & Given Expression
We are tasked with classifying the following mathematical expression as either a rational or an irrational number:
$\frac{2\sqrt{7}}{7\sqrt{7}}$
Step 1: Algebraic Simplification
Before classifying any real number, it must be reduced to its simplest form. We begin by analyzing the factors in both the numerator and the denominator.
- Numerator: $2\sqrt{7}$ (which is the product of the integer $2$ and the irrational number $\sqrt{7}$)
- Denominator: $7\sqrt{7}$ (which is the product of the integer $7$ and the irrational number $\sqrt{7}$)
[Per the fundamental properties of fractions and real numbers, any non-zero common factor present in both the numerator and the denominator can be divided out without changing the value of the expression]. Since $\sqrt{7} \approx 2.645$ and strictly $\sqrt{7} \neq 0$, we can safely cancel this common factor:
$\frac{2\sqrt{7}}{7\sqrt{7}} = \frac{2 \cdot \sqrt{7}}{7 \cdot \sqrt{7}} = \frac{2}{7} \cdot \frac{\sqrt{7}}{\sqrt{7}}$
$\frac{2}{7} \cdot 1 = \frac{2}{7}$
Step 2: Theoretical Classification
We must now evaluate the simplified expression, $\frac{2}{7}$, against the formal axioms of the real number system.
[By definition, a rational number ($\mathbb{Q}$) is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$), and the denominator $q$ is not equal to zero ($q \neq 0$)].
Analyzing our simplified fraction $\frac{2}{7}$:
- $p = 2$, which is an integer ($2 \in \mathbb{Z}$).
- $q = 7$, which is an integer ($7 \in \mathbb{Z}$) and strictly non-zero ($7 \neq 0$).
Because the simplified form perfectly satisfies the necessary and sufficient conditions of a rational number, the original expression is inherently rational.
Visual Representation of the Number System Classification
Final Solution: The given expression $\frac{2\sqrt{7}}{7\sqrt{7}}$ simplifies algebraically to $\frac{2}{7}$. Because it can be expressed as the ratio of two integers where the denominator is not zero, it is a rational number.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.4
- Q1(i): Classify the following numbers as rational or irrational: (i) $2 - \sqrt{5}$
- Q1(ii): Classify the following numbers as rational or irrational: (ii) $(3 + \sqrt{23}) - \sqrt{23}$
- Q1(iv): Classify the following numbers as rational or irrational: (iv) $\frac{1}{\sqrt{2}}$
- Q1(v): Classify the following numbers as rational or irrational: (v) $2\pi$
- Q2(i): Simplify each of the following expressions: (i) $(3 + \sqrt{3})(2 + \sqrt{2})$
- Q2(ii): Simplify each of the following expressions: (ii) $(3 + \sqrt{3})(3 - \sqrt{3})$
- Q2(iii): Simplify each of the following expressions: (iii) $(\sqrt{5} + \sqrt{2})^2$
- Q2(iv): Simplify each of the following expressions: (iv) $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})$
- Q3: Recall, $\pi$ is defined as the ratio of the circumference (say $c$) of a circle to its diameter (say $d$). That is, $\pi = \frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?
- Q4: Represent $\sqrt{9.3}$ on the number line.
- Q5(i): Rationalise the denominators of the following: (i) $\frac{1}{\sqrt{7}}$
- Q5(ii): Rationalise the denominators of the following: (ii) $\frac{1}{\sqrt{7} - \sqrt{6}}$
- Q5(iii): Rationalise the denominators of the following: (iii) $\frac{1}{\sqrt{5} + \sqrt{2}}$
- Q5(iv): Rationalise the denominators of the following: (iv) $\frac{1}{\sqrt{7} - 2}$
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
