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Number Systems Class 9 Questions And Answers are provided here. These NCERT solutions are prepared by expert team at Careers360 according to latest CBSE syllabus 2023-24. Apart from school exams, the solutions are also beneficial if you are aiming for exams like- National Talent Search Examination (NTSE), Indian National Olympiad (INO), SSC, CAT, etc. The Chapter starts with an introduction which includes rational numbers, whole numbers, and integers followed by important topics like irrational numbers, real numbers, how to represent real numbers on the number line, operations on real numbers, and many more.
With the understanding of rational numbers, we will also study how to represent square roots of 2, 3, 5, and other non-rational numbers. NCERT solutions for Class 9 Maths chapter 1 Number Systems are written keeping important basics in mind so that a student can get 100% out of it. students can find all the NCERT solutions for class 9 here. practice these to command the concepts.
Any unique real number can be represented on a number line.
If r = rational number and s = irrational number
Then (r + s), (r – s), (r × s), and (r ⁄ s) all are irrational.
Rules for positive real numbers:
√ab = √a × √b
√(a/b) = √a/√b
(√a + √b) × (√a – √b) = a−b
(a + √b) × (a − √b) = a2 − b
(√a+√b)2 = a2 + 2√ab + b
To rationalise the denominator of 1/√(a + b), one must multiply it by √(a – b) / √(a – b), where a and b are both integers.
Suppose a = real number (greater than 0) and p and q are rational numbers:
ap × bp = (ab)(p)
(ap)q = apq
ap / aq = (a)(p-q)
ap / bp = (a/b)p
Free download NCERT Solutions for Class 9 Maths Chapter 1 Number Systems for CBSE Exam.
Number systems class 9 questions and answers - Exercise 1.1
Q1 Is zero a rational number? Can you write it in the form , where p and q are integers and q ≠ 0?
Answer:
Any number that can represent in the form of is a rational number
Now, we can write 0 in the form of for eg. etc.
Therefore, 0 is a rational number.
Q2 Find six rational numbers between 3 and 4.
Answer:
There are an infinite number of rational numbers between 3 and 4. one way to take them is
Therefore, six rational numbers between 3 and 4 are
Q3 Find five rational numbers between and .
Answer:
We can write
And
Therefore, five rational numbers between and . are
Answer:
(i) TRUE
The number that is starting from 1, i.e 1, 2, 3, 4, 5, 6, .................. are natural numbers
The number that is starting from 0. i.e, 0, 1, 2, 3, 4, 5.............are whole numbers
Therefore, we can clearly see that the collection of whole numbers contains all natural numbers.
Answer:
(ii) FALSE
Because integers may be negative or positive but whole numbers are always positive. for eg. -1 is an integer but not a whole number.
Answer:
(iii) FALSE
Numbers that can be represented in the form of are a rational number.
And numbers that are starting from 0 i.e. 0,1,2,3,4,......... are whole numbers
Therefore, we can clearly see that every rational number is not a whole number for eg. is a rational number but not a whole number
Class 9 maths chapter 1 question answer - Exercise 1.2
Answer:
(i) TRUE
Since the real numbers are the collection of all rational and irrational numbers.
Answer:
(ii) FALSE
Because negative numbers are also present on the number line and no negative number can be the square root of any natural number
Answer:
(iii) FALSE
As real numbers include both rational and irrational numbers. Therefore, every real number cannot be an irrational number.
For eg. 4 is a real number but not an irrational number
Answer:
NO, Square root of all positive integers is not irrational. For the eg square of 4 is 2 which is a rational number.
Q3 Show how can be represented on the number line.
Answer:
We know that
Now,
Let OA be a line of length 2 unit on the number line. Now, construct AB of unit length perpendicular to OA. and join OB.
Now, in right angle triangle OAB, by Pythagoras theorem
Now, take O as centre and OB as radius, draw an arc intersecting number line at C. Point C represent on a number line.
Class 9 maths chapter 1 question answer - Exercise: 1.3
Q1 (i) Write the following in decimal form and say what kind of decimal expansion each has : (i)
Answer:
We can write as
Since the decimal expansion ends after a finite number of steps. Hence, it is terminating
Q1 (ii) Write the following in decimal form and say what kind of decimal expansion each has : (ii)
Answer:
We can rewrite as
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.
Q1 (iii) Write the following in decimal form and say what kind of decimal expansion each has : (iii)
Answer:
We can rewrite as
Since the decimal expansion ends after a finite number. Therefore, it is terminating
Q1 (iv) Write the following in decimal form and say what kind of decimal expansion each has : (iv)
Answer:
We can rewrite as
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.
Q1 (v) Write the following in decimal form and say what kind of decimal expansion each has: (v)
Answer:
We can rewrite as
Since decimal expansion repeats itself so it is a non-terminating recurring decimal expansion.
Q1 (vi) Write the following in decimal form and say what kind of decimal expansion each has : (vi)
Answer:
We can rewrite as
Since decimal expansion ends after finite no. of figures. Hence, it is terminating.
Answer:
It is given that
Therefore,
Similarly,
Q3 (i) Express the following in the form , where p and q are integers and q ≠ 0. (i)
Answer:
Let -(i)
Now, multiply by 10 on both sides
Therefore, form of is
Q3 (ii) Express the following in the form , where p and q are integers and q ≠ 0. (ii)
Answer:
We can write as
-(i)
Now,
Let -(ii)
Now, multiply by 10 on both sides
Now, put the value of x in equation (i). we will get
Therefore, form of is
Q3 (iii) Express the following in the form , where p and q are integers and q ≠ 0. (iii)
Answer:
Let -(i)
Now, multiply by 1000 on both sides
Therefore, form of is
Q4 Express 0.99999 .... in the form . Are you surprised by your answer?
Answer:
Let -(i)
Now, multiply by 10 on both sides
Therefore, form of is 1
The difference between 1 and 0.999999 is o.000001 which is almost negligible.
Therefore, 0.999 is too much closer to 1. Hence, we can write 0.999999.... as 1
Answer:
We can rewrite as
Therefore, there are total 16 number of digits be in the repeating block of digits in the decimal expansion of
Answer:
We can observe that when q is 2, 4, 5, 8, 10… then the decimal expansion is terminating. For example:
, denominator
, denominator
, denominator
Therefore,
It can be observed that the terminating decimal can be obtained in a condition where prime factorization of the denominator of the given fractions has the power of 2 only or 5 only or both.
Q7 Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer:
Three numbers whose decimal expansions are non-terminating non-recurring are
1) 0.02002000200002......
2) 0.15115111511115.......
3) 0.27227222722227.......
Q8 Find three different irrational numbers between the rational numbers and .
Answer:
We can write as
And as
Therefore, three different irrational numbers between the rational numbers and are
1) 0.72737475....
2) 0.750760770780...
3) 0.790780770760....
Q9 (i) Classify the following numbers as rational or irrational :
Answer:
We can rewrite in decimal form as
Now, as the decimal expansion of this number is non-terminating non-recurring.
Therefore, it is an irrational number.
Q9 (ii) Classify the following numbers as rational or irrational :
Answer:
We can rewrite as
We can clearly see that it is a rational number because we can represent it in form
Q9 (iii) Classify the following numbers as rational or irrational : 0.3796
Answer:
We can rewrite 0.3796 as
Now, we can clearly see that it is a rational number as the decimal expansion of this number is terminating and we can also write it in form.
Q9 (iv) Classify the following numbers as rational or irrational : 7.478478....
Answer:
We can rewrite 7.478478.... as
Now, as the decimal expansion of this number is non-terminating recurring. Therefore, it is a rational number.
Q9 (v) Classify the following numbers as rational or irrational : 1.101001000100001...
Answer:
In the case of number 1.101001000100001...
As the decimal expansion of this number is non-terminating non-repeating. Therefore, it is an irrational number.
Class 9 maths chapter 1 NCERT solutions - Exercise: 1.4
Q1 Visualise 3.765 on the number line, using successive magnification.
Answer:
3.765 can be visualised as in the following steps.
First, we draw a number line and mark points on it after that we will divide the number line between points 3 and 4. And then we will divide the points between 3.7 and 3.8 as the number is between both of them.
Q2 Visualise on the number line, up to 4 decimal places.
Answer:
We can rewrite as
Now, 4.2626 can be visualised as in the following steps.
Class 9 maths chapter 1 NCERT solutions - Exercise: 1.5
Q1 (i) Classify the following numbers as rational or irrational:
Answer:
Value of is 2.23606798....
Now,
Now,
Since the number is in non-terminating non-recurring. Therefore, it is an irrational number.
Q1 (ii) Classify the following numbers as rational or irrational:
Answer:
Given number is
Now, it is clearly a rational number because we can represent it in the form of
Q1 (iii) Classify the following numbers as rational or irrational:
Answer:
Given number is
As we can clearly see that it can be represented in form. Therefore, it is a rational number.
Q1 (iv) Classify the following numbers as rational or irrational:
Answer:
Given number is
Now,
Clearly, as the decimal expansion of this expression is non-terminating and non-recurring. Therefore, it is an irrational number.
Q1 (v) Classify the following numbers as rational or irrational:
Answer:
Given number is
We know that the value of
Now,
Now,
Clearly, as the decimal expansion of this expression is non-terminating and non-recurring. Therefore, it is an irrational number.
Q2 (i) Simplify each of the following expressions:
Answer:
Given number is
Now, we will reduce it into
Therefore, answer is
Q2 (ii) Simplify each of the following expressions:
Answer:
Given number is
Now, we will reduce it into
Therefore, answer is 6
Q2 (iii) Simplify each of the following expressions:
Answer:
Given number is
Now, we will reduce it into
Therefore, the answer is
Q2 (iv) Simplify each of the following expressions:
Answer:
Given number is
Now, we will reduce it into
Therefore, the answer is 3 .
Answer:
There is no contradiction.
When we measure a length with scale or any other instrument, we only obtain an approximate rational value. We never obtain an exact value.
For this reason, we cannot say that either c or d is irrational.
Therefore, the fraction is irrational. Hence, the value of is approximately equal to
Therefore, is irrational.
Q4 Represent on the number line.
Answer:
Draw a line segment OB of 9.3 unit. Then, extend it to C so that BC = 1 unit. Find the mid-point D of OC and draw a semi-circle on OC while taking D as its centre and OD as the radius. Now, Draw a perpendicular to line OC passing through point B and intersecting the semi-circle at E. Now, Take B as the centre and BE as radius, draw an arc intersecting the number line at F. the length BF is units.
Q5 (i) Rationalise the denominators of the following:
Answer:
Given number is
Now, on rationalisation, we will get
Therefore, the answer is
Q5 (ii) Rationalise the denominators of the following:
Answer:
Given number is
Now, on rationalisation, we will get
Therefore, the answer is
Q5 (iii) Rationalise the denominators of the following:
Answer:
Given number is
Now, on rationalisation, we will get
Therefore, the answer is
Q5 (iv) Rationalise the denominators of the following:
Answer:
Given number is
Now, on rationalisation, we will get
Therefore, answer is
Class 9 number systems ncert solutions - Exercise: 1.6
Q1 (i) Find :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, answer is 8
Q1 (ii) Find :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is 2
Q1 (iii) Find :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is 5
Q2 (i) Find :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is 27
Q2 (ii) Find :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is 4
Q2 (iii) Find :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is 8
Q3 (i) Simplify :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is
Q3 (ii) Simplify :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is
Q3 (iii) Simplify :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is
Q3 (iv) Simplify :
Answer:
Given number is
Now, on simplifying it we will get
Therefore, the answer is
Also Read :
Interested students can practice class 9 maths ch 1 question answer using the following exercises.
Chapter No. | Chapter Name |
Chapter 1 | Number Systems |
Chapter 2 | |
Chapter 3 | |
Chapter 4 | |
Chapter 5 | |
Chapter 6 | |
Chapter 7 | |
Chapter 8 | |
Chapter 9 | |
Chapter 10 | |
Chapter 11 | |
Chapter 12 | |
Chapter 13 | |
Chapter 14 | |
Chapter 15 |
How to Use NCERT Solutions for Class 9 Maths Chapter 1 Number Systems?
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The basic concept of the number system, rational numbers, whole numbers, and integers, irrational numbers, real numbers, representation of real numbers on the number line, operations on real numbers are the important topics of this chapter. Students can practice math 9th class chapter 1 solutions to get in-depth understanding of concepts.
Most of the students consider geometry especially triangles as the most difficult chapter in the CBSE Class 9 Maths. To solve more problems students can also refer to NCERT exemplar questions.
No, CBSE doesn’t provide NCERT solutions for any class or subject. To get a good score in CBSE exams students can follow NCERT syllabus and book.
Here, students can get detailed NCERT solutions for Class 9 Maths by clicking on the link. practicing these real numbers class 9 solutions are important as these provide confidence to students which ultimately lead to high score.
NCERT solutions are helpful for the students if they stuck while solving NCERT problems. Also, these solutions are provided in a very detailed manner which will give them conceptual clarity.
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For publishing books, newspapers, magazines and digital material, editorial and commercial strategies are set by publishers. Individuals in publishing career paths make choices about the markets their businesses will reach and the type of content that their audience will be served. Individuals in book publisher careers collaborate with editorial staff, designers, authors, and freelance contributors who develop and manage the creation of content.
Careers in journalism are filled with excitement as well as responsibilities. One cannot afford to miss out on the details. As it is the small details that provide insights into a story. Depending on those insights a journalist goes about writing a news article. A journalism career can be stressful at times but if you are someone who is passionate about it then it is the right choice for you. If you want to know more about the media field and journalist career then continue reading this article.
Individuals in the editor career path is an unsung hero of the news industry who polishes the language of the news stories provided by stringers, reporters, copywriters and content writers and also news agencies. Individuals who opt for a career as an editor make it more persuasive, concise and clear for readers. In this article, we will discuss the details of the editor's career path such as how to become an editor in India, editor salary in India and editor skills and qualities.
Individuals who opt for a career as a reporter may often be at work on national holidays and festivities. He or she pitches various story ideas and covers news stories in risky situations. Students can pursue a BMC (Bachelor of Mass Communication), B.M.M. (Bachelor of Mass Media), or MAJMC (MA in Journalism and Mass Communication) to become a reporter. While we sit at home reporters travel to locations to collect information that carries a news value.
Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.
A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.
Welding Engineer Job Description: A Welding Engineer work involves managing welding projects and supervising welding teams. He or she is responsible for reviewing welding procedures, processes and documentation. A career as Welding Engineer involves conducting failure analyses and causes on welding issues.
A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.
A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.
A Structural Engineer designs buildings, bridges, and other related structures. He or she analyzes the structures and makes sure the structures are strong enough to be used by the people. A career as a Structural Engineer requires working in the construction process. It comes under the civil engineering discipline. A Structure Engineer creates structural models with the help of computer-aided design software.
The Process Development Engineers design, implement, manufacture, mine, and other production systems using technical knowledge and expertise in the industry. They use computer modeling software to test technologies and machinery. An individual who is opting career as Process Development Engineer is responsible for developing cost-effective and efficient processes. They also monitor the production process and ensure it functions smoothly and efficiently.
An AWS Solution Architect is someone who specializes in developing and implementing cloud computing systems. He or she has a good understanding of the various aspects of cloud computing and can confidently deploy and manage their systems. He or she troubleshoots the issues and evaluates the risk from the third party.
An Azure Administrator is a professional responsible for implementing, monitoring, and maintaining Azure Solutions. He or she manages cloud infrastructure service instances and various cloud servers as well as sets up public and private cloud systems.
Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.
A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.
Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack
An Automation Test Engineer job involves executing automated test scripts. He or she identifies the project’s problems and troubleshoots them. The role involves documenting the defect using management tools. He or she works with the application team in order to resolve any issues arising during the testing process.
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