NCERT Solutions Class 9 Maths Chapter 1 Number Systems
Number System is the first chapter of Class 9 Maths NCERT Book. It covers various topics, including an introduction to number systems, irrational numbers, real numbers and their decimal expansions, operations on real numbers, and the laws of exponents for real numbers and their applications.
The chapter begins with an introduction to Number Systems in section 1.1, followed by significant topics in sections 1.2, 1.3, 1.4, and 1.5.
1.0Download Class 9 Science Chapter 1 NCERT Solutions PDF Online
ALLEN'S Experts lucidly curated the solutions to improve the students' problem-solving abilities. For a more precise idea about Number Systems NCERT solutions, students can download the below NCERT Solutions for Class 9 Maths chapter 1 pdf solution.
NCERT Solutions Class 9 Maths Chapter 1 - Number Systems
2.0NCERT Solutions Class 9 Maths Chapter 1 : Important Topics
Irrational Numbers: Numbers that can't be written as p/q.
Real Numbers and their Decimal Expansions: This section includes the decimal expansions of real numbers and how to distinguish between rational and irrational numbers.
Number Line: Representing Real Numbers on the Number Line.
Operations on Real Numbers: Here, you explore operations like addition, subtraction, multiplication and division on irrational numbers.
Laws of Exponents for Real Numbers: Use these laws of exponents to solve the questions.
3.0Class 9 Maths Chapter 1 Number Systems: Exercise Solutions
There are five exercises in chapter 1 (Number systems) of class 9 Maths. Students can find the split below:
Class 9 Maths Chapter 1 Exercises
No. Of Questions
Exercise 1.1
4
Exercise 1.2
3
Exercise 1.3
9
Exercise 1.4
5
Exercise 1.5
3
Total
24
Explore Number Systems and learn how to solve various problems only on NCERT Solutions For Class 9 Maths.
4.0Advantages of Class 9 Maths Chapter 1 NCERT Solutions – Number Systems
1. Foundation for Advanced Math: It forms the basis for understanding more complex
mathematical concepts in higher classes, including algebra, calculus, and trigonometry.
2. Real-Life Applications: Helps in solving real-world problems like calculating areas and volumes and understanding financial transactions.
3. Understanding Different Types of Numbers: Students learn about rational, irrational, and real numbers, enabling them to classify and use numbers effectively in various mathematical problems.
4. Preparation for Competitive Exams: A strong grasp of number systems is crucial for excelling in competitive exams like NTSE, Olympiads, and even entrance exams like JEE and NEET.
5. Building a Strong Mathematical Base: It provides a solid foundation for other topics in mathematics, making it easier to understand future concepts and chapters.
By mastering this chapter, students not only excel academically but also develop skills that are useful in everyday life and future studies.
5.0NCERT Questions with Solutions for Class 9 Maths Chapter 1 - Detailed Solutions
Exercise: 1.1
Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q=0 ?
Sol. Yes, zero is a rational number. We can write zero in the form p/q where p and q are integers and q=0.
So, 0 can be written as 10=20=30 etc.
Find six rational numbers between 3 and 4 .
Sol. First rational number between 3 and 4 is
=23+4=27
Similarly, other numbers are
23+27=41323+413=82523+825=164923+1649=329723297+3=64193
So, numbers are 27,413,825,1649,3297,64193
Find five rational numbers between 3/5 and 4/5.
Sol. Let
53n+1(n+1)=53×66=301854n+1(n+1)=54×66=3024
So, required rational numbers are
3019,3020,3021,3022,3023
State whether the following statements are true or false? Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Sol. (i) True, the collection of whole numbers contains all natural numbers.
(ii) False, -2 is not a whole number.
(iii) False, 21 is an integer but a rational number but not a whole number.
Exercise : 1.2
State whether the following statements are true or false? Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form m, where m is a natural number.
(iii) Every real number is an irrational number.
Sol. (i) True, since collection of real numbers consists of rationals and irrationals.
(ii) False, because no negative number can be the square root of any natural number.
(iii) False, 2 is real but not irrational.
Are the square roots of all positive integer's irrational? If not, give an example of the square root of a number that is a rational number.
Sol. No, 4=2 is a rational number.
Show how 5 can be represented on the number line.
Sol. 5 on Number line.
OABC is unit square.
So, OB=12+12=2OD=(2)2+1=3OE=(3)2+1=2OF=(2)2+1=5
Using compass we can cut arc with centre O and radius =OF on number line. ON is required result.
Exercise : 1.3
Write the following in decimal form and say what kind of decimal expansion each has :
(i) 10036
(ii) 111
(iii) 481
(iv) 133
(v) 112
(vi) 400329
Sol. (i) 10036=0.36
(Terminating)
(ii) 111=0.090909.....
(Non-Terminating Repeating)
11 1 1.00000 0.090909 .... 100−9910099199
(iii) 481=833=4.125
(Terminating decimal)
(iv) 133=0.230769230769......
=0.230769
(Non-Terminating repeating)
(v) 112=0.1818…….
=0.18 (Non-Terminating repeating)
(vi) 400329
You know that 71=0.142857. Can you predict what the decimal expansions of 72,73,74,75,76 are, without actually doing the long division? If so, how?
Sol. Yes, we can predict decimal explain without actually doing long division method as
72=2×71=2×0.142857=0.28571473=3×71=3×0.142857=0.42857174=4×71=4×0.142857=0.57142875=5×71=5×0.142857=0.71428576=6×71=6×0.142857=0.857142
Express the following in the form p/q, where p and q are integers and q=0.
(i) 0.6
(ii) 0.47
(iii) 0.001
Sol. (i) Let x=0.6666....
Multiplying both the sides by 10 10x=6.666.
Subtract (1) from (2)
10x−x=(6.6666....)−.(0.6666…...
⇒9x=6⇒x=96=32
(ii) Let x=0.47=.4777…
Multiply both sides by 10
10x=4.7
Multiply both sides by 10
100x=47.7
Subtract (1) from (2)
90x=43x=9043
(iii) Let x=0.001=0.001001001
Multiply both sides by 1000
1000x=1.001
Subtract (1) from (2)
999x=1x=9991
Express 0.99999..... in the form qp. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
Sol. Let x=0.999....
Multiply both sides by 10 we get
10x=9.99…...
Subtract (1) from (2)
9x=9⇒x=10.9999…=1=11∴p=1,q=1
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 171 ? Perform the division to check your answer.
Sol. Maximum number of digits in the repeating block of digits in decimal expansion of 171 can be 16 .
Look at several examples of rational numbers in the form p/q(q=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?
Sol. There is a property that q must satisfy rational number of form qp(q=0) where p,q are integers with no common factors other than 1 having terminating decimal representation (expansions) is that the prime factorisation of q has only powers of 2 or powers of 5 or both (i.e., q must be of the form 2m×5n ). Here m,n are whole numbers.
Write three numbers whose decimal expansions are non-terminating nonrecurring.
Sol. 0.01001000100001...
0.202002000200002...
0.003000300003...
Find three different irrational numbers between the rational numbers 75 and 119.
Sol. 75.000000 (0.714285...
Thus, 75=0.714285
Thus, 119=0.81
Three different irrational numbers between 75 and 119 are taken as
0.750750075000750000...
0.780780078000780000...
0.80800800080000800000...
Classify the following numbers as rational or irrational :
(i) 23
(ii) 225
(iii) 0.3796
(iv) 7.478478 ......
(v) 1.101001000100001......
Sol. (i) 23= Irrational number
(ii) 225=15= Rational number
(iii) 0.3796
decimal expansion is terminating
⇒.3796= Rational number
(iv) 7.478478...
=7.478 which is non-terminating recurring.
= Rational number
(v) 1.101001000100001.....
decimal expansion is non
terminating and non-repeating.
= Irrational number
Exercise: 1.4
Visualise 3.765 on the number line, using successive magnification.
Sol. n=3.765
Visualise 4.26 on the number line, up to 4 decimal places.
Sol. n=4.26
So, n=4.2626 (upto 4 decimal places)
Exercise 1.5
Classify the following numbers as rational or irrational :
(i) 2−5
(ii) (3+23)−23
(iii) 7727
(iv) 21
(v) 2π
Sol. (i) 2 is a rational number and 5 is an irrational number.
∴2−5 is an irrational number.
(ii) (3+23)−23⇒(3+23)−23=3 is a rational number.
(iii) 7727=72 is a rational number.
(iv) 21∵1 is a rational number and 2 is an irrational number.
So, is an irrational number.
(v) 2π∵2 is a rational number and π is an irrational number.
So, 2π is an irrational number.
Simplify each of the following expressions :
(i) (3+3)(2+2)
(ii) (3+3)(3−3)
(iii) (5+2)2
(iv) (5−2)(5+2)
Sol. (i) (3+3)(2+2)=3(2+2)+3(2+2)=6+32+23+6
(ii) (3+3)(3−3)=(3)2−(3)2=9−3=6
(iii) (5+2)2=(5)2+210+(2)2=7+210
(iv) (5−2)(5+2)=5−2=3
Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, π=c/d. This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Sol. There is no contradiction. When we measure a length with a scale or any other device, we only get an approximate rational value. Therefore, we may not realise that c is irrational.
Represent 9.3 on the number line.
Sol.
Let ℓ be the number line.
Draw a line segment AB=9.3 units and BC
=1 unit. Find the mid point 0 of AC.
Draw a semicircle with centre 0 and radius OA or OC.
Draw BD⊥AC intersecting the semicircle at D . Then, BD=9.3 units. Now, with centre B and radius BD , draw an arc intersecting the number line l at P.
Hence, BD=BP=9.3
Rationalise the denominators of the following :
(i) 71
(ii) 7−61
(iii) 5+21
(iv) 7−21
Sol. (i) 71=71×77=77
(ii) 7−61=7−61×7+67+6=7−67+6=17+6=7+6
(iii) 5+215+21×5−25−2=35−2
(iv) 7−21=7−21×7+27+2=7−47+2=37+2
Exercise : 1.6
Find:
(i) (64)1/2
(ii) 321/5
(iii) 1251/3
Sol. (i) (64)1/2=(82)1/2=(82×21)=81=8
(ii) 321/5=(25)1/5=(25×51)=21=2
(iii) (125)1/3=(53)1/3=53×31=5
Find:
(i) 93/2
(ii) 322/5
(iii) 163/4
(iv) 125−1/3
Sol. (i) 923=(921)3=(3)3=27
(ii) 3252=(25)52=25×52=22=4
(iii) 163/4=(24)3/4=23=8
(iv) 125−1/3=(53)−1/3=5−1=1/5
Simplify :
(i) 22/3⋅21/5
(ii) (331)7
(iii) 111/4111/2
(iv) 71/2⋅81/2
Sol. (i) 232⋅251=232+51=21510+3=21513
(ii) (331)7=(33)717=3211=3−21
(iii) 11411121=1121−41=1141=411
(iv) 721.821=(7×8)1/2=(56)1/2
NCERT Solutions for Class 9 Maths Other Chapters:-
Should I download NCERT Solutions for Class 9 Maths Chapter 1?
ALLEN's experts provide step-by-step answers to NCERT Solutions for Class 9 Maths Chapter 1. This helps the students learn all the concepts in detail and clear their doubts. Regular practice also helps them score high on math exams.
What are the core topics to study in class 9 maths chapter 1 NCERT?
In Chapter 1, Number Systems of Class 9 Maths, students will study: 1. Natural Numbers, Whole Numbers, Integers, Rational Numbers. 2. Irrational Numbers. 3. Real Numbers and their Decimal Expansions. 4. Representing Real Numbers on the Number Line. 5. Operations on Real Numbers. 6. Laws of Exponents for Real Numbers.
Is chapter 1 of class 9th Maths difficult to solve?
Chapter 1 of class 9th Maths is not easy and simple. It lies in the middle of easy and difficult because some examples and questions in this chapter are easy, and some are difficult. However, the difficulty level of anything varies from student to student. Some students find it difficult, some find it easy, and some find it easy and difficult.