9

CBSE

MATHEMATICS Vineeta Rawat MSc, LT HOD Mathematics Queen Mary’s School, Model Town Delhi

Full Marks Pvt Ltd (Progressive Educational Publishers)

New Delhi-110002

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Preface Mathematics-9 is based on the latest curriculum guidelines specified by the CBSE. It will certainly prove to be a torch-bearer for those who toil hard to achieve their goal. This All-in-one practice material has been developed keeping in mind all the requirement of the students for Board Examinations preparations like learning, practicing, revising and assessing. Salient Features of the Book: ●● Each chapter is designed in ‘Topic wise’ manner where each topic is briefly explained with sufficient Examples and Exercise. Exercise which covers Objective Type Questions and all the possible variety of Questions. ●● Answers with hints are provided separately after the exercise. ●● Importance of Each Topic and Frequently Asked Types of Questions Reference provides an idea to the students on which type they should focus more. ●● Assignment is provided at the end of each chapter. ●● CBSE & NCERT Questions have been covered in every chapter. ●● 10 Unsolved Sample Papers for mock test are given with solutions for self assessment. ●● Common Errors by the students are provided to make students aware what errors are usually done unknowingly. ●● The book has been well prepared to build confidence in students. uggestions for further improvement of the book, pointing out printing errors/mistakes which might S have crept in spite of all efforts, will be thankfully received and incorporated in the next edition. –Author

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Mathematics 9

GLIMPSE OF A CHAPTER 1

. Exercise related to each topic dealt separately and Questions included segregated into 1 Mark, 2 Marks, 3 Marks and 4 Marks Questions.

Number Systems

Topics covered 1. Basics of Number Systems and Rational Numbers

2. Irrational Numbers and Real Numbers

3. Decimal Expansions of Real Numbers

4. Visual Representations of Real Numbers on Number Line through Successive Magnifications

5. Operations on Real Numbers

6. Laws of Exponents for Real Numbers

Exercise 1.2 I. Objective Type Questions 1. Choose the correct option in the following (MCQs): (i) Which of the following is an irrational number?

12 (a) (b) (d) 81 (c) 7 13 (ii) The product of any two irrational numbers is (NCERT Exemplar) (a) always an irrational number (b) always a irrational number (c) always an integer (d) sometimes rational, sometime irrational (iii) Which of the following is not a rational number? (NCERT Exemplar) 4 (b) 5 (c) (d) 1 (a) 4 9 4

CHAPTER MAP Number Systems Rational Numbers (Q)

Real Numbers

Whole Numbers (W) (0, 1, 2, 3, 4 ...)

Natural Numbers (N) (Counting Numbers 1, 2, 3, .....)

Integers (Z)

(iv) Final which one is an irrational number? (a)

Negative Integers (–1, –2, –3, ....)

Zero (0)

Decimal Expansions of Real Numbers

(1 Mark) (NCERT Exemplar)

Visual Representations of Real Numbers on Number Line

(b)

2 8

(NCERT Exemplar) (c)

9 25

(v) Which one is different from other?

Positive Integers (1, 2, 3, 4, ...)

Operations on Real Numbers

1 5

(d)

4 9

(NCERT Exemplar)

(a) 2 (b) 1 (c) 3 (d) 10 2. Fill in the Blanks: (i) The sum of two irrational numbers is ..................... number. (ii) The difference of a rational and ..................... number is always an irrational number. (iii) A ..................... point on the number line can be found representing a real number. (iv) The reciprocal of every (non-zero) rational number is a ..................... number. (v) The product or quotient of a non-zero rational number with an irrational number is ...................... . 3. True or False (i) The product of a rational number and an irrational number is an irrational number. (ii) The sum of two irrational numbers is an irrational number. (iii) Negative of an irrational number is an irrational number. (iv) The square roots of all positive integers are irrational numbers. (NCERT) II. Very Short Answer Type Questions (1 Mark) 4. Give an example in which the sum of a rational number and an irrational number is an irrational number. III. Short Answer Type Questions-I (2 Marks) 5. Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer. [CBSE 2010, 2011] IV. Short Answer Type Questions-II (3 Marks) 6. Let x be rational and y be irrational. Is xy necessarily an irrational? Justify your answer by an example. [CBSE 2011] V. Long Answer Type Questions (4 Marks)

Laws of Exponents for Real Numbers

.E ach chapter is divided into topics and explained separately. .Chapter map representation of the chapter.

7. Construct 10 on number line. Write the steps of construction 8. Represent following irrational numbers on the number line:

(NCERT Exemplar) [Imp.] 2 , 3 , 5 , 7 , 11, 17

[Imp.]

5. operatioNs oN real NumBers In previous classes, you have already studied that rational numbers are closed under addition, subtraction, multiplication and division, i.e., if we add, subtract, multiply or divide (except by zero) two rational numbers, we still get a rational number. Also for irrational numbers, the sum, difference, product or quotient is not necessarily be an irrational number. Let us see the following examples. Example 1. Check whether 2 + 3 is rational or irrational. Solution. 2 is a rational number and 3 is an irrational number. 3 = 1.7320508..., as decimal expansion is non-terminating and non-recurring. 2 + 3 = 2 + 1.7320508... = 3.7320508..., which is also non-terminating and non-recurring

\

Hence, 2 + 3 is an irrational number. Example 2. Find out whether the following numbers are rational or irrational: 2 2 12 (i ) 2 + 2 (ii ) 3 − 2 3 + 2 (iii ) 2 3

(

(

)

(

Solution. (i ) 2 + 2

(

)(

)

2

)(

)

( )

2

= (2) 2 + 2 + 2 × 2 2 = 4 + 2 + 4 2 = 6 + 4 2 (rational + irrational) = an irrational number.

)

( )

2

2 (ii) 3 − 2 3 + 2 = (3) − 2 = 9 − 2 = 7 (a rational number) 12 6 2×3 = = = 2 3 = (rational × irrational) = an irrational number (iii) 2 3 3 3

Example 3. Add 2 2 + 5 3 and 2 − 3 3

(

) (

Solution. 2 2 + 5 3 +

) (

) (

)

[NCERT]

2 −3 3 = 2 2 + 2 + 5 3−3 3 =3 2 +2 3

Example 4. Multiply 3 5 by 4 5 . Solution. 3 5 × 4 5 = 3 × 4 × 5 × 5 = 12 × 5 = 60 Example 5. Divide 8 15 by 2 5 . Solution. 8 15 ÷ 2 5 =

8× 5 × 3 2 5

. Topic wise concepts are presented to remember them easily.

=4 3

(iv)

. Each topic is well explained with relevant example for better understanding.

. Quite effective for a quick revision before exams. Have the complete essence of the chapter. COMMON ERRORS

REVISION CHART

ERRORS

Real Numbers A collection of all the rational and irrational numbers are called real numbers.

CORRECTIONS

(i) Incorrectly assuming that all fractions are rationals.

(i) A rational number can be written as a fraction but all the fractions are not rationals. For example,

10 2

is a fraction but not a rational.

(ii) Incorrectly categorising a root as irrational when an exact root exists.

(ii) 2 , 3, 5 , ... are irrationals but 4 , 9 , 16 , ... etc. are rationals 4 p 3 p     as 4 = 2  p form , 9 =  form , 16 =  form ,...   1q  1 q 1 q etc.

(iii) Incorrectly recognising all the numbers having non-terminating decimals as irrationals.

(iii) A number whose decimal expansion is terminating or non terminating repeating (recurring) is rational while a number whose decimal expansion is non-terminating, non-repeating (non-recurring) is irrational.

(iv) Incorrectly recognising that the product of a rational number and an irrational number is an irrational always.

(iv) It is not necessarily true. For example, 0 is a rational number and 2 is an irrational number but their product, i.e., 0 × 2 = 0 is also a rational number.

(v) Incorrectly raising a power to a power when solving an exponential equation.

(v) Do not confuse in the rules for the power of a power property, the power of a product property and the products of powers property. Learn and understand all of them very well.

(vii) Incorrectly interpreting repeating decimal notation.

Irrational Numbers A number s is called an irrational number, if p it cannot be written in the form , where p q

are integers, and q ≠ 0.

and q are integers and q ≠ 0.

The decimal expansion of a rational number is either terminating or non-terminating repeating (recurring).

The decimal expansion of an irrational number is non-terminating non-repeating (non-recurring).

Representation of Real Numbers on the Number Line There is a unique real number corresponding to every point on the number line. Also, corresponding to each real number, there is a unique point on the number line.

Operations on Real Numbers If r is a rational number and s is an irrational number, then l r + s is irrational l r – s is irrational r is irrational l rs is irrational l s where s ≠ 0

Laws of Exponents for Real Numbers If a, b are positive real numbers and m, n are rational numbers, then

(vi) The student assumes an expression, say (5 + 5 ) (5 − 5 ) is irrational because there is an irrational number in each parenthesis but it is not correct. When we remove parenthesis, we get 2 (5)2 − ( 5 ) = 25 − 5 = 20 which is a rational number.

(vi) Incorrectly recognising expressions involving radicals as irrationals.

Rational Numbers A number r is called a rational number, if it p can be written in the form , where p and q q

l

l

m

n

a ×a =a

l

(am)n = amn

l

l

ambm = (ab)m

l

(a m ) n = a n

1

(vii) Remember that a bar over the decimal digits indicates that a decimal number or a set of decimal numbers are repeating.

m+n

l

am an

= am − n

a− m = am bm

1 am

 a =   b

m

m

ASSIGNMENT Time: 30 Minutes

M.M.: 30

I. Very Short Answer Type Questions 4

(81) − 2

2. What number do we obtain on rationalising the denominator of

1 7+2

?

II. Short Answer Type Questions-I 3. Simplify

y 3 and express the result in the form of a power of y.

3

p , where p and q are integers and q ≠ 0. q

4. Express 2. 8 in the form of

III. Short Answer Type Questions-II 1

5. If x = 2 +

3 , find the value of x3 +

6. Represent

4.5 on the number line.

(

)(

)(

x3

.

)(

)

7. Show that 2 + 5 2 − 5 3 + 2 3 − 2 is a rational number. 8. Find the values of a and b, if

3+ 2 3− 2

= a + b 2.

IV. Long Answer Type Questions 2

3

4

 a  b  c 9. Simplify:   ×   ×   such that a = 2, b = a2, c = b2.  b  c  a 10. If a =

3− 2 3+ 2

3+ 2

and b =

11. Find the value of

3− 2

4 (216)

−

2 3

+

, find the value of a2 + b2 – 5ab. 1 3

( 256)− 4

+

2 (243)

−

1 5

[NCERT Exemplar]

.Chapterwise Assignments appended with for self evaluation.

(v)

Rationalisation Let a > 0 be a real number and n be a positive 1

n integer. Then n a = b ⇒ a n = b ⇒ a = b and b > 0. is called the radical sign and The symbol n a is called a surd or radical. 1 , l To rationalise the denominator of a +b

we multiply this by are integers.

.Common errors have been tagged to clear confusions with cautions and answers for productive learning.

1. Find the value of

l

a −b a −b

where a and b

Syllabus Units I. II. III. IV. V. VI.

Marks 08 17

Number Systems Algebra Coordinate Geometry Geometry Mensuration Statistics & Probability

04 28 13 10 80

Total





UNIT I: NUMBER SYSTEMS 1. Real Numbers (16 Periods) 1. Review of representation of natural numbers, integers, rational numbers on the number line. Representation of terminating / non-terminating recurring decimals on the number line through successive magnification. Rational numbers as recurring/ terminating decimals. Operations on real numbers. 2. Examples of non-recurring/non-terminating decimals. Existence of non-rational numbers (irrational numbers) such as 2, 3 , and their representation on the number line. Explaining that every real number is represented by a unique point on the number line and conversely, viz. every point on the number line represents a unique real number. 3. Definition of nth root of a real number. 1 1 and (and 4. Rationalization (with precise meaning) of real numbers of the type a+b x x+ y

their combinations) where x and y are natural numbers and a and b are integers. 5. Recall of laws of exponents with integral powers. Rational exponents with positive real bases (to be done by particular cases, allowing learner to arrive at the general laws.) UNIT II: ALGEBRA 1. Polynomials (23 Periods) Definition of a polynomial in one variable, with examples and counter examples. Coefficients of a polynomial, terms of a polynomial and zero polynomial. Degree of a polynomial. Constant, linear, quadratic and cubic polynomials. Monomials, binomials, trinomials. Factors and multiples. Zeros of a polynomial. Motivate and State the Remainder Theorem with examples. Statement and proof of the Factor Theorem. Factorization of ax2 + bx + c, a ≠ 0 where a, b and c are real numbers, and of cubic polynomials using the Factor Theorem. Recall of algebraic expressions and identities. Verification of identities: (x + y + z)2 = x2 + y2 + z2 + 2xy + 2yz + 2zx (x ± y)3 = x3 ± y3 ± 3xy(x ± y) x3 ± y3 = (x ± y) (x2 + xy + y2) x3 + y3 + z3 – 3xyz = (x + y + z) (x2 + y2 + z2 – xy – yz – zx) and their use in factorization of polynomials. 2. Linear Equations in Two Variables (14 Periods) Recall of linear equations in one variable. Introduction to the equation in two variables. Focus on linear equations of the type ax + by + c = 0. Explain that a linear equation in two variables has infinitely many solutions and justify their being written as ordered pairs of real numbers, plotting them

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and showing that they lie on a line. Graph of linear equations in two variables. Examples, problems from real life, including problems on Ratio and Proportion and with algebraic and graphical solutions being done simultaneously. UNIT III: COORDINATE GEOMETRY Coordinate Geometry (6 Periods) The Cartesian plane, coordinates of a point, names and terms associated with the coordinate plane, notations, plotting points in the plane. UNIT IV: GEOMETRY 1. Introduction To Euclid’s Geometry (Not for assessment) (6 Periods) History - Geometry in India and Euclid’s geometry. Euclid’s method of formalizing observed phenomenon into rigorous Mathematics with definitions, common/ obvious notions, axioms/postulates and theorems. The five postulates of Euclid. Equivalent versions of the fifth postulate. Showing the relationship between axiom and theorem, for example: (Axiom) 1. Given two distinct points, there exists one and only one line through them. (Theorem) 2. (Prove) Two distinct lines cannot have more than one point in common. 2. Lines and Angles (13 Periods) 1. (Motivate) If a ray stands on a line, then the sum of the two adjacent angles so formed is 180° and the converse. 2. (Prove) If two lines intersect, vertically opposite angles are equal. 3. (Motivate) Results on corresponding angles, alternate angles, interior angles when a transversal intersects two parallel lines. 4. (Motivate) Lines which are parallel to a given line are parallel. 5. (Prove) The sum of the angles of a triangle is 180°. 6. (Motivate) If a side of a triangle is produced, the exterior angle so formed is equal to the sum of the two interior opposite angles. 3. Triangles (20 Periods) 1. (Motivate) Two triangles are congruent if any two sides and the included angle of one triangle is equal to any two sides and the included angle of the other triangle (SAS Congruence). 2. (Prove) Two triangles are congruent if any two angles and the included side of one triangle is equal to any two angles and the included side of the other triangle (ASA Congruence). 3. (Motivate) Two triangles are congruent if the three sides of one triangle are equal to three sides of the other triangle (SSS Congruence). 4. (Motivate) Two right triangles are congruent if the hypotenuse and a side of one triangle are equal (respectively) to the hypotenuse and a side of the other triangle. (RHS Congruence) 5. (Prove) The angles opposite to equal sides of a triangle are equal. 6. (Motivate) The sides opposite to equal angles of a triangle are equal. 7. (Motivate) Triangle inequalities and relation between ‘angle and facing side’ inequalities in triangles. 4. Quadrilaterals (10 Periods) 1. (Prove) The diagonal divides a parallelogram into two congruent triangles. 2. (Motivate) In a parallelogram opposite sides are equal, and conversely. 3. (Motivate) In a parallelogram opposite angles are equal, and conversely. 4. (Motivate) A quadrilateral is a parallelogram if a pair of its opposite sides is parallel and equal. 5. (Motivate) In a parallelogram, the diagonals bisect each other and conversely. 6. (Motivate) In a triangle, the line segment joining the mid points of any two sides is parallel to the third side and in half of it and (motivate) its converse.

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5. Area (7 Periods) Review concept of area, recall area of a rectangle. 1. (Prove) Parallelograms on the same base and between the same parallels have the same area. 2. (Motivate) Triangles on the same (or equal base) base and between the same parallels are equal in area. 6. Circles (15 Periods) Through examples, arrive at definition of circle and related concepts-radius, circumference, diameter, chord, arc, secant, sector, segment, subtended angle. 1. (Prove) Equal chords of a circle subtend equal angles at the center and (motivate) its converse. 2. (Motivate) The perpendicular from the center of a circle to a chord bisects the chord and conversely, the line drawn through the center of a circle to bisect a chord is perpendicular to the chord. 3. (Motivate) There is one and only one circle passing through three given non-collinear points. 4. (Motivate) Equal chords of a circle (or of congruent circles) are equidistant from the center (or their respective centers) and conversely. 5. (Prove) The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. 6. (Motivate) Angles in the same segment of a circle are equal. 7. (Motivate) If a line segment joining two points subtends equal angle at two other points lying on the same side of the line containing the segment, the four points lie on a circle. 8. (Motivate) The sum of either of the pair of the opposite angles of a cyclic quadrilateral is 180° and its converse. 7. Constructions (10 Periods) 1. Construction of bisectors of line segments and angles of measure 60°, 90°, 45° etc., equilateral triangles. 2. Construction of a triangle given its base, sum/difference of the other two sides and one base angle. 3. Construction of a triangle of given perimeter and base angles. UNIT V: MENSURATION 1. Areas (4 Periods) Area of a triangle using Heron’s formula (without proof) and its application in finding the area of a quadrilateral. 2. Surface Areas and Volumes (12 Periods) Surface areas and volumes of cubes, cuboids, spheres (including hemispheres) and right circular cylinders/cones.

UNIT VI: STATISTICS & PROBABILITY 1. Statistics (13 Periods) Introduction to Statistics: Collection of data, presentation of data — tabular form, ungrouped / grouped, bar graphs, histograms (with varying base lengths), frequency polygons. Mean, median and mode of ungrouped data. 2. Probability (9 Periods) History, Repeated experiments and observed frequency approach to probability. Focus is on empirical probability. (A large amount of time to be devoted to group and to individual activities to motivate the concept; the experiments to be drawn from real - life situations, and from examples used in the chapter on statistics).

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Question Paper Design for Mathematics (Class IX) Time : 3 Hours

S. No.

Max. Marks : 80

Typology of Questions

Very Short Short Short Answer-I AnswerAnswer(SA) II (SA) Objective type (VSA) 1 Mark

1.

2.

3.

4.

2 Marks 3 Marks

Long Answer (LA)

% Total Weightage Marks (approx.)

4 Marks

Remembering: Exhibit memory of previously learned material by recalling facts, terms, basic concepts, and answers.

6

2

2

1

20

25%

Understanding: Demonstrate understanding of facts and ideas by organizing, comparing, translating, interpreting, giving descriptions, and stating main ideas

6

1

1

3

23

29%

Applying: Solve problems to new situations by applying acquired knowledge, facts, techniques and rules in a different way.

5

2

2

1

19

24%

Analysing: Examine and break information into parts by identifying motives or causes. Make inferences and find evidence to support generalizations Evaluating: Present and defend opinions by making judgments about information, validity of ideas, or quality of work based on a set of criteria. Creating: Compile information together in a different way by combining elements in a new pattern or proposing alternative solutions

3

1

3

1

18

22%

20×1=20

6×2=12

8×3=24

6×4=24

80

100%

TOTAL

Internal Assessment Pen Paper Test and Multiple Assessment (5 + 5) Portfolio Lab Practical (lab activities to be done from the prescribed books)

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20 Marks 10 Marks 05 Marks 05 Marks

CONTENTS 1. Number Systems.................................................................................................................... 11 2. Polynomials............................................................................................................................ 46 3. Coordinate Geometry............................................................................................................. 67 4. Linear Equations in Two Variables........................................................................................ 83 5. Introduction to Euclid’s Geometry (As per the CBSE Syllabus this Chapter is Not for Assessment)......................................................................................................................... 6. Lines and Angles.................................................................................................................. 104 7. Triangles............................................................................................................................... 131 8. Quadrilaterals....................................................................................................................... 168 9. Areas of Parallelograms and Triangles................................................................................ 197 10. Circles.................................................................................................................................. 224 11. Constructions........................................................................................................................ 266 12. Heron’s Formula................................................................................................................... 282 13. Surface Areas and Volumes.................................................................................................. 303 14. Statistics............................................................................................................................... 348 15. Probability............................................................................................................................ 388

l

Answers to Chapterwise Assignments............................................................................. 404

Sample Papers (1 to 10).............................................................................................................. 407

l

Solutions to Sample Papers (1 to 10)................................................................................ 446

(x)

1

Number Systems

Topics Covered 1. Basics of Number Systems and Rational 2. Irrational Numbers and Real Numbers

Numbers

3. Decimal Expansions of Real Numbers

4. Visual Representations of Real Numbers on Number Line through Successive Magnifications

5. Operations on Real Numbers

6. Laws of Exponents for Real Numbers

CHAPTER MAP Number Systems Real Numbers

Rational Numbers (Q)

Integers (Z)

Whole Numbers (W) (0, 1, 2, 3, 4 ...)

Natural Numbers (N) (Counting Numbers 1, 2, 3, .....)

Zero (0)

Negative Integers (–1, –2, –3, ....)

Decimal Expansions of Real Numbers

Positive Integers (1, 2, 3, 4, ...)

Operations on Real Numbers

Visual Representations of Real Numbers on Number Line

Laws of Exponents for Real Numbers

QUICK REVISION NOTES 1.  Basics of Number Systems and Rational Numbers Basics of Number Systems The numbers form the base of mathematics. We all know that counting numbers are 1, 2, 3, 4, ... which are also called natural numbers and are denoted by N. N The natural numbers can be represented by the number ray as: 1, 2, 1

2

3

4

and their collection as N = {1, 2, 3, 4, 5, ....}.

11

5

6

3, 4, 5, ........

But while doing basic operations on natural numbers, there was no answer to satisfy the problems like 5 – 5 = ? or 16 – 16 = ? So, a new group was made by adding a number zero (0) to this group, which was able to answer the above problems. Hence, the group containing 0 was named as whole numbers and is denoted by W. Whole numbers group has now all natural numbers and 0 (zero). Whole numbers can be represented by the number ray as: 0

1

2

3

5

4

6

W 0

and their collection as W = {0, 1, 2, 3, 4, 5, ....} N 1, 2, But again for solving the problems such as more expenditure than the income, fall 3, 4, ... in the temperature below its normal range etc., still no solution was found using whole numbers. Hence, the number system was modified and now a larger system was introduced called integers. This system contained all the whole numbers and negative of natural numbers and is denoted by Z. Number ray now was changed to number line, where negative numbers were extended to the left side of zero. Those can be represented on the number line as: –4



–2

–3

–1

0

1

3

2

4

The collection of integers is written as Z = { .... , – 3, – 2, – 1, 0, 1, 2, 3, .....} The integers include natural numbers, zero and negative of natural numbers. Z W

0

N 1, 2, 3, 4...

–1, –2, –3, –4, ...

Rational Numbers The system of integers was further extended to form a bigger system, which is called rational numbers to solve the problems such as dividing a quantity in a ratio, calculation of a portion covered of a particular distance, etc. p Rational numbers are denoted by Q and are written in the form , where p and q are integers. q



3 − 8 12 , But, the numbers like , etc. can’t be defined. 0 0 0

p Hence, a rational number is defined as ‘A number which can be written in the form is called q a rational number, where p and q are integers, such that q ≠ 0’.  p \ Q =  q ; q ≠ 0, p, q ∈ Z    Like natural numbers, whole numbers and integers, rational numbers can also be represented on the number line as shown below. Q –2

12

–

3 2

Mathematics–9

R –1

–

1 2

P 0

1

3 2

2



Point P here represents the rational number

3 . 2

1 Point R here represents the rational number − . 2 3 Point Q here represents the rational number − . 2

Rational numbers include natural numbers, whole numbers and integers.

Q 21 , 3, 5

4 7 ...

,



Z W

0

N 1, 2, 3, 4, 5, ...

–1, –2, –3, –4, ...

Important Features 0 l 0 (zero) is a rational number as it can also be written in the form , where m is any integer m and m ≠ 0. p l Every integer (p) is a rational number, as it can be represented as , where 1 ≠ 0. 1 l There are infinitely many rational numbers between any two given rational numbers. l A rational number between two distinct rational numbers a and b such that a < b is

a+b 2

l For finding n rational numbers between two distinct rational numbers a and b such that a < b, b−a find d = . n +1 Then required rational numbers will be (a + d), (a + 2d), (a + 3d), ..., + (a + nd). [Imp.] Example 1. Find a rational number between 3 and 4. Solution. There are infinitely many rational numbers between 3 and 4. We can proceed in different ways to find the answer. 3+ 4 7 Method 1: A rational number between 3 and 4 is = . 2 2 Method 2: Since, a rational number between 3 and 4 is required, so we write 3 and 4 as rational numbers with the denominator 1 + 1, i.e., 2. 3×2 6 4×2 8 So, 3 = = , and 4 = = 2 2 2 2 6 8 Now, we have to find a rational number between and . 2 2 7 \ Easily we can see that a rational number between 3 and 4 is . 2 3 4 Example 2. Find five rational numbers between and . [NCERT] [Imp.] 5 5 3 4 Solution. For finding five rational numbers between and , we shall find equivalent rational numbers 5 5 3 4 each of and by multiplying 5 + 1 = 6. 5 5 3 3 6 18 4 4 6 24 So, = × = and = × = 5 5 6 30 5 5 6 30 18 24 19 20 21 22 23 and are , , , and . Hence, five rational numbers between 30 30 30 30 30 30 30

Number Systems 13

2 −7 Example 3. Represent 1 and on the number line. 5 5

Solution. Draw a line. Take a point O in the middle of it which represents 0. 2 2 As 1 = 1 + 5 5

Taking equal distance of 1 unit, mark the points P and Q respectively on the right of O. Divide the distance PQ in 5 equal parts again. Count upto two equal parts. Let’s mark this point as 2 A which represents 1 on the number line. 5 –2

B

–1

O

1

0

P

A

Q 7 2 1 = 5 5

–7 5



2

Proceed in the same way towards left from O. Point B represents

−7 5

Exercise 1.1 I. Objective Type Questions

(1 Mark)

1. Choose the correct option in the following (MCQs) (i) Every rational number is (a) a natural number (b) an integer (c) a real number (d) a whole number 2. Fill in the Blanks: ..................... is the smallest whole number. 3. True or False (i) Every integer is a whole number. (ii) Every whole number is a natural number. p (iii) Every integer can be written in the form , where p, q are integers, q ≠ 0. q (iv) There are infinite number of integers between any two integers. II. Short Answer Type Questions-I

(2 Marks)

4. Find three rational numbers between

3 7 and . 5 8

5. Find five rational numbers between

5 4 and [CBSE 2011] 7 9

[Imp.] [CBSE 2011]

Answers and Hints 1. (i) (c) 2. Zero 3. (i) False, for example : 5 is an integer but not a whole number (ii) False, since zero(0) is a whole number which is not natural number.

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Mathematics–9

(iii) True (iv) False, for example : 4 and 5 are integers but these do not exist any integer between them.

3 7 and . 5 8 3×8 7×5 or and 5×8 8×5 24 35 or and 40 40 3 \ The three rational numbers between and 5 7 25 26 27 are . , and 8 40 40 40 4. Given rational numbers are

5. Given numbers are

5 4 and . 7 9

5 2.24 2.24 9 20.16 = = × = 7 7 7 9 63 ...(1) 4 4 × 7 28 Also, = ...(2) = 9 9 × 7 63 Now using (1) and (2), five rational numbers between

4 5 21 22 23 24 25 and are . , , , , 9 7 63 63 63 63 63

  2. Irrational Numbers and Real Numbers Irrational Numbers Further, it was found that still there exist infinitely many numbers on the number line which cannot be written in the form

p , such numbers are called irrational numbers. q

B

Numbers like 2 , 3, π, 0.21010110011....., 0.353553555... etc. are irrational numbers which are the part of the number line. For example, 2 being an irrational number can be shown on the number line as in figure given alongside. If O represents 0 on the number line and triangle OAB is a right-angled triangle with OA = 1 unit and AB = 1 unit,

then,



⇒

2

O

1

1

A

P

OB = OA 2 + AB2 = 12 + 12 = 1 + 1

OB = 2 units Now, measure this distance using compass and mark it on the number line, which is point P. Thus, point P represents 2 on the number line. Hence, irrational numbers can be represented on the number line. Irrational numbers can be defined as: p ‘A number s is called an irrational, if it cannot be written in the form , where p and q are integers q and q ≠ 0’.

Real Numbers Now, no other kind of numbers are left to be exhibited on the number line. Rational Hence, it is clear that the number line has rational and irrational numbers numbers only, which together form the collection of real numbers, represented + by R. Irrational numbers The number line is now called real number line. It can be stated that a real number is either a rational number or an Real Numbers irrational number. So, it is concluded that every real number is represented by a unique point on real number line. Also, every point on the real number line represents a unique real number.

Number Systems 15

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