Acme Mathematics 8

Acme Mathematics 8 1Approved by the Government of Nepal, Ministry 8of Education, Science and Technology, Curriculum Development Centre, Sanothimi, Bhaktapur as an additional material.Please scan forE-book

2 Acme Mathematics 8Publisher : Sundar Pathshala Prakashan Pvt. Ltd.Anamnagar, Kathmandu, NepalLayout Design : Sundar Pathshala Prakashan DesktopEdition : 2063 (First)2067 (Second)2080 (Third)2083 (Fourth) Revised & UpdatedPrinted in Nepal

Acme Mathematics 8 3I feel proud to present this edition of Mathematics book. It is based on the latest syllabus, formed by Curriculum Development Centre (CDC). I have emphasized the theoretical as well as the numerical aspects of the mathematics course. The underlying concepts have been gradually and systematically developed.In each chapter, all the results and concepts of a particular topic have been put together. These are followed by a large quantity of solved examples. Quite a large number of problems have been given as exercises.I am thankful to Sundar Pathshala Prakashan Pvt. Ltd. for its contribution in bringing out this series in such a splendid form as well as thankful to staff who contributed in bringing out this series like as computer designer, art worker, etc. I also wish to thank all teachers and students who have given creative suggestions. I look forward to hearing from both teachers and students' opinions and valuable suggestion that will help improve the book in next edition.Author3rd Bhadra 2082Preface

4 Acme Mathematics 8Sets (Working hour 10) 71.1 Sets1.2 Sub-setsArithmetic (Working hour 45) 272.1 Number system 2.2 Scientific Notation2.3 Ratio and Proportion2.4 Profit and Loss2.5 Unitary method2.6 Simple Interest Mensuration (Working hour 15) 1023.1 Area of Triangle3.2 Area of Quadrilateral3.3 Area of CircleAlgebra (Working hour 30) 1224.1 Indices4.2 Algebraic Expressions4.3 Equation and GraphContent

Acme Mathematics 8 5Geometry (Working hour 31) 1815.1 Angle and Lines5.2 Plane Figure5.3 Congruency and Similarity5.4 Solid Object5.5 Coordinates5.6 Symmetry and Tesselation5.7 Transformation5.8 Bearing and Scale drawingStatistics (Working hour 10) 2946.1 Revision6.2 Pie Chart6.3 Measures of Central Value (Mean, Median, Mode)

6 Acme Mathematics 8

Acme Mathematics 8 71UNIT SETS1.1 SetsA. Revision(a) Empty (Null) set: How many girls are there in the group along side?A set containing no element is called empty set. For example, A = {Even number between 6 and 8}. It is denoted by the symbol φ or { }. A girl in the group of boys is also Null set.(b) Singleton (unit) set: A set containing only one element is called singleton set. For example, B = {the highest peak in the world}(c) Finite set: A set containing the countable number of elements is called finite set. For example,X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}(d) Infinite set : A set containing the uncountable number of elements is called infinite sets. For example,Y = {1,2, 3, 4, .........}S = {stars in the sky}(e) Equal sets: Two sets A and B are said to be equal sets if both have the same elements. For example, if A = {g, o, d} and B ={ d, o, g} then set A = set B. (f) Equivalent sets : Two sets A and B are said to be equivalent sets if the number of elements of sets are equal.For example, if A ={1, 2, 3} and B={a, b, c} then A and B are equivalent sets as n (A) = n(B). It is denoted by A~B.The diagram shows the equivalent sets A and B.AA~BB123abc

8 Acme Mathematics 8Study the inormation given below and discuss in your group. There are seven PROVINCE in our country. Some of these province touches India only. Some of these province touches China only and some of these province touches India and China both. Now all province were divided into three groups. (i) Province that touches China only on the first set.It is the set of Karnali.(ii) Province that touches India only on the second set. It is set of Lumbini and Madhesh.(iii) Province that touches China and India both on the third set. It is the set of Koshi, Bagmati, Gandaki and Sudur Pashchim.How can we represent the given information using set notation? Here,The given information can be represented by set notation. The set of \"province of Nepal\" is Universal set and represented by U.so, U = {Koshi, Madhesh, Bagmati, Gandaki, Lumbini Karnali, Sudur Pashchim} If the first set ofprovince is represented by C, then, C = { Karnali} If the second set of province is represented by I, then, I = { Lumbini, Madhesh} The relationship between the sets C, I, and U can be shown in a diagram: The relationship between these sets is shown in figures C and I.If the third set of province is represented by C, then, C = { Koshi, Bagmati, Gandaki and Sudur Pashchim} We can represent the relation between U, C and I as given along side.This visual representation of the set is known as a Venn diagram. What is the difference between the two Venn diagrams above ? Discuss.ActivityUC I.Karnali.Madhesh.LumbiniFigure (i)UC I.Karnali.Lumbini.Koshi.Gandaki.Bagmati.SudurPaschim.MadheshFigure (ii)

Acme Mathematics 8 9B. Overlapping setsConsider the two sets, A = {1, 2, 3, 4} and B = {2, 4, 6, 8}. Here, elements 2 and 4 belong to both sets A and B. So, they are called overlapping sets. The diagram gives the clear idea about it. In the diagram the overlapping part of the two sets is shown by shading. Thus, two or more than two sets are said to be overlapping sets if they have at least one common element. C. Disjoint setsConsider another example, Let, two sets are, A = {1, 3, 5, 7} and B = {2, 4, 6, 8}. Here no element belongs to both sets A and B. Such sets are called disjoint sets. The adjoining diagram gives the clear idea about it. The sets A and B are called disjoint sets if there is no common elements.D. Venn DiagramOne of the attractive and useful methods of representing sets is by the means of a diagram. The diagram is called Venn diagram, this diagram is developed by John Euler Venn. Sets and their relations can be represented in the Venn diagrams. We normally use the closed figures to represent sets. Usually a rectangle represents the universal set and a circle or an oval represent any other sets. In the figure alongside, the rectangle represents the universal set U = {1, 2, 3, 4, 5, 6} and the oval shape represents the set A = {3, 6}. The rectangle encloses points that represent the elements of the universal set. The closed curve within the rectangle encloses points that represent the elements of set A. Representation of sets in Venn diagram Consider the sets, A = {a, b, c} and B = {b, c}. The sets A and B are shown in the Venn diagram alongside. From the Venn diagram, all the elements of set B are contained in set A. So set B is a subset of set A.For any sets A and B there may be three possible cases which are shown in the following Venn diagrams.A B132468UA B13752 46 8UA.3.1.5.2.4.6UABbacU

10 Acme Mathematics 8ABUA BU A BUB is sub-set of A. A and B are overlapping sets A and B are disjoint setsE. Universal SetLet us see the following sets: A = {1, 3, 5, 7, 9}and B = {2, 4, 6, 8, 10}. Here, in the sets A and B, the elements are taken from natural numbers up to 10. Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Now, from the figure alongside, set A is a subset of U. set B is a subset of U. What can be concluded? A universal set is the set containing all the elements of a particular situation under consideration. Generally it is denoted by the english capital letter U.Classwork1. Tick the null set.(a) A set containing no element. (b) A set containing one element.(c) A set containing two element. (d) All of above.2. If A = {a}, then set A is called ....(a) Unit set (b) Union set(c) Mixed set (d) 'a' set3. Singleton set has ......element.(a) two (b) three (c) one (d) {1}4. A set B = {1, 2, 3, 4, 5, 6, ..... } is given. It is ........(a) finite set (b) infinite set(c) null set (d) NoneA B137 952 46 810U

Acme Mathematics 8 115. Set X = {multiple of 2 less than 599}, It is......(a) finite set (b) infinite set(c) null set (d) None6. Two sets are said to be equivalent if .......(a) They have equal number of members.(b) They have unequal number of members.(c) One is Null set(d) One is Singleton set.7. Example of equivalent sets are .....(a) X = {1, 2} and Y = {2, 3, 4} (b) A = {r, a, m, u} and B = {1, 2, 3, 4}(c) D = {d, 0, 9} and E = {0, 1, 2, 3} (d) None8. Sub-set of set M = {m, o, n, k, e, y} is .....(a) A = {m, o} (b) B = {m, o, n, k}(c) D = {k, e, y} (d) All of above9. Tick ( ) the empty set and cross (  ) the singleton set in the following.(a) A = The set of odd numbers between 7 and 9..(b) X = The set of even prime number.(c) B = The set of number neither prime nor composite.(d) C = The set of highest peak in the world.(e) Z = The set of even numbers between 10 and 11.10. Write any 2 sub-sets of every given sets.(a) D = {Days of a week}(b) E = {0, 2, 4, 6, 8}(c) M = {Multiple of 12 less than 60}(d) A = {January, June, July}(e) X = {0, 1} (f) Z = {z, o}(f) F = {all factors of 15} (h) G = {prime factors of 30}

12 Acme Mathematics 811. State in words.(a) k∈{k, r, i, s, h, n, a} (b) 0∉{Natural number}(c) a∈A (d) {a,e,i,o,u}(e) {Sunday, Mondau, Tuesday}12. Tick the infinite sets.(a) {1, 2, 3, 4, 5, ...., 1000} (b) {Multiplies of 2}(c) {integer between 10 and 1000} (d) {Point an a line}(e) {1, 2, 3, 4, ....}Exercise 1.11. Choose the pairs of disjoint sets.(a) {g, o, d} and {d, m, n}(b) {1, 2, 3, 4} and {a, b, c, d}(c) {r, a, m} and {S, i, t, a}(d) {H, a, r, i} and {S, u, y, o g}(e) {prime numbers} and {even numbers}(f) {a, e, i, o} and {vowels}2. State overlapping and disjoint sets in the pair of sets from the following: (a) A = {even number less than 10}, B = {2, 4, 5, 6} (b) C = {natural numbers less than 5}, D = {cube numbers less than 20} (c) X = {a, b, c, d}, Y = {x, y, z} (d) R = {countries of SAARC}, A = {Nepal, India, Bhutan} (e) M12 = {multiple of 12}, F12 = {factors of 12} (f) D = {Chitwan, Makawanpur, Bara, Parsa, Rautahat}, N = {Ram, Narayan, Yadav} 3. If N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, list the elements of the following sets and represent in Venn-diagram: (a) O = {odd numbers} (b) P = {prime numbers)(c) E = {even numbers} (d) F6 = {factors of 6} (e) M2 = {multiples of 2} (f) M6 = {multiples of 6}

Acme Mathematics 8 134. Show the following sets in Venn-diagram. Taking U as universal set: (a) A = {2, 4, 6} (b) B = {4, 8, 10} (c) G = {1, 3, 5} (d) H = {a, b, c, d} (e) I = { i, n, k} (f) K = {r, a, m, e, s, h} 5. If U = {a, b, c, d, e, f, g, h, x, y, z}, show the followings in the Venn diagram: (a) X = {a, b, c, d} and Y = {b, c, x, y, z} (b) A = {e, f, g, h} and B = {x, y, z} (c) R = {a, c, f, x, y, z} and Z = {x, y, z} (d) T = {y, z, b} and S = {a, b, y, z} (e) D = {a, b} and E = {b, a} 6. If A = {4, 5, 6} and B = {1, 2, 4}, show it in a Venn diagram. 7. If P = {2, 4, 6, 8, 10} and Q = {even numbers less than 10}, show the sets P and Q in a Venn diagram. 8. If A = {a, b, c} and B = {a, b, c, d}, show the sets A and B in Venn-diagram. 9. Draw Venn diagrams for the following pairs of sets and shade the overlapping parts:(a) X = {a, b, c, d} and Y = {b, c, e, f} (b) A = {2, 4, 6, 8, 10, 12} and B = {3, 6, 9, 12, 15, 18} (c) R = {x : x is a factors of 12.} and M = {x : x is a factors of 16.} (d) S = {letters needed to spell the word 'donkey'.} and D = {letters needed to spell the word 'monkey'.} (e) N = {mountain, terai, himalaya } and E = {Bagmati, Narayani, Gandaki} 10. Study the Venn diagram given alongside and list the elements of: (a) Set U (b) Set A (c) Set B A B1379 10 52468U

14 Acme Mathematics 811. Study the given Venn diagram and (a) List the elements of set U. (b) List the elements of set A. (c) List the elements of set B.12. A diagram is given alongside.(a) List the elements of set A.(b) List the elements of set B.(c) List the set of common elements set C.13. Two sets M = (m, o, n, k, e, y} and N = {m, o, n, k} are given.(a) List the common elements as set O.(b) Show the elements of M, N, and O in diagram.14. Study the three pair of overlapping sets.(a) (b) (c)i. List the elements of each set.ii. Write the common elements of each set, as set G, H and I.15. If set A represent animal that can fly and set B represent birds, put the following animals into the correct place in the Venn-diagram.(a) dove (b) sparrow (c) peacock (d) bat (e) crowA BA Bacgb i jdefhUA Bia hrmA B54312CDacdebE Fopk nam

Acme Mathematics 8 151.2 SubsetsIntroduction to subsets Look at the following set of friends. F is a set of three friends Kamal, Ramchandra and Krishna.So, F = {Kamal, Ramchandra, Krishna} We can use the member of the set F to make the following sets: (i) A = {Kamal} (ii) B = {Ramchandra} (iii) C = {Krishna} (iv) D = {Kamal, Ramchandra} (v) E = {Ramchandra, Krishna} (vi) K = {Krishna, Kamal} (vii) R = {Kamal, Ramchandra, Krishna} (viii) N = { } or φHere, sets A, B, C, D, E, K, R and N are contained in the set F. These sets are called the sub-sets of the given set F. Set A is said to be a subset of a set B if every member of set A is also a member of set B. For example, If A = { 2, 3, 5 } and B = { 2, 3, 5, 7 } are two sets then each member of set A is also a member of set B. Hence, set A is a subset of set B. Symbolically it is written as A ⊂ B where we read '⊂' as \" is a subset of \"or \" contained in\". Study the following cases of subsets. Case I: Proper subset Consider the sets, A = {2, 3, 5} and B = {2, 3, 5, 7}. Here, each elements of set A is also an element of set B , number of elements in set A = 3 and number of elements in set B = 4. Since number of element in set A is less than element in set B, in such a case we say that set A is the proper subset of set B and we write it as A ⊂ B.Case II: Improper subset Consider the sets, A = {apple, orange, banana} and B = {banana, apple, orange}. Here, each element of set A is also an element of set B, n(A) = 3 and n(B) = 3. BA2 375A BBanana OrangeApple

16 Acme Mathematics 8Since, n(A) = n(B) and set A and set B are equal, in such a case we say that A is an improper subset of set B and we write it as A ⊆ B. Thus, a set is an improper subset of itself. Case III: Consider the sets, B = {1, 2, 3, 4} and A = {natural number less than 1}. Here, set A has no element, which is also an element of set B and n(B) = 4 and n(A) = 0. Since n(A) < n(B), in such case, we say that set A is subset of set B and we write it as A ⊂ B or φ ⊂ B. Thus, every empty set is a subset of given set. Note: If set A is subset of set B then the set B is called the super set of the set A. It is denoted by B ⊃ A and read as 'B is super set of the set A'. Number of subsets of a given setConsider the following sets and their subsets: 1. If P = {1} then subsets of P are {1} and { } or φNumber of subsets = 2 and number of element in set P = 12. If B = {1, 2} then subsets of B are {1}, {2}, {1, 2}, { } Number of subsets = 4 and number of element in set B = 2 3. If C = {1, 2, 3} then subsets of C are {1}, {2}, {3}, {1, 2}, {2, 3}, {1, 3}, {1, 2, 3}, { }Number of subsets = 8 and number of elements in set C = 3 Now, the result in summary is given below.Set Number of elements in the set Number of subsetsX = { } 0 1 = 20P = {1} 1 2 = 21B = {1, 2} 2 4 = 22C = {1, 2, 3} 3 8 = 23.................... .................... ....................A = {.......... n elements} n .................... = 2nThus if given set has 'n' elements then its number of subsets is 2n.1234BA

Acme Mathematics 8 17Note: (i) Number of subsets = 2n and Number of proper subsets = 2n – 1 (ii) If A = {a, b} then the subsets of A are {a}, {b}, {a, b} and { } or φ. A set of all subsets of A is called the power set. Thus the power set of A = {{a},{b}, {a, b}, φ} (iii) Empty set is the subset of every given set. It is proper subset.How to make Sub-sets?Let A = {a, b}{a, b}{a, b} {a} {b} { }choose 'a'choose 'b' choose 'b'YesYes No Yes NoNoHence, {a, b}, {a}, {b}, { } are the subsets of set A.Solved ExampleExample 1: Let N = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} is a given set. List the following sets: (a) O = Set of odd numbers (b) E = Set of even numbers (c) W = Set of whole numbers (d) M2 = Set of multiples of 2 (e) M12 = Set of multiples of 12 (f) F10 = Set of factors of 10 Solution: (a) O = {1, 3, 5, 7, 9} (b) E = {2, 4, 6, 8, 10} (c) W = {1, 2, 3, 4, 5, ...... 10} (d) M2 = {2, 4, 6, 8, 10}(e) M12 = { } or φ (f) F10 = {1, 2, 5, 10}Example 2: Study the following dairy items milk paneer butter yoghurt ghee chhurpiChoose at least 2 items to prepare breakfast you like most.Solution: Here, set of dairy items, D ={milk, paneer, butter, yoghurt, ghee, chhurpi}breakfast items is subset of DSo, possible choice are {milk, yoghurt}, {milk, paneer}, {milk, ghee} and etc.

18 Acme Mathematics 8Classwork1. If A = {a, b, c, d, e, f}, B = { a, b, c} and C = {a, b, c, d} then find the following : (a) Number of elements in set A (b) Number of elements in set B (c) Number of elements in set C (d) Number of subsets of A (e) Number of subsets of B (f) Number of subsets of C (g) Number of proper subsets of B (h) Number of proper subsets of A 2. Using the data from question number 1, Write down the relation between following sets. (a) B and A (b) B and C (c) C and A (d) { a, f } and A (e) φ and B (f) {a, b, c} and {a, b, c, d} 3. The set of students of a class is given below: R = {Raju, Jyoti, Rakshya, Sunam, Januka, Suman}. List the following subsets of R. (a) whose name starts with R (b) whose name starts with J (c) whose name starts with S (d) whose name starts with K (e) whose name starts with R or S (f) whose name starts with S or J (g) whose name starts with R or J (h) whose name starts with R, J or S Exercise 1.21. Let A = { 2, 4, 6, 8, 10, 12 } be the given set list the following subsets of A: (a) Q = Set of odd numbers (b) P = Set of prime numbers (c) C = Set of composite numbers (d) F12 = Set of factors of 12 (e) M2 = Set of multiples of 2 (f) M15 = Set of multiples of 152. Write all the proper subsets of the set W = {w, x, y}. 3. Write all the subsets of the set P = {13, 17, 19}. 4. Let, N = the set of all private school's students Z = the set of all private school's students of Chitwan Q = the set of all student of a school in Chitwan T = the set of all students of grade 8 of a school in Chitwan

Acme Mathematics 8 19Then choose the true statement:(a) N ⊂ Z (b) T ⊂ Q (c) Z ⊂ N (d) Z ⊂ Q (e) T ⊂ N 5. How many subsets can be obtained from the set K = { k, u, n, d, a } ? 6. How many subsets can be obtained from the set T = { t, i, k, a, n, r, y } ? 7. Write the number of the subsets of the following sets: (a) {1} (b) {1, 2} (c) {1, 2, 3} (d) {b, c, a} (e) {g, o, i, d} (f) {1, 2, 3, 4, 5} (g) {first three prime numbers} 8. Look at the following items of breakfast list of a hostel: hot-lemon, milk, coffee, milk tea, black tea, jam, butter, bread, toast, rotiPrepare a list for a week taking suitable two items at a time. 9. Look at the following food items and make the lists for 3 days for lunch (3 items at a time): 10. Consider a set E = {letters of English alphabet}. Make a list of any 10 subsets of set E. 11. Consider a set of fruits F = {mango, banana, orange, apple}. Make a list of fruits that contains at least two fruits at once. 12. Tick() the true statements in the following: (a) If A ⊂ B then n(A) = n(B). (b) φ is proper subset of any set. (c) Every set is not a subset of itself. (d) {a, b} ⊂ {a, b, c}

20 Acme Mathematics 813. Here is the menu of \"Malewa\" restaurant at Bharatpur in Chitwan: (a) Buff momo (b) Pizza (c) Veg momo (d) Finger chips (e) Chow mein (f) Green Salad (g) Chicken soup (h) Bhatmas Sandeko (i) Mushroom Soup(j) Kaju fry (k) Chicken chilly (l) Meat Prepare your 'Khaja Menu' for 10 days taking only two items at a time.14. Given a Venn-diagram.(a) Define proper subset.(b) Write the improper subset of L.(c) If e, i, o, u are the members of set M only then, what type of set are L and M? Write with reason.15. U = {x : x ≤ 15, x∈N}, A = {x:x is a factor of 12}, and B = {x:x is multiple of 3} are given,(a) List the elements of U, A and B.(b) Write all the subsets of A.(c) Are sets A and B disjoint? Why?16. Let A = {s, e, t} is a set of letters of the word 'set',(a) Write the formula to find the number of subsets of the set A.(b) Write all possible subsets of the set A.(c) Which one of the subsets is the improper subset? Give reason.17. Set A = {x : x is prime number, 2 < x ≤ 15} and B = {y : y is odd number, 2 < x ≤ 15} are given,(a) Write the set A in listing method.(b) Show the elements of set A and B in a Venn-diagram.(c) Is the set A a proper subset of B? Write with reason.18. Two sub-sets of the universal set U = {1, 2, 3, 4, 5, 6} are A = {1, 3, 4, 5} and B = {2, 3, 5}(a) Are sets A and B overlapping or disjoint sets? Write with reasons.(b) Write any two sub-sets of set A having single element.(c) Show sets U, A and B in a Venn-diagram.L Mb ic a eUou

Acme Mathematics 8 2119. Study the given Venn-diagram and answer the following question.(a) Define improper subset.(b) How many sub sets of set B can be made?(c) What type of sets are A and B overlapping or disjoint sets? Write with reason.20. Write the answer of the following on the basis of given Venn-diagram.(a) Define overlapping set.(b) Write any one proper subset of set B.(c) If the elements 2 and 5 of A are replaced by 7 and 8, then what types of set A and B are? Why?21. The subsets A and B of universal set U are presented in the Venn-diagram.(a) Identify and write whether the sets A and B are overlapping or disjoint?(b) Illustrate the improper subset that can be made from set B.(c) How many more or less proper subsets can be made from set B than from the set A?22. In a survey it was found that Ram, Shyam, Hari, Krishna like Dashain festival (D) but Rita, Sita, Hari, like Tihar festival (T), then (d) Define overlapping sets.(e) Show the relation between 'D' and 'T' in a Venn-diagram.23. Let A = {1, 2, 3, 4, 5, 6, .................... 10} and B = {1, 3, 5, 7, 9}(f) Identify the universal set and write cardinal number?(g) Which is a subset of universal set? Give reason.(h) Show the universal set and its subset in a Venn-diagram.A B6 52 9 31U74 10 8A B1 34 6 52U7 9 8A B3 5 64U78 9

22 Acme Mathematics 8Project Work1. Objective : To make sub sets.2. Materials required :“ A-4 Size Paper“ Pen3. Activity : “ List the things in your edible item. Classify them according to same character. Name the sets and three sub sets from each sets. Present it to your class room.For example : Edible item : ..........................................................................................................................................................................................................................................

Acme Mathematics 8 231. The sets A and B are defined as,A = {Prime numbers between 10 and 24}B = {Multiple of 7 between 6 and 40}(a) Represent set A by listing method.(b) Write set B in set-builder form.(c) Are set A and B equivalent sets? Give reason.2. Sets A ={0, 2}, B = {1, 2} and C = {4} are proper subsets of U.(a) Represent set A and B in a Venn diagram.(b) Among set A and set C,which set has more subsets, give reason?3. Sets A ={x:x is even odd number less than 12} and B = {y:y is prime number less than 10} are given,(a) List the elements of sets A and B separately.(b) If universal set U contains elements of set A or set B or both, then write cardinality of the universal set U.(c) What types of set are the sets A and B? Write reason.4. A ={2, 4, 6, 8, 10} and B = {Set of even prime number}are given.(a) Write the cardinal number of set A.(b) List out the elements of set B.(c) Write the relation of set A and set B in set notation.5. A and B are two sets, A ={odd number less than 10} and B = {Multiple of 2 less than 10}.(a) Are the sets A and B are overlapping or disjoint sets?(b) Make any 4 subsets from set A.(c) Write the relation of set A and set B in set notation.Mixed Exercise

24 Acme Mathematics 86. U ={Natural numbers upto 10}, A = {Factors of 8}, B = {Multiples of 2}are given.(a) List down the elements of set A and B.(b) What type of sets A and B are? Why?(c) Draw a Venn-diagram to show the relation between U, A and B.(d) Write down the common elements of set A and B.7. A Venn-diagram is given. Answer the following questions on this basis.(a) List down the elements of set A and B.(b) 'B is proper subset of set A'. Is this statement correct? Give reason.(c) Make all the possible subsets of set B.8. Study the given Venn diagram then write the answer of the following questions:(a) Write set M and set N by listing method and description method. (b) Construct a universal set U.9. If A ={a, b, c}, then(a) Find the subsets of set A.(b) Which one among them is not a proper subset?(c) If a set B = {s, k, y} then what type of sets are A and B, disjoint or overlapping? Give reasons.10. Let A = {x:x is a factor of 15} and B = {y:y is a factor of 18} be two sets.(a) List the members of the sets A and B.(b) State whether the sets A and B are disjoint or overlapping sets? Give reason. (c) Show the members of the sets A and B in Venn diagram.11. From the given venn diagram find the following:(a) What is the relation between the two sets A and B?(b) List the elements of set A and B. (c) List the elements of set U.(d) Write down the possible subset of a set P = {1, 2, 3}.ABioua eUM N39156121824UA Babcdg h iefU

Acme Mathematics 8 2512. P = {x:x is a factor of 8} and Q = {x:x is a factor of 4} are given sets then find the following:(a) List the elements of set P.(b) List the elements of set Q.(c) What is the relation between set P and set Q?(d) Show in Venn diagram.13. The sets A and B are defined as, A = {Prime numbers less than 10} and B = {Even numbers from 1 to 10} then find the following:(a) List the elements of set A and B.(b) Are the sets A and B overlapping or disjoint sets?(c) How many subsets can be made from set A?14. U = {Natural numbers up to 10}, A = {prime number less than 10} and B = {Even numbers less than 10} are given.(a) Write down sets A and B in listing method.(b) Write down the proper subsets from set A.(c) Show the relation between the set A and B in Venn diagram.(d) Are the sets A and B disjoint or overlapping? Give reason.15. (a) Define overlapping set.(b) If A = {a, e, i, o, u}, B = {a, b, c, d}(i) Is set A and B overlapping or disjoint set?(ii) Is A is equal to B?16. The given set is G = {g, o, d}, then(a) Write all possible subsets formed by the elements of set G.(b) Distinguish the proper and improper subset from these subsets.17. Let A = {odd numbers less than 10}, B = {2, 3, 5, 7, 11}(a) List out the elements of set A.(b) Write the cardinal number of set B.(c) What types of set are A and B? Equal or equivalent? Give reason.

26 Acme Mathematics 8EvaluationTime: 41 minutes Full Marks: 171. From the venn-diagram given alongside, answer the followingquestions: (a) Write down all sets in listing method.(b) Write down a pair of disjoint sets. [1] (c) Write down a pair of overlapping sets. [1] (d) Write down the common elements of overlapping sets. [1] (e) If there was an element 'd' in the set C, then what would be the relation between the set A and C ? [1] 2. If U = 1, 2, 3, ...., 10}, A = {factors of 6}, B={factors of 8} then..)(a) Write these set A and B in listing method. [1] (b) Find the common set of A and B. [1] (c) Represent the above information in venn-diagram and what type of set they are? Write it. [2]3. Given a venn-diagram. (a) Define subset. [1](b) Write the improper subset of set A. [1] (c) If i, o, u are the only members of set A then. What type of set are A and B. Write with reason. [1]4. The subsets A and B of universal set U are presented in the Venn-diagram. (a) Indentify and write whether the sets A and B are overlapping or disjoint ? [1](b) Illustrate the improper subset can be made from set B. [1](c) How many more or less proper subjects can be madefrom set B then from the set A ? [1]5. Two sets are A = {a, c, e} and B = {a, e, i, o}.(a) Write the relation between A and B. [1] (b) What is the number of proper subsets of set B. [1] (c) Represent the sets A and set B in a Venn-diagram. [1] UA B.3.4 .6.7.8 .9.5UA B .c.e.a .f.g.h .i.d .bCUA B .o.u.a .b.c .d.m .n.e .i

Acme Mathematics 8 272UNIT Arithmetic2.1 Number systemA. Binary Number SystemACTIVITYStudy the given activities.Answer the following questions:a. How many steps are there on lighting the bulb?b. What are the ways to go in and out?The above function needs only two steps. They are 'on' and 'off' or 'in' and 'out'. Similarly the computer also works on two steps; 'on' and 'off'.The system where only two steps are used is called binary system. In this system the symbol 0 (off) and 1 (on) is used.The Hindu-Arabic number system uses ten different symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Therefore it is called a decimal (base 10) system and we can express each and every whole number in the power of 10. e.g. (a) 123 = 1 hundred + 2 tens + 3 ones = 1 × 102 + 2 × 101 + 3 × 100(b) 3209= 3 thousand + 2 hundred + 0 tens + 9 ones = 3 × 103 + 2 × 102 + 0 × 101 + 9 × 100A system of number used to represent real number using 0 and 1 with base 2 is called the binary number system. This number system is commonly used in computers.. Because the digits 0 and 1 can be described electrically as ‘OFF’ and ‘ON’, they are also known as ‘Dyadic Number System’. The numbers. in the binary system can be expanded to the power of 2. The process of expansion is similar to decimal number system

28 Acme Mathematics 8Now consider the following table:DecimalNumberBase 2 grouping Binary2 Number 4 23 22 21 200 01 122 1023 1124 10025 10126 11027 11128 100029 1001210 1010211 1011212 11002Note represents unit in decimal system.Solved ExamplesExample 1 :Change into decimal number: 1012Solution: Here, 1012 = 1 × 22 + 0 × 21 + 1 × 20= 4 + 0 + 1= 5 Example 2 :Change into decimal number: 11012Solution: Here, 11012 = 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20= 8 + 4 + 0 + 1= 13

Acme Mathematics 8 29Example 3 : Change into decimal number: 111111112Solution: Here, 111111112 = 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 = 1 × 128 + 1 × 64 + 1 × 32 + 1 × 16 + 1 × 8 + 1 × 4 + 1 × 2 +1 × 1 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1= 255 Example 4 :Convert 38 into binary system. Solution: Dividing 38 by 2, until the quotient is 0.2 38 Remainder2 19 → 02 9 → 12 4 → 12 2 → 02 1 → 00 → 1Order of Binary Numbers.Now, 38 = 10011023810 = 1001102Place value chart of 1001102 is given below:25 24 23 22 21 201 0 0 1 1 0Classwork1. Fill in the blanks in the following.(a) The binary number system uses ………………………symbols. (b) The decimal number system uses ………………………symbols. (c) The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used in ……………………system. (d) The symbols 0 and 1 are used in ……………………system..

30 Acme Mathematics 82. Tick () the correct answer in the following.(a) The binary number system is …….(i) base 2 system (ii) base 5 system (iii) base 10 system (iv) all of above(b) The decimal number system is …….(i) base 6 system (ii) base 5 system (iii) base 10 system (iv) all of all(c) The binary number system is written as…….(i) 12 (ii) 11 (iii) 110 (iv) 15(d) The decimal number system is written as…….(i) 2012 (ii) 119 (iii) 110 (iv) 15(e) The decimal number for binary number 102 is written as…….(i) 10 (ii) 2 (iii) 11 (iv) 5(f) The binary number for decimal number 13 is written as…….(i) 11012 (ii) 132 (iii) 4 (iv) 125Exercise 2.11. Show the following binary numbers. in a place value chart: (a) 1012 (b) 10112 (c) 111102 (d) 100002(e) 1010102 (f) 101012 (g) 1010112 (h) 1111112(i) 110012 (j) 1110112 (k) 1111112 (l) 1101012(m) 1101112 (n) 1111012 (o) 1111102 (p) 1000012 2. Convert the following binary numbers into decimal numbers.: (a) 102 (b) 112 (c) 1012 (d) 1112(e) 10012 (f) 1002 (g) 10112 (h) 11102(i) 101012 (j) 101010102 (k) 10101012 (l) 101112 (m) 111012 (n) 111102 (o) 100012 (p) 110012(q) 1110112 (r) 1111112 (s) 1101012 (t) 1101112(u) 1111012 (v) 1111102 (w) 1000012

Acme Mathematics 8 313. Convert the following decimal numbers. into binary numbers.: (a) 8 (b) 25 (c) 40 (d) 56 (e) 90 (f) 118(g) 120 (h) 281 (i) 315 (j) 444 (k) 523 (l) 960 4. Convert 1 + 2 + 3 + 4 + 5 to the binary number system. 5. Convert 32 + 23 into base 2 number system. 6. For what value of A, the binary number 1000A12 will represent 35? 7. For what value of P, the binary number 1P102 will represent 14? 8. For what value of P, the binary number 10P1012 will represent 45? 9. For what value of P, the binary number 110P12 will represent 25? 10. For what value of A and B, the binary number 1001A00B12 will represent 305? 11. For what value of A and B, the binary number 10A11B2 will represent 38? 12. For what value of A and B, the binary number 101A0B2 will represent 45?13. The number 110102 only use two numerals 0 and 1.(a) What type of number system is this?(b) Convert it into decimal number system.(c) Convert 145 into quinary number system.14. Answer the following question:(a) How many digits are used in binary number system and what are they?(b) Convert 84 into the binary number.(c) Convert 1010112 into decimal number.15. 10101002 is a number:(a) In which number system is the number above? Write it.(b) What are the digits used in this number system?(c) Convert the number in Denary number system.Project WorkList the field where 'binary system' is used. Present your list in the class.

32 Acme Mathematics 8B. Quinary Number SystemThe decimal (or denary) number system uses 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 different symbols and its base is 10. The binary number system uses 0 and 1 only and its base is 2. Similarly, if the number system uses 0, 1, 2, 3 and 4 different symbols and base is 5, then the system is called quinary number system. In this system, we express each and every whole number to the power of 5. e.g. 1235 = 1 × 52 + 2 × 51 + 3 × 5040215 = 4 × 53 + 0 × 52 + 2 × 51 + 1 × 50We write the quinary number system as 1235, 40215, 1235 = 1 × 52 + 2 × 51 + 3 × 50= 1 × 25 + 2 × 5 + 3 × 1 = 25 + 10 + 3 = 381040215 = 4 × 53 + 0 × 52 + 2 × 51 + 1 × 50= 4 × 125 + 0 × 25 + 2 × 5 + 1 × 1= 500 + 0 + 10 + 1 = 51110Now consider the following table:Decimal Number 54 53 52 51 50 Quinary number0 0 01 1 12 2 23 3 34 4 45 1 0 1056 1 1 1157 1 2 1258 1 3 1359 1 4 14510 2 0 20525 1 0 0 1005

Acme Mathematics 8 33Solved ExamplesExample 1 : Convert 12110 into quinary number system.Solution:5 121 Remainder5 24 → 15 4 → 40 → 4Order of Binary Numbers.∴ 12110 = 4415Check: 4415 = 4 × 52 + 4 × 51 + 1 × 50= 100 + 20 + 1 = 121Classwork1. Fill in the blanks in the following.(a) The decimal number system uses ………………………symbols. (b) The quinary number system uses ………………………symbols. (c) The symbols 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are used in ……………………system. (d) The symbols 0, 1, 2, 3, and 4 are used in ……………………system.2. Tick () the correct answer in the following.(a) The quinary number system is …….(i) base 4 system (ii) base 5 system(iii) base 10 system (iv) none of all(b) The decimal number system is …….(i) base 10 system (ii) base 5 system(iii) base 12 system (iv) all of all(c) The quinary number system is written as…….(i) 122 (ii) 110(iii) 1230 (iv) 43215(d) The decimal number system is written as…….(i) 2012 (ii) 119(iii) 110 (iv) 15

34 Acme Mathematics 8(e) The decimal number for quinary number 115 is written as…….(i) 6 (ii) 7(iii) 5 (iv) 8(f) The binary number for quinary number 135 is written as…….(i) 1012 (ii) 132(iii) 8 (iv) 125(g) The qi nary number for binary number 111112 is written as…….(i) 100012 (ii) 305(iii) 800 (iv) none of allExercise 2.21. Show the following quinary numbers. in a place value chart: (a) 3215 (b) 2415 (c) 43115 (d) 10035(e) 121235 (f) 214305 (g) 344315 (h) 2223255(i) 432015 (j) 1234125 (k) 444445 (l) 33333352. Convert the following quinary numbers. into decimal numbers.: (a) 3215 (b) 2135 (c) 2325 (d) 2345(e) 4325 (f) 42015 (g) 10025 (h) 20345(i) 10005 (j) 403215 (k) 444445 (l) 33333353. Convert the following decimal numbers into quinary numbers:(a) 18 (b) 125 (c) 140 (d) 256 (e) 490 (f) 1128 (g) 5700 (h) 3000 (i) 9887 (j) 9999 (k) 4444 (l) 5555 4. Convert the binary numbers into quinary numbers and vice versa: (a) 1012 (b) 10102 (c) 11112 (d) 101012(e) 101010102 (f) 2135 (g) 4325 (h) 12345(i) 10005 (j) 403225 (k) 44445 (l) 222225

Acme Mathematics 8 355. Convert 24 + 42 into base 5 system.6. There are 674 students in Mount Everest School.(a) Convert 674 in quinary number system.(b) What is decimal unumber of 1012(c) Change 145 to decimal number.7. 1245 is a number:(a) In which number system is the number above? Write it.(b) What are the digits used in this number system?(c) Convert the number in Devnagari number system.8. A quinary number 11015 is given, then(a) Write the number in expanded form. (b) Convert the number into binary number system. (c) Write its equivalent base-10 number system. 9. Mr. Aryal is a science teacher. He writes his one day's salary in the expanded form as 4 × 54 + 2 × 53 + 4 × 52 + 0 × 51 + 1 × 50Answer the following questions.(a) Write the short form by his fee in quinary number. (b) Convert his one day's salary into decimal number. (c) Convert this decimal number into binary number. 10. (a) What are the numbers used in quinary number system.(b) Write 625 as the power of 5.(c) Show in place value: 1565

36 Acme Mathematics 8C. Rational NumberACTIVITYStudy the number line:– 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 9Take any numbers from the number line, let they are 5 and (– 3). Then,+ 5 + (–3) = + 2+ 5 – (–3) = 5 + 3 = 8+ 5 × (–3) = – 15+ 5 ÷ (–3) = 5– 3Now, answer the following questions.(a) Are + 2, + 8 and – 15 are rational numbers?(b) Is 5– 3, a rational number?(c) Where is 5– 3 on the number line?When any two numbers are either added, subtracted or multiplied the result are also whole number. But when any two numbers are divided one another, it may or may not be whole number. Such as, 5– 3, 34, 12 etc. are not whole numbers. The numbers like 5– 3, 34, 12 etc are called rational numbers.You have studied about natural numbers, whole numbers and integers. Every number system has its own limitations and gives birth to the new number system. Here, we shall recall our previous knowledge to study the further number system and their properties.1. Natural Numbers: The numbers 1, 2, 3, ... are called counting or natural numbers. It is denoted by N where N = {1, 2, 3, 4, ...} 2. Whole Numbers: The natural numbers with zero are called whole numbers. It is denoted by W, where W = {0, 1, 2, 3, 4, …} 3. Integers: The natural numbers their negative, including zero are called integers. It is denoted by Z or I where, Z or I = {…. – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, …} The number of the form pq, where p and q are integers and q≠ 0 is called rational number. It is denoted by Q. where Q = ... 5, – 4, – 35 , 25 , 0, 13 , 27 , 2, 3 ...NWZQ

Acme Mathematics 8 37Note: (i) Every natural number is a rational number. (ii) Every whole number is a rational number. (iii) Every integer is a rational number. (iv) Every fraction is a rational number.Properties of Rational Numbers1. Properties of addition: If x, y and z are three rational numbers then, (a) x + y = y + x (b) (x + y) + z = x + (y + z) (c) 0 + x = x + 0 = x where 0 is zero (d) x + (– x) = (– x) + x = 0, where – x is called negative of x. 2. Properties of multiplication: If x, y and z are three rational numbers, then, (a) x × y = y × x (b) (x × y) × z = x × (y × z) (c) x × 0 = 0 × x = 0(d) x × 1 = 1 × x = x (e) (i) x × (y + z) = x × y + x × z (f) x × (y – z) = x × y – x × z Note: Verification of the properties is left for you.3. Properties of division: If x, y and z are rational numbers then we have (i) (x + y) ÷ z = x ÷ z + y ÷ z (ii) (x – y) ÷ z = x ÷ z – y ÷ zIf the numbers p and q in the form pq has no common factor other than 1, then pq is in its simplest or lowest form. e.g., 1015, 3451, 200300 are all equivalent to 23 in the lowest form.Reciprocal of a Rational NumberConsider a rational number 513 which is made of two integers 5 and 13, where 5 is the numerator and 13 is the denominator. If we interchange the position of integers in the numerator and the denominator, we get another rational number 135 . In general for any non-zero rational number pq, we have another non-zero rational number qp. The number qpis called the reciprocal of pq and vice-versa. The number 37 is the reciprocal of 73.

38 Acme Mathematics 8Similarly, 2– 7 is reciprocal of – 72 or 7– 2. If we denote the number pq by x, then its reciprocal qp is denoted by x–1. Now we have, (i) pq–1 = qp (ii) (x –1)–1 = x (iii) x–1 × x = qp × pq = 1, The product of a non-zero rational number and its reciprocal is always 1. The reciprocal of x is x–1. It is also denoted by 1 x i.e. x– 1 = 1 xThe reciprocal of the reciprocal of a number is the number itself.Zero as a rational number has no reciprocal. A rational number in the lowest form is said to be in the standard form, if the denominator is positive. The standard form of – 4 8 is – 12 and the standard form of 5– 7 is – 57 . D. Rational Numbers on the Number Line– 2 – 1 0C O A B D1 2The points A, B, C and D represent the numbers 3 10, 7 10, – 6 10 and 2 1 respectively. Similarly 7 10 can be represented as0 710Classwork1. Tick () the correct answer in the following.(a) ........ is the identity for the addition of rational numbers.(i) 1 (ii) 0 (iii) – 1 (iv) 12 (b) ........ is the multiplicative identity for rational numbers.(i) 1 (ii) 0 (iii) – 1 (iv) 12 (c) The additive inverse of 75 is.(i) 1 (ii) 0 (iii) – 75 (iv) 75 1010

Acme Mathematics 8 39(d) Zero has ........... reciprocal.(i) 1 (ii) 2 (iii) 3 (iv) no(e) The numbers ......... and .......... are their own reciprocals.(i) 1 and 0 (ii) 1 and – 1 (iii) – 1 and 0 (iv) none of these(f) The reciprocal of – 5 is ......... .(i) 5 (ii) 1 (iii) – 15 (iv) 15 (g) Reciprocal of 1x , where x ≠ 0 is ......... .(i) 1 (ii) x (iii) 0 (iv) none of these(h) The product of two rational numbers is always a ........... .(i) whole numbers (ii) integers(iii) natural numbers (iv) rational numbers(i) Simplify : – 45 × 37 × 1516 × – 149(i) 1 (ii) 0 (iii) 2 (iv) 12 (j) The sum of the rational numbers – 516 and 712 is (i) – 748 (ii) – 1130 (iii) 1348 (iv) 13 (k) A rational number can be expressed as a terminating decimal if its denominator has factors .............(i) 4 or 5 (ii) 2 or 3 (iii) 3 or 5 (iv) 2 or 5Exercise 2.31. Draw the number line and represent the following numbers on it:(a) 2 3 (b) 3 7 (c) 5 8 (d) 4 13 (e) 7 192. Write true or false for the following statements: (a) Every natural number is a rational number. (b) The rational number 5 2 lies to the left of zero on the number line.(c) The rational number 6 – 7 lies to the right of zero on the number line.(d) The rational number – 13 – 4 lies neither to the right nor to the left of zero on the number line.

40 Acme Mathematics 83. Add the following rational numbers: (a) 1 2 and 3 4 (b) 4 5 and 5 6 (c) 6 8 and 8 9 (d) 9 14 and 12 214. Multiply the following rational numbers: (a) 4 3 and 15 16 (b) 9 15 and 25 27 (c) 4 15 and 9 16 (d) 22 35 and 42 555. Subtract: (a) 1 7 from 2 3 (b) – 89 from 10 7 (c) 27 39 from – 15 176. The sum of two rational numbers is -8. If one of them is – 8 17 , find the other. 7. What number should be added to – 5 6 to get 2 3? 8. Find the reciprocal of the following: (a) 2 9 (b) 3 5 + 5 7 (c) 2 7 – 2 5 (d) 11 9 × 2 79. Let 3 4, and 4 5 are three rational number,(a) Verify its addition properties.(b) Verify its multiplication properties.(c) Verify its division properties.10. The product of two rational numbers is 1 18 . If one of them is – 16 81 , find the other. 11. Write the rational numbers represented by letters on the following number lines.(a)0A B C D1 2 3(b)– 0A B C D E0 1 2 3

Acme Mathematics 8 41E. Decimal Representation of Rational Numbers We know that every rational number, when expressed in decimal form, is expressed either by terminating or non-terminating decimal form. For example, (i) 1 2 = 0.5, 2 5 = 0.4, 3 10 = 0.3, 3 4 = 0.75, 45 80 = 0.5625(ii) 1 3 = 0.333…, 15 36 = 0.41666…., 2 11 = 0.181818…If the denominator contains 2 or 5 only as prime factors, then the fraction becomes a terminating decimal. In the above examples 1 2, 2 5, 3 10 , 3 4 and 45 80 are all terminating decimals.If after the decimal point, a digit or a group of digits repeats again and again, then it is called a non-terminating or recurring or repeating decimal. In the above examples 1 3, 15 36and 2 11 , are all recurring, non-terminating or repeating decimals. In the non-terminating or recurring decimal representation 1 3, we find that the digit 3 goes on repeating. Similarly, in the number 2 11 , the digits 1 and 8 repeat in this order infinitely many times. While writing such decimals, we put a bar (–) or dot (•) over the repeating part.Thus, we write, 1 3 = 0.333…, = 0.3 or 0.3.Likewise, we write 15 36 = 0.41666 … = 0.416, 2 11 = 0.181818 …. = 0.18 etc. A decimal number can be expressed in the form of a rational number. For example (i) 0.2 = 2 10 = 1 5(ii) 0.75 = 75 100 = 5 × 5 × 3 5 × 5 × 4 = 3 4(iii) 0.3 = 1 3(iv) 0.18 = 2 11 How ?

42 Acme Mathematics 8Solved ExamplesExample 1 : Convert 0.3 into rational number, Solution : Let, x = 0.3 ………….......(i) Then, 10x = 3.3 ………. (ii) [Multiplying both sides by 10] From (ii) and (i) on subtraction10x = 3.3– x = 0.39x = 3.0 or, x = 3 9 or, x = 1 3 ∴ 0.3 = 1 3Example 2 : Convert 0.18 into rational number, Solution : Let x = 0.18 ………… (i) 100x = 18.18 ………........(ii) [multiplying both sides by 100] Subtracting (i) from (ii), we have, 99x = 18 or, x = 18 99or, x = 2 11∴ 0.18 = 2 11Alternative method: (i) 0.18 = 18 – 0 99 = 18 99 = 2 11(ii) 0.183 = 183– 0 999 = 183 999 = 61 333(iii) 2.34 = 234 – 2 99 = 232 99(iv) 6.327 = 6327 – 63 990 = 6264 990 = 3132495Note: 6.327 is mixed recurring number and 3 is called anti-repetend.If number after decimal is one (i.e 0.3) multiply it by 10.If number after decimal is two(i.e 0.18) multiply it by 100.

Acme Mathematics 8 43Classwork1. Without actual division, find which of the following numbers represent a terminating decimal: (a) 3 11 (b) 17 10 (c) 7 30 (d) 41 45 (e) 1 1252. Convert the following rational numbers into decimals:(a) 17 200 (b) 52 125 (c) 2 5 (d) 135 180 (e) 3367 13000Exercise 2.41. Express as rational numbers: (a) 0.4 (b) 0.17 (c) 1.53 (d) 0.318 (e) 0.45(f) 0.3 (g) 0.8 (h) 0.5 (i) 0.12 (j) 0.15(k) 0.24 (l) 1.23 (m) 2.35 (n) 5.42 (o) 6.432. Write true or false for the following statements: (a) pq has a terminating decimal, if q is a prime. (b) Every rational number has a decimal representation. (c) Every decimal number, having a finite number of terms in the decimal part, is a rational number. 3. Express each of the following decimals in the form of a rational number in lowest terms: (a) 0.34 (b) 0.34444…. (c) 1.979797…(d) 0.212121… (e) 0.363636… (f) 0.123123123….(g) 1.23452345… (h) 0.132132132...4. Insert 3 rational numbers in between: (a) 1 4 and 1 7 (b) – 3 8 and – 23 (c) – 1 2 and 1 2(d) 5 7 and – 3 9 (e) 11 6 and 13 18

44 Acme Mathematics 8F. Irrational NumbersWe know that rational numbers can be represented in the form of p q , where q ≠ 0. Therefore, irrational numbers cannot be represented in the form of p q, where q ≠ 0, and the number whose exact roots cannot be found.For example, 2, 3, 5 etc are all irrational numbers.The numbers which are non-terminating and non-recurring decimals likee = 2.7182818284…, π = 3.141592653589…… are also irrational numbers.a. Representation of Irrational Numbers on a Number Line 2 on a number line: Draw a number line X'OX and take OA = 1 unit. Draw a right angle OAB. Put your compass at A mark an arc of 1 unit which cuts the perpendicular at B. With O as centre, join O and B. Here, OA = AB = 1 unit According to Pythagoras' theoremOB2 = OA2 + AB2or, OB = OA2 + AB2or, OB = 12 + 12or, OB = 2With O as centre and OB as radius, mark an arc which cuts the line segment XX' at point C. The length OC represents 2. Similarly, we can represent 3, 5, 6, etc. on a number line.b. Properties of Irrational Numbers 1. The sum of two irrational numbers is not always an irrational number.2. The product of two irrational numbers is not always an irrational number. 3. The four operations of addition, subtraction, multiplication and division hold good for irrational numbers as rational numbers. X'–1 0 1OBA C2 3X22

Acme Mathematics 8 45c. Real Numbers The set of numbers that includes integers, rational and irrational numbers is called the set of real numbers. The real numbers verify the properties of rational and irrational numbers. The set of real numbers is denoted by R. The set R can be represented by a Venn-diagram as given alongside.Classwork1. Tick () the correct answer in the following.(a) Every irrational number is(i) a natural number (ii) an integers(iii) a real number (iv) a whole number(b) Between two irrational numbers(i) there is no rational number(ii) there is exactly one rational number(iii) there are infinitely numbers(iv) there are only rational numbers and no irrational numbers(c) Decimal representation of a irrational number cannot be(i) terminating (ii) non-terminating(iii) non-terminating repeating (iv) non-terminating non-repeating(d) The product of any two irrational numbers is(i) always an irrational number(ii) always a rational number(iii) always an integer(iv) sometimes rational, sometime irrational(e) Which of the following is irrational?(i) 49 (ii) 123 (iii) 7 (iv) 81(f) The decimal expansion of irrational number 2 is(i) a finite decimal (ii) 1.41421(iii) non-terminating recurring (iv) non-terminating non-recuringRQZWN1, 2, 3, ...04– 1– 2– 3 – 40.614 7 – 35

46 Acme Mathematics 8(g) Which of the following is irrational?(i) 0.14 (ii) 0.1416(iii) 0.1416 (iv) 0.401401400014...(h) A irrational number between 2 and 3 is .....(i) 2 + 32 (ii) 2 × 33 (iii) 1.5 (iv) 1.8(i) 2 3 + 3 is equal to .....(i) 2 6 (ii) 6 (iii) 3 3 (iv) 4 6(j) 10 × 15 is equal to(i) 6 5 (ii) 5 6 (iii) 25 (iv) 10 5(k) The number obtained on rationalising the denominator of 17 – 2 is(i) 7 + 23 (ii) 7 – 23 (iii) 7 + 25 (iv) 7 + 245Exercise 2.51. Identify the irrational numbers.(a) 11 (b) 17 (c) 2 (d) 22 14 (e) 27 3 (f) 4 (g) 7 (h) 144 (i) – 3 7 (j) 1232. Locate the following irrational numbers in a number line: (a) – 2 (b) – 3 (c) 3 (d) 2 (e) 53. Taking (2 + 3) and (2 – 3) as two numbers verify the addition and multiplication properties of irrational numbers.4. Taking (4 + 5 ) and (4 – 5) as two numbers verify the addition and multiplication properties of irrational numbers.

Acme Mathematics 8 472.2 Scientific NotationSometimes we have to work with very large numbers and sometimes very small numbers. For example: The population of the earth, the weight of the earth, the weight of electron, etc. are such numbers. The weight of the earth is approximately 6,600,000,000,000,000,000,000 tonnes; it is difficult to remember, isn't it? We can express it as 6.6 × 1021 tonnes. Similarly the population of earth is about 8,000,000,000 which can be written as 8 × 109 and the weight of electron is 0.00,000,000,000,000,000,000,000,000,000,091 gram and can be written as 9.1 × 10–31. Thus, the large numbers expressed in the form of a × 10n where 1 ≤ a < 10 and n ∈ Z is called the numbers in scientific notation or numbers in standard form.Solved ExamplesExample 1 : Express 43000000000000 into scientific notation.Solution : Since 43000000000000 = 43 × 1000000000000= 43 × 1012= 4.3 × 1013∴ 43000000000000 = 4.3 × 1013Example 2 : The distance of the moon from the earth is 3.8 × 108 m. What is the distance in normal form? Solution : Since 3.8 × 108 = 3.8 × 100000000 m = 3810 × 100000000 m = 380000000 m = 3800000001000 km= 380000 km ∴ The distance is 380,000 kmClasswork1. Study the term a × 10b and fill in the blanks.(a) The absolute value of ‘a’ is ……………………….(b) 10 is called……………………(c) ‘b’ is exponent, it must be ……..

48 Acme Mathematics 82. Tick () the correct answer in the following.(a) Scientific notation for 7000 is …….(i) 1 × 73 (ii) 7 × 03(iii) 7 × 103 (iv) all of all(b) Scientific notation for 0.0007 is …….(i) 0 × 105 (ii) 7 × 104(iii) 7 × 10– 4 (iv) none of all(c) Scientific notation for 490000000 is …….(i) 49 × 107 (ii) 4.9 × 108(iii) 49 × 10– 8 (iv) 490 × 106(d) Scientific notation for 1230000000 is …….(i) 1.23 × 109 (ii) 12.3 × 108(iii) 123 × 107 (iv) 0.123× 106(e) Scientific notation for 0.0000212 is …….(i) 212 × 10– 4 (ii) 2.12 × 104(iii) 21.2 × 10– 5 (iv) 0.212×105Exercise 2.61. Write the following numbers in the scientific notation:(a) 25000000 (b) 7000000000(c) 1410000000000 (d) 2730000000000000(e) 1100000000000000 (f) 314000000000000000(g) 732 Million (h) 789 Billion(i) 0.0000007 (j) 0.0000000000000009 (k) 0.00005 (l) 0.0000000182. Write the following in the usual form:(a) 2 × 105 (b) 4.03 × 108 (c) 3.8 × 104(d) 0.7 × 1015 (e) 2.5 × 10– 5 (f) 0.43 × 10-3 (g) 5.67 × 10– 10 (h) 15 × 108 (i) The estimated age of the earth is 1.8 × 1012 days.

Acme Mathematics 8 493. Re-write the following figures in standard form: (a) The distance of the sun from the moon is 150000000000 metres. (b) The area of Asia continent is 44580000 square km.(c) The distance of the Mars from the earth is 199670000000 metres.(d) One square mm = 11000000 square m. (e) One cubic mm = 11000000000 cubic m.(f) One square mm = 11000000000000 square km. (g) The distance of the Jupiter from the sun is 778000 million meter. 4. Add the following: (a) 1.7 × 104 + 0.7 × 104 (b) 107 + 105 (c) 2.7 × 10 – 9 + 1.3 × 10– 12 (d) 0.2 × 10– 8 + 0.3 × 10– 9(e) 2.1 × 108 + 3.2 × 107 (f) 4.5 × 104 + 2.5 × 1055. Subtract the following:(a) 3.14 × 105 – 3.1 × 105 (b) 107 – 105(c) 4 × 10– 9 – 4 × 10–12 (d) 5.67 × 105 – 4.56 × 105(e) 0.73 × 10– 10 – 0.7 × 10– 126. Simplify:(a) (5 × 107) + (2 × 107) + 3 × 107(b) (1.3 × 105 + 2.3 × 105) – (7 × 102 – 2.5 × 104)(c) (5 × 10– 3) (6 × 104)2 × 10– 10 (d) 1600000 0.00032 × 0.00004(e) 0.279 × 0.00000161728 × 0.000031 (f) (5.5 × 10– 4) (1.2 × 107)(3.3 × 10– 6) (2 × 104)2(g) (6 × 10– 3) (7 × 106)(3 × 10– 4) (14 × 105)

50 Acme Mathematics 81. Following are some numbers that Rita felt new to her.7.5. π227 , 2.333..., 5.16247 ..., 2 , 3.14(a) Identify irrational numbers from the list.(b) Show 2in a number line.(c) Why is 5an irrational number?(d) Insert four rational numbers between 114 and 113 .2. The road distance between Kathmandu and Pokhara is 204.8 km.(a) Convert this distance into millimetre (mm).(b) Express the distance in mm in scientific notation.(c) Subtract 3.2 × 10–9 from 2.8 × 10 –7.3. A water tank has capacity 2.4 × 103 litres of water.(a) Express capacity of tank in decimal value.(b) If 1.4 × 102 litres of water is used, how much decimal value of water is left in tank?(c) Convert remaining quantity of water in tank in quinary form.4. Answer the following questions(a) Define binary number system.(b) Write the digits used in quinary number system.(c) Write the digits used in decimal number system.(d) How many students are there in your class? Write it in binary number.5. There are (1234)5 students in Sunrise school and (11011101)2 students in Moonrise school.(a) Convert the number of students of Sunrise school in decimal number system(b) Convert the number of students of Moonrise school in decimal number system.(c) Find the sum of number of students of (a) and (b) in decimal number system and convert it into binary number.Mixed Exercise