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1 BOOK 6 ( CBSE ) / Chap 6 / Integers. Chapter Facts : 1. The collection consisting of natural numbers, zero and negatives of natural numbers is called integers [..., 4, 3, 2, 1, 0, 1, 2, 3, 4,... ] 1, 2, 3, 4,... are negative integers 0 is neither positive nor negative. 1, 2, 3, 4,... are positive integers. 2. The collection : 0, 1, 2, 3,.. is called the collection of non-negative integers.. This is same as collection of whole numbers. 3. Integers can be represented on a number line. All integers to right of 0 are positive integers and those to left of 0 are negative integers. 4. Collection of integers is ordered. This means it is possible to compare any two integers. For any two integers a and b we can say if a > b, a = b or a < b. If two integers are marked on a number line the one on the left is smaller and the one on the right is greater. 5. Every integer has a successor. This can be obtained by adding 1 to it. 6. Every integer has a predecessor. This can be obtained by subtracting 1 from it. 7. When two positive integers are added, we simply add them and take the sum as positive. We get a positive integer. [e.g. (+ 3) + ( + 2) = + 5 ] 8. When two negative integers are added we add them as whole numbers by considering the numbers without their sign and then put the minus sign with the sum obtained. We get a negative integer [e.g. ( 2) + ( 3) = 5]. 9. When a positive and a negative integers are added we subtract them as whole numbers by considering the numbers without their sign and then put the sign of the bigger number with the subtraction obtained. The bigger integer is decided by ignoring the signs of the integers. [e.g. (+4) + ( 3) = + 1 and ( 4) + ( + 3) = 1]. 10. Addition and subtraction of integers can also be shown on a number line. 11. Two integers whose sum is zero are called additive inverses of ach other. A number and its opposite are additive inverses of each other. 0 is its own additive inverse. 12. The numerical value of an integer regardless of its sign is called its absolute value. It is shown symbolically by writing the number between. Absolute value of 3 is written as 3. Also 3 = 3 Copyright with the author. 1

2 Absolute value of 3 is written as - 3. Also - 3 = Absolute value of a positive integer is same as itself. 5 = Absolute value of a negative number is equal to its opposite. - 5 = Absolute value of 0 is 0. 0 = For each natural number we have two integers whose absolute value is equal to that natural number. 1 is absolute value of 1 and 1 ; 2 is absolute value of 2 and 2 etc. Formative Assessment A ) Multiple Choice Questions. In each of the following tick the correct option. 1. The greatest integer lying between 10 and 15 is (a) 11 (b) 13 (c) 9 (d) The integer with negative sign ( ) is always less than (a) 0 (b) 3 (c) 1 (d) 2 3. The statement When an integer is added to itself, the sum is greater than the integer (a) is always true (b) is never true (c) may be true or false (d) is true for non-negative integers only. 4. In which of the following pairs of integers, the first integer is not on the left of the other integer on the number line? (a) ( 1, 10) (b) ( 3, 5) (c) ( 5, 3) (d) ( 6, 0) 5. Which of the following shows the maximum rise in temperature? (a) 0 C to 10 C (b) 4 C to 8 C (c) 15 C to 8 C (d) 7 C to 0 C 6. The integer 4 units to the left of 0 on the number line is (a) +5 (b) 5 (c) +4 (d) 4 7. On the number line, the integer 3 is located (a) to the left of 0 and to right of 2 (b) to the right of 4 and to left of -1. (c) to the left of 1 but right of -3. (d) to the left of 4 and to right of The opposite of the successor of the predecessor of 50 is (a) 48 (b) 50 (c) 50 (d) Number of natural numbers lying between 6 and 6 is (a) 10 (b) 12 (c) 11 (d) 5 Copyright with the author. 2

3 10. The opposite of a negative integer (a) is always negative (b) is always positive (c) may be positive or negative (d) is zero. B ) Fill in the blanks 1. 1, 2, 3, 4,... are integers. 2. On a number line all integers to of 0 are positive integers is than every negative integer. ( greater / smaller ) 4. of a number can be obtained by moving 1 unit to the left on the number line. 5. The subtraction of an integer is the same as the addition of its. 6. Absolute value of 11 is. 7. sum of two positive integers is always, ( positive / negative ) 8. ( 6) + = 0 Fill in the blanks with >, < or = sign.( 16-17) 9. ( 3) + ( 6) ( 3) ( 6) ( 11) 57 + ( 4) C ) True / False State whether the given statements are true (T) or false (F). State whether the given statements are true (T) or false (F) is positive as well as negative. T / F 2. Every positive integer is greater than 0. T / F 3. Only positive integers have a successor. T / F 4. Exactly two non-zero integers have same absolute value. T / F 5. Sum of an integer and its negative is always zero. T / F D ) True false with Justification Say true or false for each of the following. Justify your answer 1. Collection of integers is ordered. T / F 2. Sum of a positive integer and a negative integer is always positive. T / F Copyright with the author. 3

4 E ) Matching Match the following : 1. Match each item of Column I with the corresponding item of Column II. Column I Column II (i) The opposite of - 2 (A) 0 (ii) The greatest negative integer (B) 2 (iii) The smallest positive integer (C) 2 (iv) The smallest integer greater than every negative integer (D) 1 (v) Sum of predecessor and successor of 1 (E) 1 Summative Assessment 1 mark Q s 1. Write two negative integers greater than Write two negative integers less than Write the integer which is 2 more than its additive inverse. 4. Write the integer which is 4 less than its additive inverse. 5. Write two integers whose sum is less than both the integers. 6. Write two distinct integers whose sum is equal to one of the integers. 7. Write two integers whose sum is 8 and difference is Write two integers which are less than 100 but greater than If we are at 8 on the number line, in which direction should we move to reach 0? 10. If we are at 1 on the number line, in which direction should we move to reach the integer 5? 2 mark Q s 1. Indicate using integers with appropriate sign. a) 3 C above 0 b) a withdrawal of Rs If we denote the height of a place above sea level by a positive integer and depth below the sea level by a negative integer, write the following using integers with the appropriate signs: a) 100 m above sea level b) 40 m below sea level c) at sea level Copyright with the author. 4

5 3. Give the successor of the following; a) 216 b) 418 c) 0 4. Give the predecessor of the following; a) 29 b) 48 c) 0 5. Give the opposites of a) 10 b) c) 0 6. Encircle the larger integer in each pair. a) 21 and 32 b) 27 and 27 c) 0 and 65 d) 72 and 0 7. Encircle the smaller integer in each pair. a) 43 and 42 b) 23 and 52 c) 0 and 43 d) - 45 and Write a pairs of integers which are at the same distance from 3 on the number line. 9. Compute each of the following a) 35 (20) b) ( 15) + ( 18) c) ( 20) (13) d) 23 + ( 12) 10. Compute each of the following: a) 0 + ( 5) + ( 1) b) 0 ( 4) (+4) c) 0 3 ( 3) d) 20 + ( 40) From the sum of 17 and 12 subtract To the difference of 76 and 100 add Subtract 4008 from the sum [( 1100) + ( 1001)] 14. Give the absolute value of a) 722 b) 715 c) Give the numbers whose absolute value is a) 271 b) mark Q s 1. Write the opposite of each of the following: a) Decrease in size b) Rise in water level c) Failure d) Winning by a margin of 2000 votes e) 20 C rise in temperature f) 10 m above the danger mark of river Ganga 2. Compute each of the following: a) ( 380) + ( 270) b) ( 7) ( 8) + ( 90) c) 50 ( 40) ( 2) 3. From the sum of 72 and -19 subtract the sum of -26 and -13. Copyright with the author. 5

6 4. Temperature of a place at 12:00 noon was +5 C. Temperature increased by 3 C in first hour and decreased by 1 C in the second hour. What was the temperature at 2:00 pm? 5. An aircraft was flying at a height of 4000 feet.. The pilot decreased the height by 500 feet. Then he increased the height by 700 feet. At what height is the plane flying now? 6. Point A is 1500 m below sea level and point B is 700 m above sea level. Find the vertical distance between A and B. 7. Plato was born in 429 BC and died in 348 BC. How long did he live? 8. Hypatia an Egyptian mathematician, was born in 20 BC and died in 53 AD. How long did she live? 9. Find the value of a) 2 4 b) Find the value of a) b) mark Q s 1. Compute each of the following: a) 30 + ( 25) + ( 10) b) 70 + ( 20) + ( 30) c) 1 + ( 2) + ( 3) + ( 4) d) 49 ( 40) ( 3) Adjacent figure is a vertical number line, representing integers. Observe it and locate the following points : a) If point A is + 4, then which point is 4? b) Write integers for points B and E. c) Which point marked on this number line has the least value? d) Arrange all the points in increasing order of value. 3. Following is the list of temperatures of five places in India on a particular day of the year. Srinagar : 7 C below 0 C ; Shimla 2 C below 0 C ; Ambala 3 C above 0 C ; Delhi : 8 C above 0 C ; Manali 5 C below 0 C ; (a) Write the temperatures of these places in the form of integers using appropriate sign. Copyright with the author. 6

7 (b) Plot the name of the city against its temperature on the number line representing the temperature in degree Celsius. 4. Using a number line and answer the following : (a) Which number will we reach if we move 4 units to the right of 2? (b) Which number will we reach if we move 5 units to the left of 1? (c) If we are at 1 on the number line, in which direction should we move to reach 7? (d) If we are at 6 on the number line, in which direction should we move to reach 1? 5. Using number line determine. a) ( 2) + ( 3) + ( 1) b) ( 2) ( 4) ( + 2 ) End.. ANSWERS Formative MCQ s 1 a 2 a 3 c 4 b 5 b 6 d 7 b 8 b 9 d 10 b Fill in the blanks 1 negative 2 left 3 greater 4 predecessor 5 negative positive = 10 > True False 1 F 2 T 3 F 4 T 5 T True False with Justification. 1 T. It is possible to compare any two integers. 2 F. (+ 4) + ( 5 ) = 1, which is negative. Matching (i) C, (ii) E, ( iii) D, (iv) A, (v) B Summative Copyright with the author. 7

8 One mark questions 1 9, , , 2 6 2, 0 7 8, , right 10 left Two marks questions 1 a) +5 o C b) Rs a) + 100m b) 40 m c) 0 m 3 a) 215 b) 419 c) 0 4 a) 28 b) 49 c) 1 5 a) 10 b) 200 c) 0 6 a) 21 b) 27 c) 0 d) 72 7 a) 43 b) 52 c) 0 d) and 6 9 a) 15 b) 33 c) 33 d) a) 6 b) 0 c) 0 d) a) 722 b) 715 c) 0 15 a) 271 and 271 b) 432 and 432 Three marks questions 1 a) Increase in size b) Fall in water level c) Success d) Losing by a margin of 2000 votes e) 20 C fall in temperature f) 10 m below the danger mark 2 a) 350 b) 89 c) o C feet m 7 81 years 8 73 years 9 a) 2 b) a) 4 b) 13 Four marks questions 1 a) 5 b) 20 c) 8 d) a) C b) +1 and 2 c) D d) O, C, E, B. A 3 a) Srinagar : 7 C; Shimla : 2 C Ambala : 3 C above 0 C ; Delhi : 8 C Manali 5 C Copyright with the author. 8

9 b) 4 a) +2 b) 4 c) left d0 right 5 a) 6 b) + 2 Copyright with the author. 9