Page 1 1.1 INTRODUCTION W e have learnt about whole numbers and integers in Class VI. We know that integers form a bigger collection of numbers which contains whole numbers and negative numbers. What other differences do you find between whole numbers and integers? In this chapter, we will study more about integers, their properties and operations. First of all, we will review and revise what we have done about integers in our previous class. 1.2 RECALL W e know how to represent integers on a number line. Some integers are marked on the number line given below . Can you write these marked integers in ascending order? The ascending order of these numbers is â€“ 5, â€“ 1, 3. Why did we choose â€“ 5 as the smallest number? Some points are marked with integers on the following number line. W rite these integers in descending order. The descending order of these integers is 14, 8, 3, ... The above number line has only a few integers filled. Write appropriate numbers at each dot. Chapter 1 Integers Page 2 1.1 INTRODUCTION W e have learnt about whole numbers and integers in Class VI. We know that integers form a bigger collection of numbers which contains whole numbers and negative numbers. What other differences do you find between whole numbers and integers? In this chapter, we will study more about integers, their properties and operations. First of all, we will review and revise what we have done about integers in our previous class. 1.2 RECALL W e know how to represent integers on a number line. Some integers are marked on the number line given below . Can you write these marked integers in ascending order? The ascending order of these numbers is â€“ 5, â€“ 1, 3. Why did we choose â€“ 5 as the smallest number? Some points are marked with integers on the following number line. W rite these integers in descending order. The descending order of these integers is 14, 8, 3, ... The above number line has only a few integers filled. Write appropriate numbers at each dot. Chapter 1 Integers MATHEMATICS 2 â€“3 â€“2 ABC D E F G H I J K L M N O TRY THESE 1. A number line representing integers is given below â€“3 and â€“2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O? 2. Arrange 7, â€“5, 4, 0 and â€“ 4 in ascending order and then mark them on a number line to check your answer. We have done addition and subtraction of integers in our previous class. Read the following statements. On a number line when we (i) add a positive integer, we move to the right. (ii) add a negative integer, we move to the left. (iii) subtract a positive integer, we move to the left. (iv) subtract a negative integer, we move to the right. State whether the following statements are correct or incorrect. Correct those which are wrong: (i) When two positive integers are added we get a positive integer. (ii) When two negative integers are added we get a positive integer. (iii) When a positive integer and a negative integer are added, we always get a negative integer. (iv) Additive inverse of an integer 8 is (â€“ 8) and additive inverse of (â€“ 8) is 8. (v) For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer. (vi) (â€“10) + 3 = 10 â€“ 3 (vii) 8 + (â€“7) â€“ (â€“ 4) = 8 + 7 â€“ 4 Compare your answers with the answers given below: (i) Correct. For example: (a) 56 + 73 = 129 (b) 113 + 82 = 195 etc. Construct five more examples in support of this statement. (ii) Incorrect, since (â€“ 6) + (â€“ 7) = â€“ 13, which is not a positive integer. The correct statement is: When two negative integers are added we get a negative integer. For example, (a) (â€“ 56) + (â€“ 73) = â€“ 129 (b) (â€“ 113) + (â€“ 82) = â€“ 195, etc. Construct five more examples on your own to verify this statement. Page 3 1.1 INTRODUCTION W e have learnt about whole numbers and integers in Class VI. We know that integers form a bigger collection of numbers which contains whole numbers and negative numbers. What other differences do you find between whole numbers and integers? In this chapter, we will study more about integers, their properties and operations. First of all, we will review and revise what we have done about integers in our previous class. 1.2 RECALL W e know how to represent integers on a number line. Some integers are marked on the number line given below . Can you write these marked integers in ascending order? The ascending order of these numbers is â€“ 5, â€“ 1, 3. Why did we choose â€“ 5 as the smallest number? Some points are marked with integers on the following number line. W rite these integers in descending order. The descending order of these integers is 14, 8, 3, ... The above number line has only a few integers filled. Write appropriate numbers at each dot. Chapter 1 Integers MATHEMATICS 2 â€“3 â€“2 ABC D E F G H I J K L M N O TRY THESE 1. A number line representing integers is given below â€“3 and â€“2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O? 2. Arrange 7, â€“5, 4, 0 and â€“ 4 in ascending order and then mark them on a number line to check your answer. We have done addition and subtraction of integers in our previous class. Read the following statements. On a number line when we (i) add a positive integer, we move to the right. (ii) add a negative integer, we move to the left. (iii) subtract a positive integer, we move to the left. (iv) subtract a negative integer, we move to the right. State whether the following statements are correct or incorrect. Correct those which are wrong: (i) When two positive integers are added we get a positive integer. (ii) When two negative integers are added we get a positive integer. (iii) When a positive integer and a negative integer are added, we always get a negative integer. (iv) Additive inverse of an integer 8 is (â€“ 8) and additive inverse of (â€“ 8) is 8. (v) For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer. (vi) (â€“10) + 3 = 10 â€“ 3 (vii) 8 + (â€“7) â€“ (â€“ 4) = 8 + 7 â€“ 4 Compare your answers with the answers given below: (i) Correct. For example: (a) 56 + 73 = 129 (b) 113 + 82 = 195 etc. Construct five more examples in support of this statement. (ii) Incorrect, since (â€“ 6) + (â€“ 7) = â€“ 13, which is not a positive integer. The correct statement is: When two negative integers are added we get a negative integer. For example, (a) (â€“ 56) + (â€“ 73) = â€“ 129 (b) (â€“ 113) + (â€“ 82) = â€“ 195, etc. Construct five more examples on your own to verify this statement. INTEGERS 3 TRY THESE (iii) Incorrect, since â€“ 9 + 16 = 7, which is not a negative integer . The correct statement is : When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer . The bigger integer is decided by ignoring the signs of both the integers. For example: (a) (â€“ 56) + (73) = 17 (b) (â€“ 113) + 82 = â€“ 31 (c) 16 + (â€“ 23) = â€“ 7 (d) 125 + (â€“ 101) = 24 Construct five more examples for verifying this statement. (iv) Correct. Some other examples of additive inverse are as given below: Integer Additive inverse 10 â€“10 â€“10 10 76 â€“76 â€“76 76 Thus, the additive inverse of any integer a is â€“ a and additive inverse of (â€“ a) is a. (v) Correct. Subtraction is opposite of addition and therefore, we add the additive inverse of the integer that is being subtracted, to the other integer. For example: (a) 56 â€“ 73 = 56 + additive inverse of 73 = 56 + (â€“73) = â€“17 (b) 56 â€“ (â€“73) = 56 + additive inverse of (â€“73) = 56 + 73 = 129 (c) (â€“79) â€“ 45 = (â€“79) + (â€“ 45) = â€“124 (d) (â€“100) â€“ (â€“172) = â€“100 + 172 = 72 etc. Write atleast five such examples to verify this statement. Thus, we find that for any two integers a and b, a â€“ b = a + additive inverse of b = a + (â€“ b) and a â€“ (â€“ b) = a + additive inverse of (â€“ b) = a + b (vi) Incorrect, since (â€“10) + 3 = â€“7 and 10 â€“ 3 = 7 therefore, (â€“10) + 3 ? 10 â€“ 3 (vii) Incorrect, since, 8 + (â€“7) â€“ (â€“ 4) = 8 + (â€“7) + 4 = 1 + 4 = 5 and 8 + 7 â€“ 4 = 15 â€“ 4 = 11 However, 8 + (â€“7) â€“ (â€“ 4) = 8 â€“ 7 + 4 W e have done various patterns with numbers in our previous class. Can you find a pattern for each of the following? If yes, complete them: (a) 7, 3, â€“ 1, â€“ 5, _____, _____, _____. (b) â€“ 2, â€“ 4, â€“ 6, â€“ 8, _____, _____, _____. (c) 15, 10, 5, 0, _____, _____, _____. (d) â€“ 11, â€“ 8, â€“ 5, â€“ 2, _____, _____, _____. Make some more such patterns and ask your friends to complete them. Page 4 1.1 INTRODUCTION W e have learnt about whole numbers and integers in Class VI. We know that integers form a bigger collection of numbers which contains whole numbers and negative numbers. What other differences do you find between whole numbers and integers? In this chapter, we will study more about integers, their properties and operations. First of all, we will review and revise what we have done about integers in our previous class. 1.2 RECALL W e know how to represent integers on a number line. Some integers are marked on the number line given below . Can you write these marked integers in ascending order? The ascending order of these numbers is â€“ 5, â€“ 1, 3. Why did we choose â€“ 5 as the smallest number? Some points are marked with integers on the following number line. W rite these integers in descending order. The descending order of these integers is 14, 8, 3, ... The above number line has only a few integers filled. Write appropriate numbers at each dot. Chapter 1 Integers MATHEMATICS 2 â€“3 â€“2 ABC D E F G H I J K L M N O TRY THESE 1. A number line representing integers is given below â€“3 and â€“2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O? 2. Arrange 7, â€“5, 4, 0 and â€“ 4 in ascending order and then mark them on a number line to check your answer. We have done addition and subtraction of integers in our previous class. Read the following statements. On a number line when we (i) add a positive integer, we move to the right. (ii) add a negative integer, we move to the left. (iii) subtract a positive integer, we move to the left. (iv) subtract a negative integer, we move to the right. State whether the following statements are correct or incorrect. Correct those which are wrong: (i) When two positive integers are added we get a positive integer. (ii) When two negative integers are added we get a positive integer. (iii) When a positive integer and a negative integer are added, we always get a negative integer. (iv) Additive inverse of an integer 8 is (â€“ 8) and additive inverse of (â€“ 8) is 8. (v) For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer. (vi) (â€“10) + 3 = 10 â€“ 3 (vii) 8 + (â€“7) â€“ (â€“ 4) = 8 + 7 â€“ 4 Compare your answers with the answers given below: (i) Correct. For example: (a) 56 + 73 = 129 (b) 113 + 82 = 195 etc. Construct five more examples in support of this statement. (ii) Incorrect, since (â€“ 6) + (â€“ 7) = â€“ 13, which is not a positive integer. The correct statement is: When two negative integers are added we get a negative integer. For example, (a) (â€“ 56) + (â€“ 73) = â€“ 129 (b) (â€“ 113) + (â€“ 82) = â€“ 195, etc. Construct five more examples on your own to verify this statement. INTEGERS 3 TRY THESE (iii) Incorrect, since â€“ 9 + 16 = 7, which is not a negative integer . The correct statement is : When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer . The bigger integer is decided by ignoring the signs of both the integers. For example: (a) (â€“ 56) + (73) = 17 (b) (â€“ 113) + 82 = â€“ 31 (c) 16 + (â€“ 23) = â€“ 7 (d) 125 + (â€“ 101) = 24 Construct five more examples for verifying this statement. (iv) Correct. Some other examples of additive inverse are as given below: Integer Additive inverse 10 â€“10 â€“10 10 76 â€“76 â€“76 76 Thus, the additive inverse of any integer a is â€“ a and additive inverse of (â€“ a) is a. (v) Correct. Subtraction is opposite of addition and therefore, we add the additive inverse of the integer that is being subtracted, to the other integer. For example: (a) 56 â€“ 73 = 56 + additive inverse of 73 = 56 + (â€“73) = â€“17 (b) 56 â€“ (â€“73) = 56 + additive inverse of (â€“73) = 56 + 73 = 129 (c) (â€“79) â€“ 45 = (â€“79) + (â€“ 45) = â€“124 (d) (â€“100) â€“ (â€“172) = â€“100 + 172 = 72 etc. Write atleast five such examples to verify this statement. Thus, we find that for any two integers a and b, a â€“ b = a + additive inverse of b = a + (â€“ b) and a â€“ (â€“ b) = a + additive inverse of (â€“ b) = a + b (vi) Incorrect, since (â€“10) + 3 = â€“7 and 10 â€“ 3 = 7 therefore, (â€“10) + 3 ? 10 â€“ 3 (vii) Incorrect, since, 8 + (â€“7) â€“ (â€“ 4) = 8 + (â€“7) + 4 = 1 + 4 = 5 and 8 + 7 â€“ 4 = 15 â€“ 4 = 11 However, 8 + (â€“7) â€“ (â€“ 4) = 8 â€“ 7 + 4 W e have done various patterns with numbers in our previous class. Can you find a pattern for each of the following? If yes, complete them: (a) 7, 3, â€“ 1, â€“ 5, _____, _____, _____. (b) â€“ 2, â€“ 4, â€“ 6, â€“ 8, _____, _____, _____. (c) 15, 10, 5, 0, _____, _____, _____. (d) â€“ 11, â€“ 8, â€“ 5, â€“ 2, _____, _____, _____. Make some more such patterns and ask your friends to complete them. MATHEMATICS 4 â€“10 â€“5 0 5 10 15 20 25 Lahulspiti Srinagar Shimla Ooty Bangalore EXERCISE 1.1 1. Following number line shows the temperature in degree celsius (°C) at different places on a particular day. (a) Observe this number line and write the temperature of the places marked on it. (b) What is the temperature difference between the hottest and the coldest places among the above? (c) What is the temperature difference between Lahulspiti and Srinagar? (d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar? 2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jackâ€™s scores in five successive rounds were 25, â€“ 5, â€“ 10, 15 and 10, what was his total at the end? 3. At Srinagar temperature was â€“ 5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day? 4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them? 5. Mohan deposits Rs 2,000 in his bank account and withdraws Rs 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohanâ€™s account after the withdrawal. 6. Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from A? Page 5 1.1 INTRODUCTION W e have learnt about whole numbers and integers in Class VI. We know that integers form a bigger collection of numbers which contains whole numbers and negative numbers. What other differences do you find between whole numbers and integers? In this chapter, we will study more about integers, their properties and operations. First of all, we will review and revise what we have done about integers in our previous class. 1.2 RECALL W e know how to represent integers on a number line. Some integers are marked on the number line given below . Can you write these marked integers in ascending order? The ascending order of these numbers is â€“ 5, â€“ 1, 3. Why did we choose â€“ 5 as the smallest number? Some points are marked with integers on the following number line. W rite these integers in descending order. The descending order of these integers is 14, 8, 3, ... The above number line has only a few integers filled. Write appropriate numbers at each dot. Chapter 1 Integers MATHEMATICS 2 â€“3 â€“2 ABC D E F G H I J K L M N O TRY THESE 1. A number line representing integers is given below â€“3 and â€“2 are marked by E and F respectively. Which integers are marked by B, D, H, J, M and O? 2. Arrange 7, â€“5, 4, 0 and â€“ 4 in ascending order and then mark them on a number line to check your answer. We have done addition and subtraction of integers in our previous class. Read the following statements. On a number line when we (i) add a positive integer, we move to the right. (ii) add a negative integer, we move to the left. (iii) subtract a positive integer, we move to the left. (iv) subtract a negative integer, we move to the right. State whether the following statements are correct or incorrect. Correct those which are wrong: (i) When two positive integers are added we get a positive integer. (ii) When two negative integers are added we get a positive integer. (iii) When a positive integer and a negative integer are added, we always get a negative integer. (iv) Additive inverse of an integer 8 is (â€“ 8) and additive inverse of (â€“ 8) is 8. (v) For subtraction, we add the additive inverse of the integer that is being subtracted, to the other integer. (vi) (â€“10) + 3 = 10 â€“ 3 (vii) 8 + (â€“7) â€“ (â€“ 4) = 8 + 7 â€“ 4 Compare your answers with the answers given below: (i) Correct. For example: (a) 56 + 73 = 129 (b) 113 + 82 = 195 etc. Construct five more examples in support of this statement. (ii) Incorrect, since (â€“ 6) + (â€“ 7) = â€“ 13, which is not a positive integer. The correct statement is: When two negative integers are added we get a negative integer. For example, (a) (â€“ 56) + (â€“ 73) = â€“ 129 (b) (â€“ 113) + (â€“ 82) = â€“ 195, etc. Construct five more examples on your own to verify this statement. INTEGERS 3 TRY THESE (iii) Incorrect, since â€“ 9 + 16 = 7, which is not a negative integer . The correct statement is : When one positive and one negative integers are added, we take their difference and place the sign of the bigger integer . The bigger integer is decided by ignoring the signs of both the integers. For example: (a) (â€“ 56) + (73) = 17 (b) (â€“ 113) + 82 = â€“ 31 (c) 16 + (â€“ 23) = â€“ 7 (d) 125 + (â€“ 101) = 24 Construct five more examples for verifying this statement. (iv) Correct. Some other examples of additive inverse are as given below: Integer Additive inverse 10 â€“10 â€“10 10 76 â€“76 â€“76 76 Thus, the additive inverse of any integer a is â€“ a and additive inverse of (â€“ a) is a. (v) Correct. Subtraction is opposite of addition and therefore, we add the additive inverse of the integer that is being subtracted, to the other integer. For example: (a) 56 â€“ 73 = 56 + additive inverse of 73 = 56 + (â€“73) = â€“17 (b) 56 â€“ (â€“73) = 56 + additive inverse of (â€“73) = 56 + 73 = 129 (c) (â€“79) â€“ 45 = (â€“79) + (â€“ 45) = â€“124 (d) (â€“100) â€“ (â€“172) = â€“100 + 172 = 72 etc. Write atleast five such examples to verify this statement. Thus, we find that for any two integers a and b, a â€“ b = a + additive inverse of b = a + (â€“ b) and a â€“ (â€“ b) = a + additive inverse of (â€“ b) = a + b (vi) Incorrect, since (â€“10) + 3 = â€“7 and 10 â€“ 3 = 7 therefore, (â€“10) + 3 ? 10 â€“ 3 (vii) Incorrect, since, 8 + (â€“7) â€“ (â€“ 4) = 8 + (â€“7) + 4 = 1 + 4 = 5 and 8 + 7 â€“ 4 = 15 â€“ 4 = 11 However, 8 + (â€“7) â€“ (â€“ 4) = 8 â€“ 7 + 4 W e have done various patterns with numbers in our previous class. Can you find a pattern for each of the following? If yes, complete them: (a) 7, 3, â€“ 1, â€“ 5, _____, _____, _____. (b) â€“ 2, â€“ 4, â€“ 6, â€“ 8, _____, _____, _____. (c) 15, 10, 5, 0, _____, _____, _____. (d) â€“ 11, â€“ 8, â€“ 5, â€“ 2, _____, _____, _____. Make some more such patterns and ask your friends to complete them. MATHEMATICS 4 â€“10 â€“5 0 5 10 15 20 25 Lahulspiti Srinagar Shimla Ooty Bangalore EXERCISE 1.1 1. Following number line shows the temperature in degree celsius (°C) at different places on a particular day. (a) Observe this number line and write the temperature of the places marked on it. (b) What is the temperature difference between the hottest and the coldest places among the above? (c) What is the temperature difference between Lahulspiti and Srinagar? (d) Can we say temperature of Srinagar and Shimla taken together is less than the temperature at Shimla? Is it also less than the temperature at Srinagar? 2. In a quiz, positive marks are given for correct answers and negative marks are given for incorrect answers. If Jackâ€™s scores in five successive rounds were 25, â€“ 5, â€“ 10, 15 and 10, what was his total at the end? 3. At Srinagar temperature was â€“ 5°C on Monday and then it dropped by 2°C on Tuesday. What was the temperature of Srinagar on Tuesday? On Wednesday, it rose by 4°C. What was the temperature on this day? 4. A plane is flying at the height of 5000 m above the sea level. At a particular point, it is exactly above a submarine floating 1200 m below the sea level. What is the vertical distance between them? 5. Mohan deposits Rs 2,000 in his bank account and withdraws Rs 1,642 from it, the next day. If withdrawal of amount from the account is represented by a negative integer, then how will you represent the amount deposited? Find the balance in Mohanâ€™s account after the withdrawal. 6. Rita goes 20 km towards east from a point A to the point B. From B, she moves 30 km towards west along the same road. If the distance towards east is represented by a positive integer then, how will you represent the distance travelled towards west? By which integer will you represent her final position from A? INTEGERS 5 7. In a magic square each row, column and diagonal have the same sum. Check which of the following is a magic square. 5 â€“1 â€“ 4 1 â€“10 0 â€“5 â€“2 7 â€“ 4 â€“3 â€“2 0 3 â€“3 â€“ 6 4 â€“7 (i) (ii) 8. V erify a â€“ (â€“ b) = a + b for the following values of a and b. (i) a = 21, b = 18 (ii) a = 118, b = 125 (iii) a = 75, b = 84 (iv) a = 28, b = 11 9. Use the sign of >, < or = in the box to make the statements true. (a) (â€“ 8) + (â€“ 4) (â€“8) â€“ (â€“ 4) (b) (â€“ 3) + 7 â€“ (19) 15 â€“ 8 + (â€“ 9) (c) 23 â€“ 41 + 11 23 â€“ 41 â€“ 11 (d) 39 + (â€“ 24) â€“ (15) 36 + (â€“ 52) â€“ (â€“ 36) (e) â€“ 231 + 79 + 51 â€“399 + 159 + 81 10. A water tank has steps inside it. A monkey is sitting on the topmost step (i.e., the first step). The water level is at the ninth step. (i) He jumps 3 steps down and then jumps back 2 steps up. In how many jumps will he reach the water level? (ii) After drinking water, he wants to go back. For this, he jumps 4 steps up and then jumps back 2 steps down in every move. In how many jumps will he reach back the top step? (iii) If the number of steps moved down is represented by negative integers and the number of steps moved up by positive integers, represent his moves in part (i) and (ii) by completing the following; (a) â€“ 3 + 2 â€“ ... = â€“ 8 (b) 4 â€“ 2 + ... = 8. In (a) the sum (â€“ 8) represents going down by eight steps. So, what will the sum 8 in (b) represent? 1.3 PROPERTIES OF ADDITION AND SUBTRACTION OF INTEGERS 1.3.1 Closure under Addition W e have learnt that sum of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is again a whole number. W e know that, this property is known as the closure property for addition of the whole numbers.Read More

211 videos|109 docs|45 tests

### Introduction to Integers

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### Addition and Subtraction of Integers using Number Line

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### NCERT Solutions(Part - 1) - Integers

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### NCERT Solutions(Part - 2) - Integers

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### Closure Property of Integers under Addition and Subtraction

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### Worksheet Questions - Integers

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- Test: Integers - 1
- Test | 10 ques | 10 min