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Chapter 1 Integers 1.1 PROPER TIES OF ADDITION AND SUBTRACTION OF INTEGERS We have learnt about whole numbers and integers in Class VI. We have also learnt about additi...

Chapter 1 Integers 1.1 PROPER TIES OF ADDITION AND SUBTRACTION OF INTEGERS We have learnt about whole numbers and integers in Class VI. We have also learnt about addition and subtraction of integers. 1.1.1 Closure under Addition We have learnt that sum of two whole numbers is again a whole number. For example, 17 + 24 = 41 which is again a whole number. We know that, this property is known as the closure property for addition of the whole numbers. Let us see whether this property is true for integers or not. Following are some pairs of integers. Observe the following table and complete it. Statement Observation (i) 17 + 23 = 40 Result is an integer (ii) (–10) + 3 = _____ ______________ (iii) (– 75) + 18 = _____ ______________ (iv) 19 + (– 25) = – 6 Result is an integer (v) 27 + (– 27) = _____ ______________ (vi) (– 20) + 0 = _____ ______________ (vii) (– 35) + (– 10) = _____ ______________ What do you observe? Is the sum of two integers always an integer? Did you find a pair of integers whose sum is not an integer? Since addition of integers gives integers, we say integers are closed under addition. In general, for any two integers a and b, a + b is an integer. 2024-25 2 MATHEMATICS 1.1.2 Closure under Subtraction What happens when we subtract an integer from another integer? Can we say that their difference is also an integer? Observe the following table and complete it: Statement Observation (i) 7 – 9 = – 2 Result is an integer (ii) 17 – (– 21) = _______ ______________ (iii) (– 8) – (–14) = 6 Result is an integer (iv) (– 21) – (– 10) = _______ ______________ (v) 32 – (–17) = _______ ______________ (vi) (– 18) – (– 18) = _______ ______________ (vii) (– 29) – 0 = _______ ______________ What do you observe? Is there any pair of integers whose difference is not an integer? Can we say integers are closed under subtraction? Yes, we can see that integers are closed under subtraction. Thus, if a and b are two integers then a – b is also an intger. Do the whole numbers satisfy this property? 1.1.3 Commutative Property We know that 3 + 5 = 5 + 3 = 8, that is, the whole numbers can be added in any order. In other words, addition is commutative for whole numbers. Can we say the same for integers also? We have 5 + (– 6) = –1 and (– 6) + 5 = –1 So, 5 + (– 6) = (– 6) + 5 Are the following equal? (i) (– 8) + (– 9) and (– 9) + (– 8) (ii) (– 23) + 32 and 32 + (– 23) (iii) (– 45) + 0 and 0 + (– 45) Try this with five other pairs of integers. Do you find any pair of integers for which the sums are different when the order is changed? Certainly not. We say that addition is commutative for integers. 2024-25  INTEGERS 3 In general, for any two integers a and b, we can say a+b=b+a l We know that subtraction is not commutative for whole numbers. Is it commutative for integers? Consider the integers 5 and (–3). Is 5 – (–3) the same as (–3) –5? No, because 5 – ( –3) = 5 + 3 = 8, and (–3) – 5 = – 3 – 5 = – 8. Take atleast five different pairs of integers and check this. We conclude that subtraction is not commutative for integers. 1.1.4 Associative Property Observe the following examples: Consider the integers –3, –2 and –5. Look at (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2). In the first sum (–3) and (–2) are grouped together and in the second (–5) and (–3) are grouped together. We will check whether we get different results. (–5) + [(–3) + (–2)] [(–5) + (–3)] + (–2) In both the cases, we get –10. i.e., (–5) + [(–3) + (–2)] = [(–5) + (–2)] + (–3) Similarly consider –3 , 1 and –7. ( –3) + [1 + (–7)] = –3 + __________ = __________ [(–3) + 1] + (–7) = –2 + __________ = __________ Is (–3) + [1 + (–7)] same as [(–3) + 1] + (–7)? Take five more such examples. You will not find any example for which the sums are different. Addition is associative for integers. In general for any integers a, b and c, we can say a + (b + c) = (a + b) + c 2024-25  4 MATHEMATICS 1.1.5 Additive Identity When we add zero to any whole number, we get the same whole number. Zero is an additive identity for whole numbers. Is it an additive identity again for integers also? Observe the following and fill in the blanks: (i) (– 8) + 0 = – 8 (ii) 0 + (– 8) = – 8 (iii) (–23) + 0 = _____ (iv) 0 + (–37) = –37 (v) 0 + (–59) = _____ (vi) 0 + _____ = – 43 (vii) – 61 + _____ = – 61 (viii) _____ + 0 = _____ The above examples show that zero is an additive identity for integers. You can verify it by adding zero to any other five integers. In general, for any integer a a+0=a=0+a TRY THESE 1. Write a pair of integers whose sum gives (a) a negative integer (b) zero (c) an integer smaller than both the integers. (d) an integer smaller than only one of the integers. (e) an integer greater than both the integers. 2. Write a pair of integers whose difference gives (a) a negative integer. (b) zero. (c) an integer smaller than both the integers. (d) an integer greater than only one of the integers. (e) an integer greater than both the integers. E XAMPLE 1 Write down a pair of integers whose (a) sum is –3 (b) difference is –5 (c) difference is 2 (d) sum is 0 S OLUTION (a) (–1) + (–2) = –3 or (–5) + 2 = –3 (b) (–9) – (– 4) = –5 or (–2) – 3 = –5 (c) (–7) – (–9) = 2 or 1 – (–1) = 2 (d) (–10) + 10 = 0 or 5 + (–5) = 0 Can you write more pairs in these examples? 2024-25  INTEGERS 5 EXERCISE 1.1 1. Write down a pair of integers whose: (a) sum is –7 (b) difference is –10 (c) sum is 0 2. (a) Write a pair of negative integers whose difference gives 8. (b) Write a negative integer and a positive integer whose sum is –5. (c) Write a negative integer and a positive integer whose difference is –3. 3. In a quiz, team A scored – 40, 10, 0 and team B scored 10, 0, – 40 in three successive rounds. Which team scored more? Can we say that we can add integers in any order? 4. Fill in the blanks to make the following statements true: (i) (–5) + (– 8) = (– 8) + (............) (ii) –53 +............ = –53 (iii) 17 +............ = 0 (iv) [13 + (– 12)] + (............) = 13 + [(–12) + (–7)] (v) (– 4) + [15 + (–3)] = [– 4 + 15] +............ 1.2 MULTIPLICATION OF INTEGERS We can add and subtract integers. Let us now learn how to multiply integers. 1.2.1 Multiplication of a Positive and a Negative Integer We know that multiplication of whole numbers is repeated addition. For example, 5 + 5 + 5 = 3 × 5 = 15 Can you represent addition of integers in the same way? TRY T HESE We have from the following number line, (–5) + (–5) + (–5) = –15 Find: 4 × (– 8), 8 × (–2), –20 –15 –10 –5 0 3 × (–7), 10 × (–1) But we can also write using number line. (–5) + (–5) + (–5) = 3 × (–5) Therefore, 3 × (–5) = –15 2024-25  6 MATHEMATICS Similarly (– 4) + (– 4) + (– 4) + (– 4) + (– 4) = 5 × (– 4) = –20 –20 –16 –12 –8 –4 0 And (–3) + (–3) + (–3) + (–3) = __________ = __________ Also, (–7) + (–7) + (–7) = __________ = __________ Let us see how to find the product of a positive integer and a negative integer without using number line. Let us find 3 × (–5) in a different way. First find 3 × 5 and then put minus sign (–) before the product obtained. You get –15. That is we find – (3 × 5) to get –15. Similarly, 5 × (– 4) = – (5×4) = – 20. Find in a similar way, 4 × (– 8) = _____ = _____, 3 × (– 7) = _____ = _____ 6 × (– 5) = _____ = _____, 2 × (– 9) = _____ = _____ Using this method we thus have, TRY THESE 10 × (– 43) = _____ – (10 × 43) = – 430 Find: Till now we multiplied integers as (positive integer) × (negative integer). (i) 6 × (–19) Let us now multiply them as (negative integer) × (positive integer). (ii) 12 × (–32) We first find –3 × 5. (iii) 7 × (–22) To find this, observe the following pattern: We have, 3 × 5 = 15 2 × 5 = 10 = 15 – 5 1 × 5 = 5 = 10 – 5 0×5=0=5–5 So, –1 × 5 = 0 – 5 = –5 –2 × 5 = –5 – 5 = –10 –3 × 5 = –10 – 5 = –15 We already have 3 × (–5) = –15 So we get (–3) × 5 = –15 = 3 × (–5) Using such patterns, we also get (–5) × 4 = –20 = 5 × (– 4) Using patterns, find (– 4) × 8, (–3) × 7, (– 6) × 5 and (– 2) × 9 Check whether, (– 4) × 8 = 4 × (– 8), (– 3) × 7 = 3 × (–7), (– 6) × 5 = 6 × (– 5) 2024-25  INTEGERS 7 and (– 2) × 9 = 2 × (– 9) Using this we get, (–33) × 5 = 33 × (–5) = –165 We thus find that while multiplying a positive integer and a negative integer, we multiply them as whole numbers and put a minus sign (–) before the product. We thus get a negative integer. TRY THESE 1. Find: (a) 15 × (–16) (b) 21 × (–32) (c) (– 42) × 12 (d) –55 × 15 2. Check if (a) 25 × (–21) = (–25) × 21 (b) (–23) × 20 = 23 × (–20) Write five more such examples. In general, for any two positive integers a and b we can say a × (– b) = (– a) × b = – (a × b) 1.2.2 Multiplication of two Negative Integers Can you find the product (–3) × (–2)? Observe the following: –3 × 4 = – 12 –3 × 3 = –9 = –12 – (–3) –3 × 2 = – 6 = –9 – (–3) –3 × 1 = –3 = – 6 – (–3) –3 × 0 = 0 = –3 – (–3) –3 × –1 = 0 – (–3) = 0 + 3 = 3 –3 × –2 = 3 – (–3) = 3 + 3 = 6 Do you see any pattern? Observe how the products change. Based on this observation, complete the following: –3 × –3 = _____ –3 × – 4 = _____ Now observe these products and fill in the blanks: – 4 × 4 = –16 – 4 × 3 = –12 = –16 + 4 – 4 × 2 = _____ = –12 + 4 2024-25 8 MATHEMATICS – 4 × 1 = _____ – 4 × 0 = _____ TRY THESE – 4 × (–1) = _____ (i) Starting from (–5) × 4, find (–5) × (– 6) – 4 × (–2) = _____ (ii) Starting from (– 6) × 3, find (– 6) × (–7) – 4 × (–3) = _____ From these patterns we observe that, (–3) × (–1) = 3 = 3 × 1 (–3) × (–2) = 6 = 3 × 2 (–3) × (–3) = 9 = 3 × 3 and (– 4) × (–1) = 4 = 4 × 1 So, (– 4) × (–2) = 4 × 2 = _____ (– 4) × (–3) = _____ = _____ So observing these products we can say that the product of two negative integers is a positive integer. We multiply the two negative integers as whole numbers and put the positive sign before the product. Thus, we have (–10) × (–12) = + 120 = 120 Similarly (–15) × (– 6) = + 90 = 90 In general, for any two positive integers a and b, (– a) × (– b) = a × b TRY THESE Find: (–31) × (–100), (–25) × (–72), (–83) × (–28) Game 1 (i) Take a board marked from –104 to 104 as shown in the figure. (ii) Take a bag containing two blue and two red dice. Number of dots on the blue dice indicate positive integers and number of dots on the red dice indicate negative integers. (iii) Every player will place his/her counter at zero. (iv) Each player will take out two dice at a time from the bag and throw them. (v) After every throw, the player has to multiply the numbers marked on the dice. 2024-25  INTEGERS 9 104 103 102 101 100 99 98 97 96 95 94 83 84 85 86 87 88 89 90 91 92 93 82 81 80 79 78 77 76 75 74 73 72 61 62 63 64 65 66 67 68 69 70 71 60 59 58 57 56 55 54 53 52 51 50 39 40 41 42 43 44 45 46 47 48 49 38 37 36 35 34 33 32 31 30 29 28 17 18 19 20 21 22 23 24 25 26 27 16 15 14 13 12 11 10 9 8 7 6 –5 –4 –3 –2 –1 0 1 2 3 4 5 –6 –7 –8 –9 –10 –11 –12 –13 –14 –15 –16 –27 –26 –25 –24 –23 –22 –21 –20 –19 –18 –17 –28 –29 –30 –31 –32 –33 –34 –35 –36 –37 –38 – 49 – 48 – 47 – 46 – 45 – 44 – 43 – 42 – 41 – 40 –39 – 50 –51 –52 –53 –54 –55 –56 –57 –58 –59 – 60 – 71 – 70 – 69 – 68 – 67 – 66 – 65 – 64 – 63 – 62 – 61 –72 –73 –74 –75 –76 –77 –78 –79 –80 –81 –82 – 93 – 92 – 91 – 90 – 89 – 88 – 87 – 86 – 85 – 84 – 83 – 94 – 95 – 96 – 97 – 98 – 99 – 100 –101 –102 –103 –104 (vi) If the product is a positive integer then the player will move his counter towards 104; if the product is a negative integer then the player will move his counter towards –104. (vii) The player who reaches either -104 or 104 first is the winner. 2024-25 10 MATHEMATICS 1.3 PROPER TIES OF MULTIPLICATION OF INTEGERS 1.3.1 Closure under Multiplication 1. Observe the following table and complete it: Statement Inference (–20) × (–5) = 100 Product is an integer (–15) × 17 = – 255 Product is an integer (–30) × 12 = _____ (–15) × (–23) = _____ (–14) × (–13) = _____ 12 × (–30) = _____ What do you observe? Can you find a pair of integers whose product is not an integer? No. This gives us an idea that the product of two integers is again an integer. So we can say that integers are closed under multiplication. In general, a × b is an integer, for all integers a and b. Find the product of five more pairs of integers and verify the above statement. 1.3.2 Commutativity of Multiplication We know that multiplication is commutative for whole numbers. Can we say, multiplication is also commutative for integers? Observe the following table and complete it: Statement 1 Statement 2 Inference 3 × (– 4) = –12 (– 4) × 3 = –12 3 × (– 4) = (– 4) × 3 (–30) × 12 = _____ 12 × (–30) = _____ (–15) × (–10) = 150 (–10) × (–15) = 150 (–35) × (–12) = _____ (–12) × (–35) = (–17) × 0 = _____ __________ = _____ (–1) × (–15) = 2024-25  INTEGERS 11 What are your observations? The above examples suggest multiplication is commutative for integers. Write five more such examples and verify. In general, for any two integers a and b, a×b=b×a 1.3.3 Multiplication by Zero We know that any whole number when multiplied by zero gives zero. Observe the following products of negative integers and zero. These are obtained from the patterns done earlier. (–3) × 0 = 0 0 × (– 4) = 0 – 5 × 0 = _____ 0 × (– 6) = _____ This shows that the product of a negative integer and zero is zero. In general, for any integer a, a×0=0×a=0 1.3.4 Multiplicative Identity We know that 1 is the multiplicative identity for whole numbers. Check that 1 is the multiplicative identity for integers as well. Observe the following products of integers with 1. (–3) × 1 = –3 1×5=5 (– 4) × 1 = _____ 1 × 8 = _____ 1 × (–5) = _____ 3 × 1 = _____ 1 × (– 6) = _____ 7 × 1 = _____ This shows that 1 is the multiplicative identity for integers also. In general, for any integer a we have, a×1=1×a=a What happens when we multiply any integer with –1? Complete the following: (–3) × (–1) = 3 3 × (–1) = –3 0 is the additive identity whereas 1 is the (– 6) × (–1) = _____ multiplicative identity for integers. We get (–1) × 13 = _____ additive inverse of an integer a when we multiply (–1) × (–25) = _____ (–1) to a, i.e. a × (–1) = (–1) × a = – a 18 × (–1) = _____ What do you observe? Can we say –1 is a multiplicative identity of integers? No. 2024-25 12 MATHEMATICS 1.3.5 Associativity for Multiplication Consider –3, –2 and 5. Look at [(–3) × (–2)] × 5 and (–3) × [(–2) × 5]. In the first case (–3) and (–2) are grouped together and in the second (–2) and 5 are grouped together. We see that [(–3) × (–2)] × 5 = 6 × 5 = 30 and (–3) × [(–2) × 5] = (–3) × (–10) = 30 So, we get the same answer in both the cases. Thus, [(–3) × (–2)] × 5 = (–3) × [(–2) × 5] Look at this and complete the products: [(7) × (– 6)] × 4 = __________ × 4 = __________ 7 × [(– 6) × 4] = 7 × __________ = __________ Is [7 × (– 6)] × 4 = 7 × [(– 6) × 4]? Does the grouping of integers affect the product of integers? No. In general, for any three integers a, b and c (a × b) × c = a × (b × c) Take any five values for a, b and c each and verify this property. Thus, like whole numbers, the product of three integers does not depend upon the grouping of integers and this is called the associative property for multiplication of integers. 1.3.6 Distributive Property We know 16 × (10 + 2) = (16 × 10) + (16 × 2) [Distributivity of multiplication over addition] Let us check if this is true for integers also. Observe the following: (a) (–2) × (3 + 5) = –2 × 8 = –16 and [(–2) × 3] + [(–2) × 5] = (– 6) + (–10) = –16 So, (–2) × (3 + 5) = [(–2) × 3] + [(–2) × 5] (b) (– 4) × [(–2) + 7] = (– 4) × 5 = –20 and [(– 4) × (–2)] + [(– 4) × 7] = 8 + (–28) = –20 So, (– 4) × [(–2) + 7] = [(– 4) × (–2)] + [(– 4) × 7] (c) (– 8) × [(–2) + (–1)] = (– 8) × (–3) = 24 and [(– 8) × (–2)] + [(– 8) × (–1)] = 16 + 8 = 24 So, (– 8) × [(–2) + (–1)] = [(– 8) × (–2)] + [(– 8) × (–1)] 2024-25  INTEGERS 13 Can we say that the distributivity of multiplication over addition is true for integers also? Yes. In general, for any integers a, b and c, a × (b + c) = a × b + a × c Take atleast five different values for each of a, b and c and verify the above Distributive property. TRY THESE (i) Is 10 × [(6 + (–2)] = 10 × 6 + 10 × (–2)? (ii) Is (–15) × [(–7) + (–1)] = (–15) × (–7) + (–15) × (–1)? Now consider the following: Can we say 4 × (3 – 8) = 4 × 3 – 4 × 8? Let us check: 4 × (3 – 8) = 4 × (–5) = –20 4 × 3 – 4 × 8 = 12 – 32 = –20 So, 4 × (3 – 8) = 4 × 3 – 4 × 8. Look at the following: ( –5) × [( – 4) – ( – 6)] = ( –5) × 2 = –10 [( –5) × ( – 4)] – [( –5) × ( – 6)] = 20 – 30 = –10 So, ( –5) × [( – 4) – ( – 6)] = [( –5) × ( – 4)] – [ ( –5) × ( – 6)] Check this for ( –9) × [ 10 – ( –3)] and [( –9) × 10 ] – [ ( –9) × ( –3)] You will find that these are also equal. In general, for any three integers a, b and c, a × (b – c) = a × b – a × c Take atleast five different values for each of a, b and c and verify this property. TRY THESE (i) Is 10 × (6 – (–2)] = 10 × 6 – 10 × (–2)? (ii) Is (–15) × [(–7) – (–1)] = (–15) × (–7) – (–15) × (–1)? 2024-25 14 MATHEMATICS E XERCISE 1.2 1. Find each of the following products: (a) 3 × (–1) (b) (–1) × 225 (c) (–21) × (–30) (d) (–316) × (–1) (e) (–15) × 0 × (–18) (f) (–12) × (–11) × (10) (g) 9 × (–3) × (– 6) (h) (–18) × (–5) × (– 4) (i) (–1) × (–2) × (–3) × 4 (j) (–3) × (–6) × (–2) × (–1) 2. Verify the following: (a) 18 × [7 + (–3)] = [18 × 7] + [18 × (–3)] (b) (–21) × [(– 4) + (– 6)] = [(–21) × (– 4)] + [(–21) × (– 6)] 3. (i) For any integer a, what is (–1) × a equal to? (ii) Determine the integer whose product with (–1) is (a) –22 (b) 37 (c) 0 4. Starting from (–1) × 5, write various products showing some pattern to show (–1) × (–1) = 1. 1.4 DIVISION OF INTEGERS We know that division is the inverse operation of multiplication. Let us see an example for whole numbers. Since 3 × 5 = 15 So 15 ÷ 5 = 3 and 15 ÷ 3 = 5 Similarly, 4 × 3 = 12 gives 12 ÷ 4 = 3 and 12 ÷ 3 = 4 We can say for each multiplication statement of whole numbers there are two division statements. Can you write multiplication statement and its corresponding divison statements for integers? l Observe the following and complete it. Multiplication Statement Corresponding Division Statements 2 × (– 6) = (–12) (–12) ÷ (– 6) = 2 , (–12) ÷ 2 = (– 6) (– 4) × 5 = (–20) (–20) ÷ 5 = (– 4) , (–20) ÷ (– 4) = 5 (– 8) × (–9) = 72 72 ÷ _____ = _____ , 72 ÷ _____ = _____ (–3) × (–7) = _____ _____ ÷ (–3) = ____ , _____________ _____________ , _____________ (– 8) × 4 = ____ _____________ , _____________ 5 × (– 9) = _____ _____________ , _____________ (–10) × (–5) = 2024-25  INTEGERS 15 From the above we observe that : TRY THESE (–12) ÷ 2 = (– 6) Find: (–20) ÷ 5 = (– 4) (a) (–100) ÷ 5 (b) (–81) ÷ 9 (–32) ÷ 4 = (– 8) (c) (–75) ÷ 5 (d) (–32) ÷ 2 (– 45) ÷ 5 = (– 9) We observe that when we divide a negative integer by a positive integer, we divide them as whole numbers and then put a minus sign (–) before the quotient. l We also observe that: Can we say that 72 ÷ (–8) = –9 and 50 ÷ (–10) = –5 (– 48) ÷ 8 = 48 ÷ (– 8)? Let us check. We know that 72 ÷ (–9) = – 8 50 ÷ (–5) = –10 (– 48) ÷ 8 = – 6 So we can say that when we divide a positive integer by a negative and 48 ÷ (– 8) = – 6 integer, we first divide them as whole numbers and then put a minus So (– 48) ÷ 8 = 48 ÷ (– 8) sign (–) before the quotient. Check this for In general, for any two positive integers a and b (i) 90 ÷ (– 45) and (–90) ÷ 45 a ÷ (– b) = (– a) ÷ b where b ≠ 0 (ii) (–136) ÷ 4 and 136 ÷ (– 4) TRY THESE Find: (a) 125 ÷ (–25) (b) 80 ÷ (–5) (c) 64 ÷ (–16) l Lastly, we observe that (–12) ÷ (– 6) = 2; (–20) ÷ (– 4) = 5; (–32) ÷ (– 8) = 4; (– 45) ÷ (–9) = 5 So, we can say that when we divide a negative integer by a negative integer, we first divide them as whole numbers and then put a positive sign (+). In general, for any two positive integers a and b (– a) ÷ (– b) = a ÷ b where b ≠ 0 TRY THESE Find: (a) (–36) ÷ (– 4) (b) (–201) ÷ (–3) (c) (–325) ÷ (–13) 1.5 PROPER TIES OF D IVISION OF INTEGERS Observe the following table and complete it: What do you observe? We observe that integers are not closed under division. Statement Inference Statement Inference (– 8) ÷ (– 4) = 2 Result is an integer –8 (– 8) ÷ 3 = ________________ 3 –4 (– 4) ÷ (– 8) = Result is not an integer 3 ÷ (– 8) = 3 ________________ –8 –8 2024-25  16 MATHEMATICS Justify it by taking five more examples of your own. l We know that division is not commutative for whole numbers. Let us check it for integers also. You can see from the table that (– 8) ÷ (– 4) ≠ (– 4) ÷ (– 8). Is (– 9) ÷ 3 the same as 3 ÷ (– 9)? Is (– 30) ÷ (– 6) the same as (– 6) ÷ (– 30)? Can we say that division is commutative for integers? No. You can verify it by taking five more pairs of integers. l Like whole numbers, any integer divided by zero is meaningless and zero divided by an integer other than zero is equal to zero i.e., for any integer a, a ÷ 0 is not defined but 0 ÷ a = 0 for a ≠ 0. l When we divide a whole number by 1 it gives the same whole number. Let us check whether it is true for negative integers also. Observe the following : (– 8) ÷ 1 = (– 8) (–11) ÷ 1 = –11 (–13) ÷ 1 = –13 (–25) ÷ 1 = ____ (–37) ÷ 1 = ____ (– 48) ÷ 1 = ____ This shows that negative integer divided by 1 gives the same negative integer. So, any integer divided by 1 gives the same integer. In general, for any integer a, a÷1=a l What happens when we divide any integer by (–1)? Complete the following table (– 8) ÷ (–1) = 8 11 ÷ (–1) = –11 13 ÷ (–1) = ____ (–25) ÷ (–1) = ____ (–37) ÷ (–1) = ____ – 48 ÷ (–1) = ____ What do you observe? We can say that if any integer is divided by (–1) it does not give the same integer. l Can we say[(–16) ÷ 4] ÷ (–2) is the same as (–16) ÷ [4 ÷ (–2)]? We know that [(–16) ÷ 4] ÷ (–2) = (– 4) ÷ (–2) = 2 and (–16) ÷ [4 ÷ (–2)] = (–16) ÷ (–2) = 8 TRY THESE So [(–16) ÷ 4] ÷ (–2) ≠ (–16) ÷ [4 ÷ (–2)] Is (i) 1 ÷ a = 1? (ii) a ÷ (–1) = – a? for any integer a. Can you say that division is associative for integers? No. Take different values of a and check. Verify it by taking five more examples of your own. 2024-25  INTEGERS 17 E XAMPLE 2 In a test (+5) marks are given for every correct answer and (–2) marks are given for every incorrect answer. (i) Radhika answered all the questions and scored 30 marks though she got 10 correct answers. (ii) Jay also answered all the questions and scored (–12) marks though he got 4 correct answers. How many incorrect answers had they attempted? S OLUTION (i) Marks given for one correct answer = 5 So, marks given for 10 correct answers = 5 × 10 = 50 Radhika’s score = 30 Marks obtained for incorrect answers = 30 – 50 = – 20 Marks given for one incorrect answer = (–2) Therefore, number of incorrect answers = (–20) ÷ (–2) = 10 (ii) Marks given for 4 correct answers = 5 × 4 = 20 Jay’s score = –12 Marks obtained for incorrect answers = –12 – 20 = – 32 Marks given for one incorrect answer = (–2) Therefore number of incorrect answers = (–32) ÷ (–2) = 16 E XAMPLE 3 A shopkeeper earns a profit of ` 1 by selling one pen and incurs a loss of 40 paise per pencil while selling pencils of her old stock. (i) In a particular month she incurs a loss of ` 5. In this period, she sold 45 pens. How many pencils did she sell in this period? (ii) In the next month she earns neither profit nor loss. If she sold 70 pens, how many pencils did she sell? S OLUTION (i) Profit earned by selling one pen = ` 1 Profit earned by selling 45 pens = ` 45, which we denote by + ` 45 Total loss given = ` 5, which we denote by – ` 5 Profit earned + Loss incurred = Total loss Therefore, Loss incurred = Total Loss – Profit earned = ` (– 5 – 45) = ` (–50) = –5000 paise Loss incurred by selling one pencil = 40 paise which we write as – 40 paise So, number of pencils sold = (–5000) ÷ (– 40) = 125 2024-25 18 MATHEMATICS (ii) In the next month there is neither profit nor loss. So, Profit earned + Loss incurred = 0 i.e., Profit earned = – Loss incurred. Now, profit earned by selling 70 pens = ` 70 Hence, loss incurred by selling pencils = ` 70 which we indicate by – ` 70 or – 7,000 paise. Total number of pencils sold = (–7000) ÷ (– 40) = 175 pencils. E XERCISE 1.3 1. Evaluate each of the following: (a) (–30) ÷ 10 (b) 50 ÷ (–5) (c) (–36) ÷ (–9) (d) (– 49) ÷ (49) (e) 13 ÷ [(–2) + 1] (f ) 0 ÷ (–12) (g) (–31) ÷ [(–30) + (–1)] (h) [(–36) ÷ 12] ÷ 3 (i) [(– 6) + 5)] ÷ [(–2) + 1] 2. Verify that a ÷ (b + c) ≠ (a ÷ b) + (a ÷ c) for each of the following values of a, b and c. (a) a = 12, b = – 4, c = 2 (b) a = (–10), b = 1, c = 1 3. Fill in the blanks: (a) 369 ÷ _____ = 369 (b) (–75) ÷ _____ = –1 (c) (–206) ÷ _____ = 1 (d) – 87 ÷ _____ = 87 (e) _____ ÷ 1 = – 87 (f) _____ ÷ 48 = –1 (g) 20 ÷ _____ = –2 (h) _____ ÷ (4) = –3 4. Write five pairs of integers (a, b) such that a ÷ b = –3. One such pair is (6, –2) because 6 ÷ (–2) = (–3). 5. The temperature at 12 noon was 10°C above zero. If it decreases at the rate of 2°C per hour until midnight, at what time would the temperature be 8°C below zero? What would be the temperature at mid-night? 6. In a class test (+ 3) marks are given for every correct answer and (–2) marks are given for every incorrect answer and no marks for not attempting any question. (i) Radhika scored 20 marks. If she has got 12 correct answers, how many questions has she attempted incorrectly? (ii) Mohini scores –5 marks in this test, though she has got 7 correct answers. How many questions has she attempted incorrectly? 7. An elevator descends into a mine shaft at the rate of 6 m/min. If the descent starts from 10 m above the ground level, how long will it take to reach – 350 m. 2024-25  INTEGERS 19 W HAT HAVE WE DISCUSSED? 1. We now study the properties satisfied by addition and subtraction. (a) Integers are closed for addition and subtraction both. That is, a + b and a – b are again integers, where a and b are any integers. (b) Addition is commutative for integers, i.e., a + b = b + a for all integers a and b. (c) Addition is associative for integers, i.e., (a + b) + c = a + (b + c) for all integers a, b and c. (d) Integer 0 is the identity under addition. That is, a + 0 = 0 + a = a for every integer a. 2. We studied, how integers could be multiplied, and found that product of a positive and a negative integer is a negative integer, whereas the product of two negative integers is a positive integer. For example, – 2 × 7 = – 14 and – 3 × – 8 = 24. 3. Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative. 4. Integers show some properties under multiplication. (a) Integers are closed under multiplication. That is, a × b is an integer for any two integers a and b. (b) Multiplication is commutative for integers. That is, a × b = b × a for any integers a and b. (c) The integer 1 is the identity under multiplication, i.e., 1 × a = a × 1 = a for any integer a. (d) Multiplication is associative for integers, i.e., (a × b) × c = a × (b × c) for any three integers a, b and c. 5. Under addition and multiplication, integers show a property called distributive property. That is, a × (b + c) = a × b + a × c for any three integers a, b and c. 6. The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier. 7. We also learnt how to divide integers. We found that, (a) When a positive integer is divided by a negative integer, the quotient obtained is negative and vice-versa. (b) Division of a negative integer by another negative integer gives positive as quotient. 8. For any integer a, we have (a) a ÷ 0 is not defined (b) a ÷ 1 = a 2024-25