MATHEMATICS 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 (i) 17 + 23 = 40 (ii) (–10) + 3 = _____ (iii) (– 75) + 18 = _____ (iv) 19 + (– 25) = – 6 (v) 27 + (– 27) = _____ (vi) (– 20) + 0 = _____ (vii) (– 35) + (– 10) = _____ Observation Result is an integer Result is an integer 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. 1.3.2 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 (i) 7 – 9 = – 2 (ii) 17 – (– 21) = _______ (iii) (– 8) – (–14) = 6 (iv) (– 21) – (– 10) = _______ (v) 32 – (–17) = _______ (vi) (– 18) – (– 18) = _______ (vii) (– 29) – 0 = _______ Observation Result is an integer Result is an integer 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? INTEGERS 1.3.3 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. In general, for any two integers a and b, we can say a + b =b + a • 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. 1.3.4 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) MATHEMATICS 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 1.3.5 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 INTEGERS EXAMPLE 1 Write down a pair of integers whose (a) sum is –3 (b) difference is –5 (c) difference is 2 (d) sum is 0 SOLUTION (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? EXERCISE 1.2 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] + ............ 1.4 MULTIPLICATION OF INTEGERS We can add and subtract integers. Let us now learn how to multiply integers. 1.4.1 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? INTEGERS 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 (v) After every throw, the player has to multiply the numbers marked on the dice. (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 MATHEMATICS 1.4.3 Product of three or more Negative Integers We observed that the product of two negative integers is a positive integer. Euler in his book Ankitung zur What will be the product of three negative integers? Four negative integers?Algebra(1770), was one of Let us observe the following examples:the first mathematicians to attempt to prove(a) (– 4) × (–3) = 12 (–1) × (–1) = 1 (b) (– 4) × (–3) × (–2) = [(– 4) × (–3)] × (–2) = 12 × (–2) = – 24 (c) (– 4) × (–3) × (–2) × (–1) = [(– 4) × (–3) × (–2)] × (–1) = (–24) × (–1) (d) (–5) × [(–4) × (–3) × (–2) × (–1)] = (–5) × 24 = –120 From the above products we observe that (a) the product of two negative integers A Special Case is a positive integer; Consider the following statements and (b) the product of three negative integers the resultant products: is a negative integer. (–1) × (–1) = +1 (c) product of four negative integers is (–1) × (–1) × (–1) = –1a positive integer. (–1) × (–1) × (–1) × (–1) = +1What is the product of five negative integers in (–1) × (–1) × (–1) × (–1) × (–1) = –1(d)? This means that if the integerSo what will be the product of six negative integers? (–1) is multiplied even number of times, the product is +1 and if the integer (–1)We further see that in (a) and (c) above, is multiplied odd number of times, thethe number of negative integers that are multiplied are even [two and four respectively] product is –1. You can check this by and the product obtained in (a) and (c) are making pairs of (–1) in the statement. positive integers. The number of negative This is useful in working out products of integers that are multiplied in (b) and (d) is odd integers. and the products obtained in (b) and (d) are negative integers. We find that if the number of negative integers in a product is even, then the product is a positive integer; if the number of negative integers in a product is odd, then the product is a negative integer. Justify it by taking five more examples of each kind. THINK, DISCUSS AND WRITE (i) The product (–9) × (–5) × (– 6)×(–3) is positive whereas the product (–9) × ( –5) × 6 × (–3) is negative. Why? (ii) What will be the sign of the product if we multiply together: (a) 8 negative integers and 3 positive integers? (b) 5 negative integers and 4 positive integers? MATHEMATICS 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, 1.5.3 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, 1.5.4 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, 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. INTEGERS 1.5.5 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. 1.5.6 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)] MATHEMATICS [(–30) × 13] + [(–30) × (–3)] = –390 + 90 = –300 So, (–30) × [13 + (–3)] = [(–30) × 13] + [(–30) × (–3)] EXAMPLE 4 In a class test containing 15 questions, 4 marks are given for every correct answer and (–2) marks are given for every incorrect answer. (i) Gurpreet attempts all questions but only 9 of her answers are correct. What is her total score? (ii) One of her friends gets only 5 answers correct. What will be her score? SOLUTION (i) Marks given for one correct answer = 4 So, marks given for 9 correct answers = 4 × 9 = 36 Marks given for one incorrect answer = – 2 So, marks given for 6 (= 15 – 9) incorrect answers = (–2) × 6 = –12 Therefore, Gurpreet’s total score = 36 + ( –12) = 24 (ii) Marks given for one correct answer = 4 So, marks given for 5 correct answers = 4 × 5 = 20 Marks given for one incorrect answer = (–2) So, marks given for 10 (=15 – 5) incorrect answers = (–2) × 10 = –20 Therefore, her friend’s total score = 20 + ( –20) = 0 EXAMPLE 5 Suppose we represent the distance above the ground by a positive integer and that below the ground by a negative integer, then answer the following: (i) An elevator descends into a mine shaft at the rate of 5 metre per minute. What will be its position after one hour? (ii) If it begins to descend from 15 m above the ground, what will be its position after 45 minutes? SOLUTION (i) Since the elevator is going down, so the distance covered by it will be represented by a negative integer. Change in position of the elevator in one minute = – 5 m Position of the elevator after 60 minutes = (–5) × 60 = – 300 m, i.e., 300 m below ground level. (ii) Change in position of the elevator in 45 minutes = (–5) × 45 = –225 m, i.e., 225 m below ground level. So, the final position of the elevator = –225 + 15 = –210 m, i.e., 210 m below ground level. INTEGERS answered all the questions and scored (–12) marks though he got 4 correct answers. How many incorrect answers had they attempted? SOLUTION (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 EXAMPLE 7 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? SOLUTION (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 pencils. (ii) In the next month there is neither profit nor loss. So, Profit earned + Loss incurred = 0 INTEGERS WHAT HAVE WE DISCUSSED? 1. Integers are a bigger collection of numbers which is formed by whole numbers and their negatives. These were introduced in Class VI. 2. You have studied in the earlier class, about the representation of integers on the number line and their addition and subtraction. 3. 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. 4. 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. 5. Product of even number of negative integers is positive, whereas the product of odd number of negative integers is negative. 6. 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. 7. 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. MATHEMATICS 8. The properties of commutativity, associativity under addition and multiplication, and the distributive property help us to make our calculations easier. 9. 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 a negative integer and vice-versa. (b) Division of a negative integer by another negative integer gives a positive integer as quotient. 10. For any integer a, we have (a) a ÷ 0 is not defined (b) a ÷ 1 = a