Welcome dear students! Today we are going to learn about Integers from Class 7 Maths. We have already learnt about whole numbers and integers in Class 6, along with addition and subtraction of integers. Let us explore the properties of integers in detail.
First, we examine Closure under Addition. We know that the sum of two whole numbers is a whole number, like 17 + 24 = 41. This is the closure property for whole numbers. Let us check if it holds for integers. Consider these pairs: 17 + 23 = 40, which is an integer. (–10) + 3 = –7, an integer. (–75) + 18 = –57, an integer. 19 + (–25) = –6, an integer. 27 + (–27) = 0, an integer. (–20) + 0 = –20, an integer. (–35) + (–10) = –45, an integer. What do you observe? Is the sum of two integers always an integer? Yes. Since addition of integers always gives an integer, we say integers are closed under addition. In general, for any two integers a and b, a + b is an integer.
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Next, Closure under Subtraction. What happens when we subtract one integer from another? Let us observe: 7 – 9 = –2, an integer. 17 – (–21) = 38, an integer. (–8) – (–14) = 6, an integer. (–21) – (–10) = –11, an integer. 32 – (–17) = 49, an integer. (–18) – (–18) = 0, an integer. (–29) – 0 = –29, an integer. Is there any pair whose difference is not an integer? No. Integers are closed under subtraction. Thus, if a and b are integers, a – b is also an integer. Do whole numbers satisfy this? No, because 3 – 5 = –2, which is not a whole number.
Now, the Commutative Property. We know 3 + 5 = 5 + 3 = 8. Addition is commutative for whole numbers. Does it work for integers? 5 + (–6) = –1 and (–6) + 5 = –1, so they are equal. Check these: (–8) + (–9) = –17 and (–9) + (–8) = –17. (–23) + 32 = 9 and 32 + (–23) = 9. (–45) + 0 = –45 and 0 + (–45) = –45. Try five other pairs. You will never find different sums when order changes. Addition is commutative for integers. In general, a + b = b + a.
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Is subtraction commutative for integers? Consider 5 and (–3). 5 – (–3) = 8, but (–3) – 5 = –8. They are not equal. Take five different pairs and check. Subtraction is not commutative for integers.
Let us study the Associative Property. Consider integers –3, –2 and –5. Compare (–5) + [(–3) + (–2)] and [(–5) + (–3)] + (–2). First: (–5) + (–5) = –10. Second: (–8) + (–2) = –10. Both give –10. Now try –3, 1 and –7. (–3) + [1 + (–7)] = –3 + (–6) = –9. [(–3) + 1] + (–7) = –2 + (–7) = –9. They are equal. Addition is associative for integers. In general, a + (b + c) = (a + b) + c.
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Let us practice. Write a pair of integers whose sum gives a negative integer, like (–5) + (–2) = –7. A pair whose sum gives zero: 8 + (–8) = 0. A pair whose sum is smaller than both: (–4) + (–5) = –9. A pair whose sum is smaller than only one: 3 + (–6) = –3. A pair whose sum is greater than both: 4 + 5 = 9. For differences: a pair giving a negative integer: 2 – 7 = –5. A pair giving zero: 6 – 6 = 0. A pair smaller than both: (–5) – 3 = –8. A pair greater than only one: 5 – (–2) = 7. A pair greater than both: 8 – (–3) = 11.
Now, Example 1. Write a pair whose sum is –3: (–1) + (–2) = –3 or (–5) + 2 = –3. Difference is –5: (–9) – (–4) = –5 or (–2) – 3 = –5. Difference is 2: (–7) – (–9) = 2 or 1 – (–1) = 2. Sum is 0: (–10) + 10 = 0 or 5 + (–5) = 0. You can write many more pairs.
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Next, Additive Identity. Adding zero to any whole number gives the same number. Does it work for integers? (–8) + 0 = –8. 0 + (–8) = –8. (–23) + 0 = –23. 0 + (–37) = –37. 0 + (–59) = –59. 0 + (–43) = –43. –61 + 0 = –61. –43 + 0 = –43. Zero is the additive identity for integers. In general, a + 0 = a = 0 + a.
Now, Exercise 1.1. Question 1: Pair with sum –7: (–4) + (–3) = –7. Pair with difference –10: 2 – 12 = –10. Pair with sum 0: 9 + (–9) = 0. Question 2: Pair of negative integers with difference 8: (–2) – (–10) = 8. Negative and positive with sum –5: (–8) + 3 = –5. Negative and positive with difference –3: 2 – 5 = –3.
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Question 3: Team A scores –40, 10, 0. Total = –40 + 10 + 0 = –30. Team B scores 10, 0, –40. Total = 10 + 0 + (–40) = –30. Both scored the same. Yes, we can add integers in any order, showing commutativity. Question 4: Fill in the blanks. (i) (–5) + (–8) = (–8) + (–5). (ii) –53 + 0 = –53. (iii) 17 + (–17) = 0. (iv) [13 + (–12)] + (–7) = 13 + [(–12) + (–7)]. (v) (–4) + [15 + (–3)] = [–4 + 15] + (–3).
Now we begin Section 1.2: MULTIPLICATION OF INTEGERS. We know addition and subtraction. Let us learn multiplication. First, 1.2.1 Multiplication of a Positive and a Negative Integer. Multiplication is repeated addition. 5 + 5 + 5 = 3 × 5 = 15. On a number line starting at 0, moving left by 5 units three times lands at –15. So, (–5) + (–5) + (–5) = –15, which is 3 × (–5). Thus, 3 × (–5) = –15. Similarly, (–4) + (–4) + (–4) + (–4) + (–4) = 5 × (–4) = –20. (–3) + (–3) + (–3) + (–3) = 4 × (–3) = –12. (–7) + (–7) + (–7) = 3 × (–7) = –21.
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To multiply without a number line, find 3 × (–5). First calculate 3 × 5 = 15, then put a minus sign: –(3 × 5) = –15. Similarly, 5 × (–4) = –(5 × 4) = –20. Using this: 4 × (–8) = –(4 × 8) = –32. 3 × (–7) = –(3 × 7) = –21. 6 × (–5) = –(6 × 5) = –30. 2 × (–9) = –(2 × 9) = –18. 10 × (–43) = –(10 × 43) = –430.
Now multiply a negative integer by a positive integer. Find –3 × 5 using a pattern: 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 know 3 × (–5) = –15. So (–3) × 5 = 3 × (–5). Similarly, (–5) × 4 = 5 × (–4) = –20. Using patterns: (–4) × 8 = –32, (–3) × 7 = –21, (–6) × 5 = –30, (–2) × 9 = –18. Check: (–4) × 8 = 4 × (–8) = –32. (–3) × 7 = 3 × (–7) = –21. (–6) × 5 = 6 × (–5) = –30. (–2) × 9 = 2 × (–9) = –18.
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Thus, (–33) × 5 = 33 × (–5) = –165. When multiplying a positive and a negative integer, multiply as whole numbers and put a minus sign. In general, for positive integers a and b, a × (–b) = (–a) × b = –(a × b). Try These: 6 × (–19) = –114. 12 × (–32) = –384. 7 × (–22) = –154. Find: 15 × (–16) = –240. 21 × (–32) = –672. (–42) × 12 = –504. –55 × 15 = –825. Check: 25 × (–21) = –525 and (–25) × 21 = –525. (–23) × 20 = –460 and 23 × (–20) = –460.
Now, 1.2.2 Multiplication of two Negative Integers. Find (–3) × (–2). Observe the pattern: –3 × 4 = –12. –3 × 3 = –9 = –12 – (–3). –3 × 2 = –6 = –9 – (–3). –3 × 1 = –3 = –6 – (–3). –3 × 0 = 0 = –3 – (–3). So, –3 × (–1) = 0 – (–3) = 3. –3 × (–2) = 3 – (–3) = 6. Complete: –3 × (–3) = 9. –3 × (–4) = 12.
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Now for –4: –4 × 4 = –16. –4 × 3 = –12 = –16 + 4. –4 × 2 = –8 = –12 + 4. –4 × 1 = –4 = –8 + 4. –4 × 0 = 0 = –4 + 4. –4 × (–1) = 4 = 0 + 4. –4 × (–2) = 8 = 4 + 4. –4 × (–3) = 12 = 8 + 4. We observe (–3) × (–1) = 3 = 3 × 1. (–3) × (–2) = 6 = 3 × 2. (–3) × (–3) = 9 = 3 × 3. (–4) × (–1) = 4 = 4 × 1. So, (–4) × (–2) = 4 × 2 = 8. (–4) × (–3) = 4 × 3 = 12. The product of two negative integers is positive. Multiply as whole numbers and put a positive sign. (–10) × (–12) = 120. (–15) × (–6) = 90. In general, (–a) × (–b) = a × b. Find: (–31) × (–100) = 3100. (–25) × (–72) = 1800. (–83) × (–28) = 2324.
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Try These: Starting from (–5) × 4 = –20, decreasing the second number by 1 increases the product by 5. So (–5) × (–6) = 30. Starting from (–6) × 3 = –18, following the same pattern, (–6) × (–7) = 42.
Let us describe Game 1. Imagine a board with numbers from –104 to 104. You have a bag with two blue and two red dice. Blue dots mean positive integers, red dots mean negative integers. Each player starts at zero. Take two dice, multiply the numbers. If the product is positive, move towards 104. If negative, move towards –104. First to reach either end wins.
Now Section 1.3: PROPERTIES OF MULTIPLICATION OF INTEGERS. 1.3.1 Closure under Multiplication. Observe: (–20) × (–5) = 100. (–15) × 17 = –255. (–30) × 12 = –360. (–15) × (–23) = 345. (–14) × (–13) = 182. 12 × (–30) = –360. All results are integers. Integers are closed under multiplication. In general, a × b is an integer for all integers a and b.
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1.3.2 Commutativity of Multiplication. 3 × (–4) = –12 and (–4) × 3 = –12. (–30) × 12 = –360 and 12 × (–30) = –360. (–15) × (–10) = 150 and (–10) × (–15) = 150. (–35) × (–12) = 420 and (–12) × (–35) = 420. (–17) × 0 = 0 and 0 × (–17) = 0. (–1) × (–15) = 15 and (–15) × (–1) = 15. Multiplication is commutative for integers. In general, a × b = b × a.
1.3.3 Multiplication by Zero. (–3) × 0 = 0. 0 × (–4) = 0. –5 × 0 = 0. 0 × (–6) = 0. Product of any integer and zero is zero. In general, a × 0 = 0 × a = 0.
1.3.4 Multiplicative Identity. (–3) × 1 = –3. 1 × 5 = 5. (–4) × 1 = –4. 1 × 8 = 8. 1 × (–5) = –5. 3 × 1 = 3. 1 × (–6) = –6. 7 × 1 = 7. 1 is the multiplicative identity. In general, a × 1 = 1 × a = a. Zero is additive identity, 1 is multiplicative identity. Multiplying by –1 gives the additive inverse: a × (–1) = (–1) × a = –a.
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Check: (–3) × (–1) = 3. 3 × (–1) = –3. (–6) × (–1) = 6. (–1) × 13 = –13. (–1) × (–25) = 25. 18 × (–1) = –18. –1 is not a multiplicative identity.
1.3.5 Associativity for Multiplication. Consider –3, –2, 5. [(–3) × (–2)] × 5 = 6 × 5 = 30. (–3) × [(–2) × 5] = (–3) × (–10) = 30. They match. Check: [7 × (–6)] × 4 = (–42) × 4 = –168. 7 × [(–6) × 4] = 7 × (–24) = –168. Multiplication is associative. In general, (a × b) × c = a × (b × c).
1.3.6 Distributive Property. Check: (–2) × (3 + 5) = (–2) × 8 = –16. [(–2) × 3] + [(–2) × 5] = (–6) + (–10) = –16. Equal. (–4) × [(–2) + 7] = (–4) × 5 = –20. [(–4) × (–2)] + [(–4) × 7] = 8 + (–28) = –20. Equal. (–8) × [(–2) + (–1)] = (–8) × (–3) = 24. [(–8) × (–2)] + [(–8) × (–1)] = 16 + 8 = 24. Equal. Distributivity holds. In general, a × (b + c) = a × b + a × c.
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Check Try These: 10 × [6 + (–2)] = 40. 10 × 6 + 10 × (–2) = 60 – 20 = 40. Yes. (–15) × [(–7) + (–1)] = (–15) × (–8) = 120. (–15) × (–7) + (–15) × (–1) = 105 + 15 = 120. Yes. Now over subtraction: 4 × (3 – 8) = 4 × (–5) = –20. (4 × 3) – (4 × 8) = 12 – 32 = –20. Equal. (–5) × [(–4) – (–6)] = (–5) × 2 = –10. [(–5) × (–4)] – [(–5) × (–6)] = 20 – 30 = –10. Equal. Check: (–9) × [10 – (–3)] = (–9) × 13 = –117. [(–9) × 10] – [(–9) × (–3)] = –90 – 27 = –117. Equal. In general, a × (b – c) = a × b – a × c. Check: 10 × [6 – (–2)] = 80. 10 × 6 – 10 × (–2) = 60 – (–20) = 80. Yes. (–15) × [(–7) – (–1)] = 90. (–15) × (–7) – (–15) × (–1) = 105 – 15 = 90. Yes.
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Exercise 1.2. Question 1: (a) 3 × (–1) = –3. (b) (–1) × 225 = –225. (c) (–21) × (–30) = 630. (d) (–316) × (–1) = 316. (e) (–15) × 0 × (–18) = 0. (f) (–12) × (–11) × 10 = 1320. (g) 9 × (–3) × (–6) = 162. (h) (–18) × (–5) × (–4) = –360. (i) (–1) × (–2) × (–3) × 4 = –24. (j) (–3) × (–6) × (–2) × (–1) = 36. Question 2: (a) 18 × [7 + (–3)] = 18 × 4 = 72. [18 × 7] + [18 × (–3)] = 126 + (–54) = 72. Verified. (b) (–21) × [(–4) + (–6)] = (–21) × (–10) = 210. [(–21) × (–4)] + [(–21) × (–6)] = 84 + 126 = 210. Verified. Question 3: (i) (–1) × a = –a. (ii) Integer for –22 is 22. For 37 is –37. For 0 is 0. Question 4: Pattern: (–1) × 5 = –5, (–1) × 4 = –4, (–1) × 3 = –3, (–1) × 2 = –2, (–1) × 1 = –1, (–1) × 0 = 0. Next: (–1) × (–1) = 1.
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Section 1.4: DIVISION OF INTEGERS. Division is inverse of multiplication. 3 × 5 = 15 implies 15 ÷ 5 = 3 and 15 ÷ 3 = 5. Complete the table: 2 × (–6) = –12 gives (–12) ÷ (–6) = 2, (–12) ÷ 2 = –6. (–4) × 5 = –20 gives (–20) ÷ 5 = –4, (–20) ÷ (–4) = 5. (–8) × (–9) = 72 gives 72 ÷ (–8) = –9, 72 ÷ (–9) = –8. (–3) × (–7) = 21 gives 21 ÷ (–3) = –7, 21 ÷ (–7) = –3. (–8) × 4 = –32 gives (–32) ÷ (–8) = 4, (–32) ÷ 4 = –8. 5 × (–9) = –45 gives (–45) ÷ 5 = –9, (–45) ÷ (–9) = 5. (–10) × (–5) = 50 gives 50 ÷ (–10) = –5, 50 ÷ (–5) = –10.
Observing: (–12) ÷ 2 = –6. (–20) ÷ 5 = –4. (–32) ÷ 4 = –8. (–45) ÷ 5 = –9. Dividing negative by positive: divide as whole numbers, put minus sign. Find: (–100) ÷ 5 = –20. (–81) ÷ 9 = –9. (–75) ÷ 5 = –15. (–32) ÷ 2 = –16. Check: (–48) ÷ 8 = –6 and 48 ÷ (–8) = –6. Equal. 90 ÷ (–45) = –2 and (–90) ÷ 45 = –2. Equal. (–136) ÷ 4 = –34 and 136 ÷ (–4) = –34. Equal.
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Also: 72 ÷ (–8) = –9, 72 ÷ (–9) = –8. 50 ÷ (–10) = –5, 50 ÷ (–5) = –10. Dividing positive by negative: divide as whole numbers, put minus sign. In general, a ÷ (–b) = (–a) ÷ b where b ≠ 0. Find: 125 ÷ (–25) = –5. 80 ÷ (–5) = –16. 64 ÷ (–16) = –4. Lastly: (–12) ÷ (–6) = 2. (–20) ÷ (–4) = 5. (–32) ÷ (–8) = 4. (–45) ÷ (–9) = 5. Dividing negative by negative: divide as whole numbers, put positive sign. In general, (–a) ÷ (–b) = a ÷ b where b ≠ 0. Find: (–36) ÷ (–4) = 9. (–201) ÷ (–3) = 67. (–325) ÷ (–13) = 25.
Section 1.5: PROPERTIES OF DIVISION OF INTEGERS. Table: (–8) ÷ (–4) = 2 (integer). (–4) ÷ (–8) = –1/2 (not integer). (–8) ÷ 3 = –8/3 (not integer). 3 ÷ (–8) = –3/8 (not integer). Integers are not closed under division. Check: 1 ÷ a = 1? No, only if a = 1. a ÷ (–1) = –a? Yes, for any integer a. Division is not commutative. (–8) ÷ (–4) ≠ (–4) ÷ (–8). (–9) ÷ 3 = –3, but 3 ÷ (–9) = –1/3. Not equal. (–30) ÷ (–6) = 5, but (–6) ÷ (–30) = 1/5. Not equal. Division is not commutative.
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Any integer divided by zero is undefined. 0 ÷ a = 0 for a ≠ 0. Dividing by 1: (–8) ÷ 1 = –8. (–11) ÷ 1 = –11. (–13) ÷ 1 = –13. (–25) ÷ 1 = –25. (–37) ÷ 1 = –37. (–48) ÷ 1 = –48. In general, a ÷ 1 = a. Dividing by –1: (–8) ÷ (–1) = 8. 11 ÷ (–1) = –11. 13 ÷ (–1) = –13. (–25) ÷ (–1) = 25. (–37) ÷ (–1) = 37. –48 ÷ (–1) = 48. It gives the additive inverse, not the same integer. Associativity? [(–16) ÷ 4] ÷ (–2) = (–4) ÷ (–2) = 2. (–16) ÷ [4 ÷ (–2)] = (–16) ÷ (–2) = 8. Not equal. Division is not associative.
Example 2: (+5) for correct, (–2) for incorrect. (i) Radhika: 10 correct = 5 × 10 = 50. Score = 30. Marks for incorrect = 30 – 50 = –20. Number incorrect = (–20) ÷ (–2) = 10. (ii) Jay: 4 correct = 5 × 4 = 20. Score = –12. Marks for incorrect = –12 – 20 = –32. Number incorrect = (–32) ÷ (–2) = 16.
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Example 3: Profit ₹1/pen, loss 40 paise/pencil. (i) Loss ₹5 = –500 paise. Sold 45 pens = +₹45 = +4500 paise. Total = Profit + Loss. –500 = 4500 + Loss. Loss = –5000 paise. Pencils sold = (–5000) ÷ (–40) = 125. (ii) Neither profit nor loss. Sold 70 pens = +₹70 = +7000 paise. Loss must be –7000 paise. Pencils sold = (–7000) ÷ (–40) = 175.
Exercise 1.3. Question 1: (a) (–30) ÷ 10 = –3. (b) 50 ÷ (–5) = –10. (c) (–36) ÷ (–9) = 4. (d) (–49) ÷ 49 = –1. (e) 13 ÷ [(–2) + 1] = 13 ÷ (–1) = –13. (f) 0 ÷ (–12) = 0. (g) (–31) ÷ [(–30) + (–1)] = (–31) ÷ (–31) = 1. (h) [(–36) ÷ 12] ÷ 3 = (–3) ÷ 3 = –1. (i) [(–6) + 5] ÷ [(–2) + 1] = (–1) ÷ (–1) = 1. Question 2: (a) a=12, b=–4, c=2. 12 ÷ (–2) = –6. (12 ÷ –4) + (12 ÷ 2) = –3 + 6 = 3. –6 ≠ 3. (b) a=–10, b=1, c=1. –10 ÷ 2 = –5. (–10 ÷ 1) + (–10 ÷ 1) = –10 + –10 = –20. –5 ≠ –20.
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Question 3: (a) 369 ÷ 1 = 369. (b) (–75) ÷ 75 = –1. (c) (–206) ÷ (–206) = 1. (d) –87 ÷ (–1) = 87. (e) –87 ÷ 1 = –87. (f) –48 ÷ 48 = –1. (g) 20 ÷ (–10) = –2. (h) –12 ÷ 4 = –3. Question 4: Pairs for a ÷ b = –3: (9, –3), (12, –4), (–15, 5), (–18, 6), (21, –7). Question 5: Start 10°C. Rate –2°C/hr. Target –8°C. Drop = 10 – (–8) = 18°C. Time = 18 ÷ 2 = 9 hrs. 12 noon + 9 hrs = 9 PM. At midnight (3 hrs later), drop = 6°C. Temp = –8 – 6 = –14°C. Question 6: (+3) correct, (–2) incorrect. (i) Radhika: 12 correct = 36. Score 20. Incorrect marks = 20 – 36 = –16. Incorrect count = (–16) ÷ (–2) = 8. (ii) Mohini: 7 correct = 21. Score –5. Incorrect marks = –5 – 21 = –26. Incorrect count = (–26) ÷ (–2) = 13. Question 7: Rate 6 m/min down. Start +10 m. Target –350 m. Distance = 10 – (–350) = 360 m. Time = 360 ÷ 6 = 60 minutes = 1 hour.
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What Have We Discussed? Integers are closed under addition and subtraction. Addition is commutative and associative. Zero is the additive identity. Product of positive and negative is negative. Product of two negatives is positive. Even number of negatives gives positive product, odd gives negative. Multiplication is closed, commutative, and associative. 1 is multiplicative identity. Distributive property holds: a × (b + c) = a × b + a × c. Division: positive ÷ negative = negative. Negative ÷ negative = positive. a ÷ 0 is undefined. a ÷ 1 = a. These properties make calculations easier.
Thank you for listening! Keep revising and practicing. Goodbye! [CHAPTER_COMPLETE]