Integers

Integers are the collection of whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, .... This chapter studies properties of operations on integers. Integers are closed under addition, subtraction and multiplication, which means the result is always an integer. Addition and multiplication are commutative and associative, while subtraction is not commutative, since 5 - 8 = -3 but 8 - 5 = 3. Zero is the additive identity and 1 is the multiplicative identity, and multiplication distributes over addition, as in -2 x (3 + 5) = (-2 x 3) + (-2 x 5). The chapter gives clear sign rules. The product of a positive and a negative integer is negative, and the product of two negative integers is positive, so (-4) x (-5) = 20. A product of an even number of negative integers is positive, while an odd number gives a negative result. Division follows the same sign rules, but integers are not closed under division and division by zero is not defined. These ideas apply to temperatures, profit and loss, and negative marking.

Detailed Worksheet: Integers Section A - Definitions (10 marks) 1. What are integers? Write three positive and three negative integers between -10 and 10. (2 marks) 2. State the closure property for addition and multiplication of integers, with an example of each. (2 marks) 3. What is the additive identity and the multiplicative identity for integers? Give an example of each. (2 marks) 4. State the rule for the sign of the product of (i) two negative integers (ii) one positive and one negative integer. (2 marks) 5. Write the distributive property of multiplication over addition and verify it for -3, 4 and -6. (2 marks) Section B - Calculations and Applications (15 marks) 6. Find: (i) (-25) + 17 (ii) (-12) - (-30) (iii) 35 - (-15) (iv) (-8) + (-13) + 21 (v) 0 - (-52) (vi) (-40) - 18. (3 marks) 7. Find the products: (i) 3 x (-1) (ii) (-1) x 225 (iii) (-21) x (-30) (iv) (-316) x (-1) (v) (-15) x 0 x (-18) (vi) (-12) x (-11) x 10. (3 marks) 8. Evaluate: (i) (-30) divided by 10 (ii) 50 divided by (-5) (iii) (-36) divided by (-9) (iv) (-31) divided by [(-30) + (-1)] (v) [(-36) divided by 12] divided by 3 (vi) 0 divided by (-12). (3 marks) 9. In a class test with 10 questions, 3 marks are given for every correct answer and -2 marks for every incorrect answer. Gurpreet attempts all questions and gets 4 correct. What is her score? Radhika gets 5 correct; what is her score? (3 marks) 10. The temperature at 12 noon was 10 degrees C above zero. It decreases at the rate of 2 degrees C per hour until midnight. (i) At what time is it 8 degrees C below zero? (ii) What is the temperature at midnight? (3 marks) Section C - Diagrams (10 marks) 11. Using a number line, find (i) (-3) + 5 (ii) 4 - 7 (iii) (-2) + (-4) (iv) 3 x (-2) shown as repeated jumps. (4 marks) 12. Draw a vertical number line like a thermometer and mark the temperatures of Srinagar -2 degrees C, Shimla 5 degrees C, Bengaluru 22 degrees C and Kufri -4 degrees C. Find the difference between the hottest and coldest place. (3 marks) 13. Draw a lift diagram for a mine shaft. A cage goes down at 6 m per minute starting 10 m above ground. Mark its position after 5 minutes and after 15 minutes. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Observe the pattern: 3 x 2 = 6, 3 x 1 = 3, 3 x 0 = 0, 3 x (-1) = -3, ... and (-3) x 2 = -6, (-3) x 1 = -3, (-3) x 0 = 0, .... (i) Continue both patterns for three more steps. (ii) What rule do you get for the product of two negative integers? (iii) Find (-1) x (-1) x (-1) x (-1) x (-1). (iv) Is the product of six negative integers positive or negative? (5 marks) 15. Check each property with your own examples and state whether it is true: (i) subtraction is commutative (ii) division is associative (iii) integers are closed under division (iv) a x (b - c) = a x b - a x c (v) a divided by 1 = a. (5 marks) 16. A shopkeeper earns a profit of Rs 1 by selling one pen and incurs a loss of 40 paise per pencil while selling pencils of her old stock. (i) In a month she has a loss of Rs 5. She sold 45 pens; how many pencils did she sell? (ii) In the next month she has neither profit nor loss and sells 70 pens; how many pencils did she sell? (iii) Represent profit and loss with integers. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Write the sign rule used at each step and use brackets around negative integers.