4 VIII CBSE Mathematics – Rational Numbers Instruction: This booklet can be used while watching videos. Keep filling the sheet as the videos proceed. 1. Introduction Have you ever thought about a life without numbers??? Can you really imagine that? A life without knowing when you are born!! A life without mobile phone numbers!! A life in which how much money you have in your wallet!! Question 1. The numbers which are used for counting objects are called _____________. Question 2. The number which is part of whole numbers and not a part of natural numbers is ________. Question 3. The set of positive and negative integers together with 0 is called ___________. Question 4. When we divide an integer by another integer, the resulting number is always an integer. (True/False) a. Rational numbers p Numbers which are represented in the form where p and q are integers and q ≠ 0 are q rational numbers. Set of numbers and their notation Set of numbers Notation Natural numbers N Whole numbers W Integers I or Z Rational numbers Q Question 5. All the integers are rational numbers. Justify the statement. Copyright © Think and Learn Pvt. Ltd. 5 VIII CBSE Mathematics – Rational Numbers 2. Properties of rational numbers a. Closure property Operation Closed (Yes/No) Addition 3 2 Yes + = ________ 7 5 4 2 − + = __________ 9 3 Conclusion: Rational numbers are closed under addition Subtraction 3 2 − = ________ 7 5 −4 2 − = __________ 9 3 Conclusion: Multiplication 2 −4 × = ________ 3 5 4 7 × = _________ 7 2 Conclusion: Division 5 2 ÷ = _______ 3 9 2 ÷ 0 = _________ 3 Conclusion: b. Commutative property (i) Commutative property for whole numbers and integers. For any two whole numbers a and b, a + b = b + a and a × b = ___________ But, a − b ≠ b − a and a ÷ b ≠ ________ Therefore, addition and multiplication are commutative for whole numbers. Subtraction and division are not commutative for whole numbers. Copyright © Think and Learn Pvt. Ltd. 6 VIII CBSE Mathematics – Rational Numbers Question 6. Define commutative property for integers. (ii) Commutative property for rational numbers. Operation Commutative (Yes/No) Addition 3 2 + = ________ 7 5 2 3 + = ________ 5 7 Conclusion: Addition is commutative for rational numbers. Subtraction 3 2 − = ________ 7 5 2 3 − = __________ 5 7 Conclusion: Multiplication 2 −4 × = ________ 3 5 −4 2 × = _________ 5 3 Conclusion: Division 5 2 ÷ = _______ 3 9 2 5 ÷ = _________ 9 3 Conclusion: Copyright © Think and Learn Pvt. Ltd. 7 VIII CBSE Mathematics – Rational Numbers c. Associative property (i) Associative property for whole numbers and integers For any three whole numbers a, b and c; (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c) Therefore, addition and multiplication are associative for whole numbers. But subtraction and division are not associative for whole numbers. Similarly, Addition and multiplication are associative for integers. But subtraction and division are not associative for integers. (ii) Associative property for rational numbers Operation Associative (Yes/No) Addition 2 −1 3 ( + ) + = ___________ 3 4 8 2 −1 3 + ( + ) = __________ 3 4 8 Conclusion: Addition is associative for rational numbers Subtraction 3 2 1 ( − ) − = ________ 5 3 6 3 2 1 − ( − ) = ________ 5 3 6 Conclusion: Multiplication −7 1 2 ( × ) × = ________ 3 4 5 −7 1 2 × ( × ) = ________ 3 4 5 Conclusion: Division 1 1 1 ( ÷ ) ÷ = _______ 2 3 4 1 1 1 ÷ ( ÷ ) = ________ 2 3 4 Conclusion: d. Distributive property of multiplication over addition For any three rational numbers a, b and c; a × (b + c) = a × b + a × c Copyright © Think and Learn Pvt. Ltd. 8 VIII CBSE Mathematics – Rational Numbers 3 2 5 Question 7. Verify distributive property of multiplication over addition taking the numbers , and . 4 3 7 Solution: 3 2 5 × ( + ) = 4 3 7 3 2 3 5 × + × = 4 3 4 7 Question 8. Simplify the following using commutative or associative property. 6 5 16 14 × × × 16 14 21 15 Question 9. Simplify the following using distributive property. 2 5 2 3 × + × 7 8 7 8 Copyright © Think and Learn Pvt. Ltd. 9 VIII CBSE Mathematics – Rational Numbers e. Additive identity Question 10. What do you get when ‘zero’ is added to a number? Therefore, zero is called the identity for addition or additive identity for rational numbers. f. Multiplicative identity Question 11. What do you get when a number is multiplied with 1? Therefore, 1 is called the multiplicative identity for rational numbers. 3. Negatives and reciprocals a. Negative of a number If sum of two numbers is zero, then one number is called negative of the other number. Question 12. Negative of a number is also called _____________________. b. Reciprocal of a number If the product of two rational numbers is 1 then one rational number is called the reciprocal or multiplicative inverse of other rational number. a Question 13. If is a rational number, then _____ is its reciprocal. b Question 14. Given: (x + y + z) × (ab + 1) = 1 Find the value of z if x is additive inverse of y and a is multiplicative inverse of b. Copyright © Think and Learn Pvt. Ltd. 10 VIII CBSE Mathematics – Rational Numbers Did you know? There is no rational number which when multiplied by 0 gives 1. Thus, zero has no reciprocal. 4. Representation of rational numbers on the number line Question 15. Mark -3 on the number line given below. 1 Question 16. Mark − on the number line. 2 4 10 Question 17. Mark and on the number line. 7 7 8 Question 18. Mark − on the number line. 5 Helping hand: The denominator of the rational number indicates the number of equal parts into which each unit is to be divided. The numerator indicates how many such parts are to be considered. Copyright © Think and Learn Pvt. Ltd. 11 VIII CBSE Mathematics – Rational Numbers 5. Rational numbers between two rational numbers Question 19. How many natural numbers are there in between -5 and 3? Question 20. Write 6 rational numbers between 2 and 4. Number of rational numbers between two rational number is not infinite. Any number of rational numbers can be inserted between two rational numbers. 1 1 Question 21. Find 4 rational numbers between and . 4 6 Helping hand: Take any two rational numbers and find their mean. It is also a rational number. Copyright © Think and Learn Pvt. Ltd. 13 VIII CBSE Mathematics – Rational Numbers Instructions: Solve these problems after watching the videos. Questions from 1 to 7 are MCQ’s 11 1. For what value of ‘a’ the number − is not a rational number. a A) 0 B) 1 C) 2 D) 3 2. Rational numbers are not closed under A) Addition C) Division B) Multiplication D) Subtraction a 7 3. If additive inverse of is − . Then value of (b − a) can be, b 29 A) 22 B) -22 C) -36 D) 73 4. If the multiplicative inverse of 15/3 is a/b, then value of (a + b) can be, A) 3 B) 4 C) 5 D) 6 a c 5. A rational number is greater than if (for all positive a, b, c, d) b d A) a > c C) b < d B) ad < bc D) ad > bc 6. If x + y = 2 + (3)0.5 , then select the correct statements for relation between x and y A) x is rational and y is irrational C) x can be rational or irrational B) y is rational and x is irrational D) none of these 61 7. If the sum of three rational numbers is − then what is the sum of the additive inverses of the 15 three rational numbers? 15 61 A) C) 61 15 15 61 B) − D) − 61 15 Questions from 8 to are 20 subjective. 8. Division of two irrational numbers can be rational. (True/False) 9. If a + 5 = 10 + b, then any one of a or b can be irrational. Tell whether above statement is correct or not. Then explain why so? 1 2 10. Write any three rational and irrational numbers between and 2 3 11. Arrange the following numbers in ascending order 22 21 6 6 1 , π, , , , 3, 7 6 21 22 3 12. Use the following figure and find the value of (A + B + C + D + E) + (J + I + H + G + F) Copyright © Think and Learn Pvt. Ltd. 14 VIII CBSE Mathematics – Rational Numbers 13. Explain whether rational numbers are close under addition, subtraction, multiplication and division or not. Give at least one example for each operation. 1 3 14. Rakesh earns Rs. 20000 per month. He spends of his income on food; of the remainder on 4 10 5 house rent and of the remainder on education of children. How much money is he left with? 21 15. Set of operations is being done on number A. i) A is multiplied by 5 to get B ii) 5 is added to B to get C. iii) 10 is subtracted from C to get D. iv) 1 is subtracted from D to get 9. Find the value of A. 16. Take any positive number find its inverse. Is the inverse is also positive? Same way what will be inverse of a negative number. 1 17. If you have 50 chocolates and you distributed th of the chocolates among your friends then 12 number of chocolates remaining with you is ______. 1 5 18. The sum of two rational numbers is − . If one of the numbers is , find the other number. 2 6 33 627 19. A tin holds liters, then how many such tins will be required to hold litres of oil? 2 2 20.