Rational Numbers

A rational number is any number that can be written as p/q, where p and q are integers and q is not zero, such as 3/4, -5/7, 0 and 2. In this chapter you will study the properties of rational numbers under addition, subtraction, multiplication and division. You will see that rational numbers are closed under addition, subtraction and multiplication, that addition and multiplication are commutative and associative, and that multiplication distributes over addition. You will learn the roles of 0 as the additive identity and 1 as the multiplicative identity, and find the additive inverse (negative) and the multiplicative inverse (reciprocal) of a rational number. You will represent rational numbers such as 7/4 and -5/6 on a number line by dividing each unit into equal parts. Finally, you will discover that between any two rational numbers there are infinitely many rational numbers, and find them by taking means or by writing equivalent fractions with larger denominators.

Detailed Worksheet: Rational Numbers Section A - Definitions (10 marks) 1. Define a rational number. Is every integer a rational number? Give a reason. (2 marks) 2. Under which operations are rational numbers closed? Why are rational numbers not closed under division? (2 marks) 3. What are the additive identity and the multiplicative identity for rational numbers? Give one example of each. (2 marks) 4. Define the additive inverse and the reciprocal of a rational number. Write both for -3/5. (2 marks) 5. State the distributive property of multiplication over addition for rational numbers, with an example. (2 marks) Section B - Calculations and Applications (15 marks) 6. Using appropriate properties, find (-2/3) x (3/5) + 5/2 - (3/5) x (1/6). (3 marks) 7. Verify the associative property of addition for -2/3, 3/4 and -1/6. (3 marks) 8. Using appropriate properties, find (2/5) x (-3/7) - 1/14 - (3/7) x (3/5). (3 marks) 9. Find five rational numbers between 2/3 and 4/5. (3 marks) 10. (i) Use the distributive property to find (3/7) x (7/9 + 14/15). (ii) Find the number which, when multiplied by -8/13, gives 1. (3 marks) Section C - Diagrams (10 marks) 11. Draw a number line and represent 7/4 and -5/4 on it, showing how each unit is divided. On a second number line, represent 2/3 and -1/3. (4 marks) 12. Draw a number line from 0 to 1 divided into fifths and mark 1/5, 2/5, 3/5 and 4/5. Then show, by dividing further, three rational numbers between 1/5 and 2/5. (3 marks) 13. Draw a diagram of nested regions showing natural numbers, whole numbers, integers and rational numbers, and place 5, 0, -3, 2/7 and -1.5 in the correct regions. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. (i) Is subtraction commutative for rational numbers? Check with 1/2 and 1/3. (ii) Is division associative? Check with 1/2, 1/3 and 1/4. (iii) Name the property used in -3/5 x (2/7 x 4/9) = (-3/5 x 2/7) x 4/9. (5 marks) 15. State whether each statement is true or false, with a reason: (i) 1/2 divided by 1/3 equals 1/6. (ii) Zero has a reciprocal. (iii) The reciprocal of -1 is 1. (iv) Every rational number has an additive inverse. (v) There are only a finite number of rational numbers between 1/2 and 1. (5 marks) 16. Ramesh had 3/4 of a cake. He gave 1/3 of what he had to Seema and then 1/4 of the remaining cake to Gopi. What fraction of the whole cake did Gopi get, and what fraction was left with Ramesh? Show which operations on rational numbers you used. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Name the property used at each step wherever asked, and write answers in their lowest terms.