Number Systems

This chapter builds the system of real numbers. You will revise natural numbers, whole numbers, integers and rational numbers, which can be written as p/q with p and q integers and q not zero. You will meet irrational numbers, which cannot be written in the form p/q, such as sqrt(2), sqrt(3), pi and 0.10110111011110..., where sqrt(n) stands for the square root of n. Rational and irrational numbers together make up the real numbers, and every real number is represented by a unique point on the number line. You will locate sqrt(2) and sqrt(3) on the number line using the Pythagoras theorem. You will learn that the decimal expansion of a rational number is either terminating, as in 3/8 = 0.375, or non-terminating recurring, as in 1/7 = 0.142857142857..., while that of an irrational number is non-terminating non-recurring. You will convert recurring decimals to p/q form, perform operations on real numbers, rationalise denominators, and apply the laws of exponents to real numbers with rational exponents, such as 64^(1/2) = 8.

Detailed Worksheet: Number Systems Section A - Definitions (10 marks) 1. Define a rational number and an irrational number with two examples of each. (2 marks) 2. What are real numbers? Is every real number a rational number? Explain. (2 marks) 3. What kinds of decimal expansions do rational numbers have? Give one example of each kind. (2 marks) 4. What is meant by rationalising the denominator? Rationalise 1/sqrt(7). (2 marks) 5. State the laws of exponents for real numbers x and y, with rational powers m and n. (2 marks) Section B - Calculations and Applications (15 marks) 6. Write the decimal expansions of 1/11, 7/8 and 3/13, and state the type of each. (3 marks) 7. Express in the form p/q: (i) 0.6666... (ii) 0.47777... (iii) 1.272727... (3 marks) 8. Rationalise the denominators: (i) 1/(sqrt(7) - sqrt(6)) (ii) 1/(sqrt(5) + sqrt(2)) (iii) 1/(2 + sqrt(3)). (3 marks) 9. Find: (i) 64^(1/2) (ii) 32^(2/5) (iii) 125^(-1/3) (iv) 16^(3/4). (3 marks) 10. Simplify: (i) (3 + sqrt(3))(2 + sqrt(2)) (ii) (sqrt(5) + sqrt(2))^2 (iii) (sqrt(11) - sqrt(7))(sqrt(11) + sqrt(7)). (3 marks) Section C - Diagrams (10 marks) 11. Locate sqrt(2) and sqrt(3) on the number line using the Pythagoras theorem. Show all construction steps. (4 marks) 12. Draw the square root spiral for sqrt(2), sqrt(3) and sqrt(4), starting from a unit length. (3 marks) 13. Use successive magnification to show the position of 3.765 on the number line. (3 marks) Section D - Analysis and Higher-order Thinking (15 marks) 14. Show that 0.9999... = 1. Explain why this is not a contradiction, and find p/q for 0.001001001... (5 marks) 15. Find three different irrational numbers between 5/7 and 9/11, and explain how you know they are irrational. Also insert three rational numbers between them. (5 marks) 16. Explain why the sum of a rational number and an irrational number is irrational, and give examples to show that the sum or product of two irrational numbers may be rational or irrational. (5 marks) Instructions: Time allowed 2 hours. Attempt all sections. Show every step of simplification and construction clearly.