Chapter 13 Lesson Real Numbers and 13-7A the Number Line BIG IDEA On a real number line, both rational and irrational numbers can be graphed. As we noted in Lesson 13-7, every real number is either a rational number or an irrational number. In this lesson you will explore some important properties of rational and irrational numbers. Rational Numbers on the Number Line Recall that a rational number is a number that can be expressed as a simple fraction. When a rational number is written as a fraction, it can be rewritten as a decimal by dividing the numerator by the denominator. _5 The result will either__ be terminating, such as = 0.625, or repeating, 10 8 such as _ = 0. 90 . This makes it possible to graph the number on a 11 number line. GUIDED Example 1 Graph 0.8 3 on a number line. __ __ Solution Let x = 0.83 . Change 0.8 3 into a fraction. _ Then 10x = 8.3_ 3 x = 0.8 3 ? x = 7.5 Subtract _7.5 _? _? x = = = 9 90 6 _? ? To graph 6 , divide the__ interval 0 to 1 into equal spaces. Locate the point corresponding to 0.8 3 . 0 1 When two different rational numbers are graphed on a number line, there are always many rational numbers whose graphs are between them. 1 Using Algebra to Prove Lesson 13-7A Example 2 _11 _20 Find a rational number between 13 and 23 . Solution 1 Find a common denominator by multiplying 13 · 23, which is 299. _11 _253 _20 _260 = ; = . 13 299 23 299 _253 _260 _255 So pick a fraction between and , such as . 299 299 299 255 299 11 253 260 20 13 = 299 299 = 23 Solution 2 Change the fractions to decimals. _11 = 0.8461... 13 _20 = 0.8695... 23 Pick a number, such as 0.85, that is between the two numbers. 0.85 11 20 13 ≈ 0.846 23 ≈ 0.870 In Example 2, we found a number between two rational numbers, but is it possible that there are two rational numbers so close together that no other rational number is between them? The answer is no. We prove this by showing that the mean of two numbers is always between those two numbers. Here is the proof: Let a and b be rational numbers with a < b. By its defi nition, the mean a + b a + b of a and b is _ . We fi rst need to show that a < _ . Starting with 2 2 a < b we can show this using properties of rational numbers. a < b Given a b _ < _ Multiplication Property of Inequality 2 2 a a b a _ + _ < _ + _ Addition Property of Inequality 2 2 2 2 2a b + a _ < _ Distributive Property 2 2 b + a a < _ Equal Fractions Property 2 a + b a < _ Commutative Property 2 a + b The proof that _ < b is similar. You are asked to complete it in 2 a + b a + b a + b Question 16. Since a < _ and _ < b, _ is between a and b. 2 2 2 QY1 Therefore, it is always possible to fi nd a rational number between two distinct rational numbers. Find a rational number 2 3 between _ and _ . QY1 3 4 Real Numbers and the Number Line 2 Chapter 13 Some Irrational Numbers on the Number Line Every irrational number can be written as an infi nite, non-repeating decimal. For this reason, you cannot write their exact decimal representation. However, you can still know exactly where some irrational numbers are graphed on the number line. Activity MATERIALS a straightedge, a protractor, scissors Step 1 Create a right triangle NUM with the right angle at M, MU = 1 cm, and N MN = 3 cm. Step 2 Calculate NU. 3 cm Step 3 Cut out NUM. Step 4 Draw a number line showing coordinates from –2 to 4. Make each unit 1 cm in length. __ ___ M U Step 5 Using side NU , graph the exact value of NU = √10 . 1 cm QY2 QY2 ___ √10 is between which two consecutive integers? Questions COVERING THE IDEAS In 1–4, tell whether the statement is always true, sometimes but not always true, or never true. 1. It is possible to write an exact decimal for a rational number. 2. It is possible to write an exact decimal for an irrational number. 3. It is possible to locate a rational number exactly on a number line. 4. A real number is either rational or irrational. In 5–7, graph each rational number on a number line. 4 5. 8 _ 7 _ 6. -0. 5 __ 7. 3. 27 In 8–10, nd a number between the given numbers. 5 6 8. - _ , - _ 7 7 5 7 9. __ , __ 12 15 3 4 10. 2 _ , 2 _ 5 5 ___ 11. Explain how you could show an exact length for √17 . 3 Using Algebra to Prove Lesson 13-7A APPLYING THE MATHEMATICS 12. Find a number between 2 5 a. _ and _ . Call your value a. 3 6 2 _ b. 3 and a. Call this value b. 2 _ c. 3 and b. Call this value c. 2 d. _ and c. 3 A e. How long could you continue this pattern? (Hint: The answer is not 26 times.) 13. a. Find AC in right ABC at the right. b. Explain___ how to use the result in Part a to fi nd an exact length for √40 . __ __ 6 x 14. a. Draw an accurate rectangle with length √8 cm and width √2 cm. b. Use your calculator to determine the area of the rectangle. 15. H and I are graphed on the number line below. B C H I 2 0 2 4 a. Fill in the Blanks H = ? and I = ? . b. List ten rational numbers between H and I. c. List fi ve irrational numbers between H and I. a + b 16. Complete the proof from the lesson by showing that _ < b. 2 (Hint: The structure of your proof should be similar to that of the a + b proof that a < _ .) 2 QY ANSWERS 1. Answers vary. Sample: _7 _8 10 , 11 . 2. 3 and 4 Real Numbers and the Number Line 4.