Rational and Irrational Numbers

1. L E S S O N 9 2 . Recall that a rational number is a number that can be written as a quotient , where a and b are integers and b ≠ 0. If nis a positive integer and is not a perfect square, then n and –n are irrational numbers. a b Rational and Irrational Numbers An irrational number is a number that cannot be written as a quotient of two integers.

2. L E S S O N 9 2 . Rational and Irrational Numbers a b Recall that a rational number is a number that can be written as a quotient , where a and b are integers and b ≠ 0. An irrational number is a number that cannot be written as a quotient of two integers. If nis a positive integer and is not a perfect square, then n and –n are irrational numbers. Together, rational numbers and irrational numbers make up the set of real numbers.

3. L E S S O N 9 2 . Rational numbers Irrational numbers Integers Whole numbers Rational and Irrational Numbers Together, rational numbers and irrational numbers make up the set of real numbers. The Venn diagram shows the relationships among numbers in the real number system. • The decimal form of a rational number is either terminating or repeating. • The decimal form of an irrational number does not terminate or repeat.

4. L E S S O N Rational and Irrational Numbers 9 2 3 4 . 1 11 3 1 EXAMPLE Classifying Real Numbers Rational or Irrational and why? Number Type of Decimal Decimal Form

5. Evaluate the expression when a = 9 b = 15 and c = 16. Tell whether the answer is rationalorirrational 5. 6. 7.

6. Stop here for day 1

7. L E S S O N Rational and Irrational Numbers 9 2 . ? 2 2 . 2 . 2 ≈ 1.4142 So, 2 < 2. 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2 EXAMPLE Comparing Real Numbers Graph the pair of numbers on a number line. Then copy and complete the statement with <,>, or =. Use a calculator to approximate the square root and write any fractions as decimals. Then graph the numbers on a number line and compare. SOLUTION

8. L E S S O N Rational and Irrational Numbers 9 2 . 1 2 1 2 1 2 1 2 ? 1 2 1 2 ≈ 0.7071 = 0.5 . . 3 EXAMPLE So, >. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Comparing Real Numbers Graph the pair of numbers on a number line. Then copy and complete the statement with <,>, or =.

9. L E S S O N Rational and Irrational Numbers 9 2 . Order the decimals 0.47, 0.474, 0.47, and 0.477 from least to greatest. 1 2 0.474 = 0.47 = 0.47 = 3 EXAMPLE Ordering Decimals Write each decimal out to six decimal places. Notice that the first two digits after the decimal point are the same for each number. Use the second pair of digits to order the decimals. 0.477 From least to greatest, the order of the numbers is 0.474474…, 0.474747…, 0.4770, and 0.477777….

10. L E S S O N Rational and Irrational Numbers 9 2 . 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 Practice • Tell whether the number is rational or irrational. • 1.2. • Graph the pair of numbers on a number line. Then copy and complete the statement with <, >, or =. • 3. • Order the numbers from least to greatest. • 4.

11. L E S S O N Rational and Irrational Numbers 9 2 h 0.019 9.5 0.019 h 0.019 . Waves For large ocean waves, the wind speed s in knots and the height of the waves hin feet are related by the equation s= . If the waves are about 9.5 feet tall, what must the wind speed be? (1 knot is equivalent to 1.15 miles per hour.) s = = = 500 4 EXAMPLE Using an Irrational Number SOLUTION Write original equation. Substitute 9.5 for h. Divide. ≈ 22.36 Approximate square root. ANSWER The wind speed must be about 22.36 knots.