1. Rational and Irrational Numbers 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. abab An irrational number is a number that cannot be written as a quotient of two integers. If n is a positive integer and is not a perfect square, then n and – n are irrational numbers.

2. Rational and Irrational Numbers 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. abab Together, rational numbers and irrational numbers make up the set of real numbers. An irrational number is a number that cannot be written as a quotient of two integers. If n is a positive integer and is not a perfect square, then n and – n are irrational numbers.

3. Rational and Irrational Numbers 9 2. Irrational numbers Rational numbers Integers Whole numbers  The decimal form of a rational number is either terminating or repeating.  The decimal form of an irrational number does not terminate or repeat. 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.

4. EXAMPLE 1 Classifying Real Numbers Number Rational and Irrational Numbers 9 2. Rational or Irrational and why? Decimal Form Type of Decimal 3434 1 11 3

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

6. Stop here for day 1

7. EXAMPLE 2 Comparing Real Numbers Graph the pair of numbers on a number line. Then copy and complete the statement with, or =. Rational and Irrational Numbers 9 2. 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 1.51.51.11.11.21.21.31.31.41.41.71.71.61.611.81.821.91.92.12.1 2 ≈ 1.4142. So, 2 < 2. 2 ? 2 2.

8. EXAMPLE 3 Comparing Real Numbers Graph the pair of numbers on a number line. Then copy and complete the statement with, or =. Rational and Irrational Numbers 9 2. ? 1212 1212 0.50.50.10.10.20.20.30.30.40.40.70.70.60.600.80.810.90.91.1 ≈ 0.7071 1212.. = 0.5 1212 So, >. 1212 1212

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

10. Rational and Irrational Numbers 9 2. 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. 3.22.82.933.13.43.32.73.53.73.63.8

11. EXAMPLE 4 Using an Irrational Number Rational and Irrational Numbers 9 2. SOLUTION The wind speed must be about 22.36 knots. ANSWER s = h 0.019 Waves For large ocean waves, the wind speed s in knots and the height of the waves h in 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.) h 0.019 9.5 0.019 = 500 = ≈ 22.36 Write original equation. Substitute 9.5 for h. Divide. Approximate square root.