1. The Real Numbers 3-7 Warm Up Warm Up Lesson Presentation Lesson Presentation Problem of the Day Problem of the Day Lesson Quizzes Lesson Quizzes

2. The Real Numbers 3-7 Warm Up Each square root is between two integers. Name the two integers. Use a calculator to find each value. Round to the nearest tenth. 10 and 11 –4 and –3 1.4 –11.1 1. 119 2. – 15 3. 2 4. – 123

3. The Real Numbers 3-7 Learn to determine if a number is rational or irrational.

4. The Real Numbers 3-7 irrational number real number Density Property Vocabulary

5. The Real Numbers 3-7 Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat. 3 = 3.8 4545 = 0.6 2323 1.44 = 1.2

6. The Real Numbers 3-7 Irrational numbers can only be written as decimals that do not terminate or repeat. If a whole number is not a perfect square, then its square root is an irrational number. 2 ≈1.4142135623730950488016… A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution!

7. The Real Numbers 3-7 The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

8. The Real Numbers 3-7 Example 1: Classifying Real Numbers Write all names that apply to each number. 5 is a whole number that is not a perfect square. 5 irrational, real –12.75 is a terminating decimal. –12.75 rational, real 16 2 whole, integer, rational, real = = 2 4242 16 2 A. B. C.

9. The Real Numbers 3-7 Check It Out Write all names that apply to each number. 9 whole, integer, rational, real –35.9 is a terminating decimal. –35.9 rational, real 81 3 whole, integer, rational, real = = 3 9393 81 3 D. E. F. 9 = 3

10. The Real Numbers 3-7 State if each number is rational, irrational, or not a real number. 21 irrational 0303 rational 0303 = 0 Example 2: Determining the Classification of All Numbers A. B.

11. The Real Numbers 3-7 not a real number Example 2: Determining the Classification of All Numbers –4 4949 rational 2323 = 2323 4949 C. D. State if each number is rational, irrational, or not a real number.

12. The Real Numbers 3-7 23 is a whole number that is not a perfect square. 23 irrational 9090 undefined, so not a real number Check It Out: Example 2 E. F. State if each number is rational, irrational, or not a real number.

13. The Real Numbers 3-7 not a real number –7 64 81 rational 8989 = 8989 64 81 G. H. Check It Out: Example 2 State if each number is rational, irrational, or not a real number.

14. The Real Numbers 3-7 The Density Property of real numbers states that between any two real numbers is another real number. This property is also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between –2 and –3.

15. The Real Numbers 3-7 Example 3: Applying the Density Property of Real Numbers 2525 3 + 3 ÷ 2 3535 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.5 = 6 ÷ 2 1212 = 7 ÷ 2 = 3 3 1212 333 1515 2525 433 3535 4545 Find a real number between 3 and 3. 3535 2525 A real number between 3 and 3 is 3. 3535 2525 1212

16. The Real Numbers 3-7 Check It Out 3737 4 + 4 ÷ 2 4747 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.7 = 8 ÷ 2 1 2 = 9 ÷ 2 = 4 4 1212 444444 2727 3737 4747 5757 1717 6767 Find a real number between 4 and 4. 4747 3737 A real number between 4 and 4 is 4. 4747 3737 1212

17. The Real Numbers 3-7 Lesson Quiz

18. The Real Numbers 3-7 Lesson Quiz Write all names that apply to each number. 1. 2. – State if each number is rational, irrational, or not a real number. 3. 4. Find a real number between –2 and –2. 3838 3434 5. 2 4 9 16 2 25 0 not a real number rational real, irrational real, integer, rational Possible answer –2. 5858