1. Evaluating Algebraic Expressions 4-8 The Real Numbers NS1.4 Differentiate between rational and irrational numbers. California Standards

2. Evaluating Algebraic Expressions 4-8 The Real Numbers Animal Reptile Biologists classify animals based on shared characteristics. The horned lizard is an animal, a reptile, a lizard, and a gecko. You already know that some numbers can be classified as whole numbers, integers, or rational numbers. The number 2 is a whole number, an integer, and a rational number. It is also a real number. Lizard Gecko

3. Evaluating Algebraic Expressions 4-8 The Real Numbers The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers. Irrational numbersRational numbers Real Numbers Integers Whole numbers

4. Evaluating Algebraic Expressions 4-8 The Real Numbers Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 3 = 3.8 4545 = 0.6 2323 1.44 = 1.2

5. Evaluating Algebraic Expressions 4-8 The Real Numbers A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Caution! Irrational numbers can be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational.

6. Evaluating Algebraic Expressions 4-8 The Real Numbers Additional Example 1: Classifying Real Numbers Write all classifications 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.

7. Evaluating Algebraic Expressions 4-8 The Real Numbers Check It Out! Example 1 Write all classifications 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 A. B. C. 9 = 3

8. Evaluating Algebraic Expressions 4-8 The Real Numbers A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

9. Evaluating Algebraic Expressions 4-8 The Real Numbers State if each number is rational, irrational, or not a real number. 21 irrational 0303 rational 0303 = 0 Additional Example 2: Determining the Classification of All Numbers A. B.

10. Evaluating Algebraic Expressions 4-8 The Real Numbers not a real number Additional Example 2: Determining the Classification of All Numbers 4040 C. State if each number is rational, irrational, or not a real number.

11. Evaluating Algebraic Expressions 4-8 The Real Numbers 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 A. B. State if each number is rational, irrational, or not a real number.

12. Evaluating Algebraic Expressions 4-8 The Real Numbers 64 81 rational 8989 = 8989 64 81 C. Check It Out! Example 2 State if each number is rational, irrational, or not a real number.

13. Evaluating Algebraic Expressions 4-8 The Real Numbers The Density Property of real numbers states that between any two real numbers is another real number. This property is not true when you limit yourself to whole numbers or integers. For instance, there is no integer between –2 and –3.

14. Evaluating Algebraic Expressions 4-8 The Real Numbers Additional 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 Check: Use a graph.

15. Evaluating Algebraic Expressions 4-8 The Real Numbers Check It Out! Example 3 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 1212 = 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 Check: Use a graph.