Warm Up 1-28-09 Each square root is between two integers. Name the two integers. Estimate each value. Round to the nearest tenth. 1. 119 10 and 11 2. – 15 –4 and –3 3. 2 1.4 4. – 123 –11.1

Learn to determine if a number is rational or irrational.

Vocabulary irrational number real number Density Property

Biologists classify animals based on shared characteristics 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. Animal Reptile Lizard Gecko

Recall that rational numbers can be written as fractions Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat. 4 5 23 3 = 3.8 = 0.6 1.44 = 1.2

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!

The set of real numbers consists of the set of rational numbers and the set of irrational numbers. Integers Whole numbers

What are the different types of numbers? Real Numbers Rationals Irrationals Integers Naturals Wholes

Fill In Your Real Number Chart Counting “Natural” Numbers: 1, 2, 3, 4, 5, 6, . . . Whole Numbers: 0, 1, 2, 3, 4, . . . Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . . Rational Numbers: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1, perfect squares Real Numbers: all numbers Irrationals: π, non-repeating decimal, nonperfect squares

Classifying Real Numbers Write all names that apply to each number (whole, integer, rational, irrational, real)

Example 1 A. 5 5 is a whole number that is not a perfect square. irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 = = 2 4 2 16 2 C. whole, integer, rational, real

Example 2 A. 9 9 = 3 whole, integer, rational, real B. –35.9 –35.9 is a terminating decimal. rational, real 81 3 = = 3 9 3 81 3 C. whole, integer, rational, real

Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number.

Example 3 A. 21 irrational 0 3 0 3 = 0 B. rational

Example 3 continued.. C. –4 not a real number 4 9 2 3 = 4 9 D. rational

Example 4 A. 23 23 is a whole number that is not a perfect square. irrational 9 0 B. not a number, so not a real number

Example 4 Continued… C. –7 not a real number 64 81 8 9 = 64 81 D. rational

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.

Find a real number between a set of numbers There are many solutions. Let’s try to find the solution that is halfway between the two numbers

Find a real number between 3 and 3 . 3 5 2 5 Example 5 Find a real number between 3 and 3 . 3 5 2 5 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 2 5 3 + 3 ÷ 2 3 5 5 5 = 6 ÷ 2 1 2 = 7 ÷ 2 = 3 3 1 5 2 5 4 3 5 4 5 3 1 2 A real number between 3 and 3 is 3 . 3 5 2 5 1 2

Find a real number between 4 and 4 . 4 7 3 7 Example 6 Find a real number between 4 and 4 . 4 7 3 7 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 3 7 4 + 4 ÷ 2 4 7 7 7 = 8 ÷ 2 1 2 = 9 ÷ 2 = 4 4 2 7 3 7 4 7 5 7 1 7 6 7 4 1 2 A real number between 4 and 4 is 4 . 4 7 3 7 1 2

Lesson Summary Write all names that apply to each number. 1. 2 2. – 16 2 real, irrational real, integer, rational State if each number is rational, irrational, or not a real number. 25 0 4. 3. 4 • 9 not a real number rational 5. Find a real number between –2 and –2 . 3 8 3 4 Possible answer –2 . 5 8