Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

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. 1. 119 10 and 11 2. – 15 –4 and –3 3. 2 1.4 4. – 123 –11.1

Problem of the Day The circumference of a circle is approximately 3.14 times its diameter. A circular path 1 meter wide has an inner diameter of 100 meters. How much farther is it around the outer edge of the path than the inner edge? 6.28 m

Sunshine State Standards MA.8.A.6.2 …Compare mathematical expressions involving real numbers and radical expressions.

Vocabulary irrational number real number Density Property

Biologists classify animals based on shared characteristics Biologists classify animals based on shared characteristics. The cardinal is an animal, a vertebrate, a bird, and a passerine. Animals You already know that some numbers can be classified as whole numbers, integers, or rational numbers. Vertebrates Birds Passerines

Recall that rational numbers can be written as fractions and 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 ≈1.41421356237… 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.

Additional Example 1: Classifying Real Numbers Write all names that apply to each number. 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

Write all names that apply to each number. Check It Out: Example 1 Write all names that apply to each number. A. 9 9 = 3; natural, whole, integer, rational, real 3 5 B. – rational, real C. 3.724 rational, real

Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 0 3 0 3 = 0 A. rational B. –4 not a real number

Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 4 9 2 3 = 4 9 C. rational

Check It Out: Example 2 State if the number is rational, irrational, or not a real number. Justify your answer. A. –7 √–7 is not a real number because it is the square root of a negative number. 6 0 B. – –6 0 is not a real number because division by 0 is not defined.

Check It Out: Example 2 State if the number is rational, irrational, or not a real number. Justify your answer. C. √59 + 5 √59 + 5 = √64 = 8 is a rational number.

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.

Additional Example 3: Applying the Density Property of Real Numbers 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

Check It Out: Example 3 Find a real number between 5 and 5 . 2 8 1 8 1 2 1 8 5 + 5 2 8 83 8 = 1 2 83 8 = 3 16 = 5 3 16 Possible answer: 5

Lesson Quiz for Student Response Systems Lesson Quizzes Standard Lesson Quiz Lesson Quiz for Student Response Systems 19 19

Lesson Quiz 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. Justify your answer. 25 0 4. 3. 4 • 9 not a real number;  by 0 rational; 2  3 is 6 5. Find a real number between –2 and –2 . 3 8 3 4 Possible answer –2 . 5 8

Lesson Quiz for Student Response Systems 1. Identify all names that apply to . A. real, irrational B. real, rational C. not real, irrational D. not real, rational 21 21

Lesson Quiz for Student Response Systems 2. Identify the name that applies to . A. irrational B. rational C. not a real number D. none 22 22

Lesson Quiz for Student Response Systems 3. Identify a real number between . A. –4 B. C. D. 23 23