4 8 The Real Numbers Warm Up Problem

4 -8 The Real Numbers Warm Up Problem of the Day Lesson Presentation Lesson Quizzes

4 -8 The Real Numbers Warm Up Each square root is between two integers. Name the two integers. 1. 119 2. – 15 10 and 11 – 4 and – 3 Use a calculator to find each value. Round to the nearest tenth. 3. 4. – 2 1. 4 123 – 11. 1

4 -8 The Real Numbers 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

4 -8 The Real Numbers Sunshine State Standards MA. 8. A. 6. 2 …Compare mathematical expressions involving real numbers and radical expressions.

4 -8 The Real Numbers Vocabulary irrational number real number Density Property

4 -8 The Real Numbers Biologists classify animals based on shared characteristics. The cardinal is an animal, a vertebrate, a bird, and a passerine. Animals Vertebrates Birds Passerines You already know that some numbers can be classified as whole numbers, integers, or rational numbers.

4 -8 The Real Numbers Recall that rational numbers can be written as fractions and as decimals that either terminate or repeat. 3 4 = 3. 8 5 23 = 0. 6 1. 44 = 1. 2

4 -8 The Real Numbers 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… Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

4 -8 The Real Numbers The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

4 -8 The Real Numbers Additional Example 1: Classifying Real Numbers Write all names that apply to each number. A. 5 is a whole number that is not a perfect square. irrational, real 5 B. – 12. 75 is a terminating decimal. rational, real C. 16 2 16 4 = =2 2 2 whole, integer, rational, real

4 -8 The Real Numbers Check It Out: Example 1 Write all names that apply to each number. A. 9 9 = 3; natural, whole, integer, rational, real 3 – B. 5 rational, real C. 3. 724 rational, real

4 -8 The Real Numbers Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. A. 0 3 0 =0 3 rational B. – 4 not a real number

4 -8 The Real Numbers Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. C. 4 9 rational 2 3 2 4 = 3 9

4 -8 The Real Numbers 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 B. – 0 – 6 is not a real number because division by 0 is not 0 defined.

4 -8 The Real Numbers Check It Out: Example 2 State if the number is rational, irrational, or not a real number. Justify your answer. C. √ 59 + 5 = √ 64 = 8 is a rational number.

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 also true for rational numbers, but not for whole numbers or integers. For instance, there is no integer between – 2 and – 3.

4 -8 The Real Numbers Additional Example 3: Applying the Density Property of Real Numbers 2 3 Find a real number between 3 and 3. 5 5 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 2 3 5 3 +3 ÷ 2 =6 ÷ 2 5 5 5 1 2 3 4 =7÷ 2=3 1 2 4 3 5 13 5 35 32 3 2 1 A real number between 3 and 3 is 3. 5 5 2 3

4 -8 The Real Numbers Check It Out: Example 3 1 2 Find a real number between 5 and 5. 8 8 1 83 1 2 1 5 +5 = 2 8 8 8 2 83 = 8 3 =5 16 Possible answer: 5 3 16

4 -8 The Real Numbers Lesson Quizzes Standard Lesson Quiz for Student Response Systems

4 -8 The Real Numbers Lesson Quiz Write all names that apply to each number. 1. 2. – 16 2 2 real, integer, rational real, irrational State if each number is rational, irrational, or not a real number. Justify your answer. 3. 25 0 not a real number; by 0 4. 4 • 9 rational; 2 3 is 6 5. Find a real number between – 2 3 and – 2 3. Possible answer – 25. 8 4 8

4 -8 The Real Numbers 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 .

4 -8 The Real Numbers Lesson Quiz for Student Response Systems 2. Identify the name that applies to A. irrational B. rational C. not a real number D. none .

4 -8 The Real Numbers Lesson Quiz for Student Response Systems 3. Identify a real number between A. – 4 B. C. D. .