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Transcript

3-7 The Real Numbers Warm Up Problem of the Day Lesson Presentation Lesson Quizzes 3-7 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 3-7 The Real Numbers Learn to determine if a number is rational or irrational. 3-7 The Real Numbers Vocabulary irrational number real number Density Property 3-7 The Real Numbers Recall that rational numbers can be written as fractions. Rational numbers can also be written as decimals that either terminate or repeat. 3 4 = 3.8 5 2 = 0.6 3 1.44 = 1.2 3-7 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 ≈1.4142135623730950488016… Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. 3-7 The Real Numbers The set of real numbers consists of the set of rational numbers and the set of irrational numbers. 3-7 The Real Numbers 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 –12.75 is a terminating decimal. rational, real C. 16 2 16 4 = =2 2 2 whole, integer, rational, real 3-7 The Real Numbers Check It Out Write all names that apply to each number. D. 9 9 =3 whole, integer, rational, real E. F. –35.9 –35.9 is a terminating decimal. rational, real 81 81 9 = =3 3 3 3 whole, integer, rational, real 3-7 The Real Numbers Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. A. 21 irrational B. 0 3 rational 0 =0 3 3-7 The Real Numbers Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. C. –4 not a real number D. 4 9 rational 2 3 2 4 = 3 9 3-7 The Real Numbers Check It Out: Example 2 State if each number is rational, irrational, or not a real number. E. 23 23 is a whole number that is not a perfect square. irrational F. 9 0 undefined, so not a real number 3-7 The Real Numbers Check It Out: Example 2 State if each number is rational, irrational, or not a real number. G. –7 not a real number H. 64 81 rational 8 9 8 64 = 9 81 3-7 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. 3-7 The Real Numbers 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. 32 +33 ÷2 =6 5 ÷2 5 5 5 3 1 2 3 4 =7÷2=3 1 2 3 5 3 5 13 5 35 4 32 3 2 1 A real number between 3 and 3 is 3 . 5 5 2 3-7 The Real Numbers Check It Out 3 4 Find a real number between 4 and 4 . 7 7 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2. 43 +44 7 7 ÷2 1 2 =8 3 7 ÷2 7 4 47 47 4 7 14 7 42 5 =9÷2=4 1 2 6 4 7 47 4 1 A real number between 4 3 and 4 is 4 . 7 2 7 3-7 The Real Numbers Lesson Quiz 3-7 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. 3. 25 4. 0 not a real number 4 • 9 rational 5. Find a real number between –2 3 and –2 3 . Possible answer –25 . 8 4 8