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3-10 The Real Numbers Learn to determine if a number is rational or irrational. Pre-Algebra 3-10 The Real Numbers Pre-Algebra 3-10 The Real Numbers Biologists classify animals based on shared characteristics. The gray lesser mouse lemur is an animal, a mammal, a primate, and a lemur. Animals Mammals Primates Lemurs Pre-Algebra 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. 3-10 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 Pre-Algebra 23 = 0.6 1.44 = 1.2 3-10 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… Helpful Hint A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Pre-Algebra 3-10 The Real Numbers The set of real numbers consists of the set of rational numbers and the set of irrational numbers. Real Numbers Rational numbers Integers Whole numbers Pre-Algebra Irrational numbers 3-10 The Real Numbers Examples 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 2 4 = =2 2 whole, integer, rational, real Pre-Algebra 3-10 The Real Numbers Examples 2: Determining the Classification of All Numbers State if the number is rational, irrational, or not a real number. A. 15 15 is a whole number that is not a perfect square. irrational B. 0 3 rational Pre-Algebra 0 3 =0 3-10 The Real Numbers Examples 2: Determining the Classification of All Numbers State if the number is rational, irrational, or not a real number. C. –9 not a real number D. 4 9 rational Pre-Algebra 2 32 3 = 4 9 3-10 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. Pre-Algebra 3-10 The Real Numbers Examples 3: Applying the Density Property of Real Numbers 2 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. Find a real number between 3 2 5 3 5 5 5 1 2 3 +3 ÷2 =6 ÷2 =7÷2=3 3 1 2 3 4 3 5 3 5 13 5 3 5 4 32 3 2 1 A real number between 3 and 3 is 3 . 5 5 2 Pre-Algebra 3-10 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 Pre-Algebra 23 = 0.6 1.44 = 1.2 3-10 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… Helpful Hint A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Pre-Algebra 3-10 The Real Numbers The set of real numbers consists of the set of rational numbers and the set of irrational numbers. Real Numbers Rational numbers Integers Whole numbers Pre-Algebra Irrational numbers 3-10 The Real Numbers Try This: Example 1 Write all names that apply to each number. A. 9 9 =3 whole, integer, rational, real B. C. Pre-Algebra –35.9 –35.9 is a terminating decimal. rational, real 81 3 81 3 9 = =3 3 whole, integer, rational, real 3-10 The Real Numbers Try This: Examples 2 State if the number is rational, irrational, or not a real number. A. 23 23 is a whole number that is not a perfect square. irrational B. 9 0 not a number, so not a real number Pre-Algebra 3-10 The Real Numbers Try This: Examples 2 State if the number is rational, irrational, or not a real number. C. –7 not a real number D. 64 81 rational Pre-Algebra 8 98 9 64 = 81 3-10 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. Pre-Algebra 3-10 The Real Numbers Try This: Example 3 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. 3 7 4 7 4 +4 ÷2 1 2 7 7 =8 ÷2 3 4 5 1 2 =9÷2=4 6 4 7 4 7 4 7 14 7 4 7 4 7 42 4 1 A real number between 4 3 and 4 is 4 . 7 2 7 Pre-Algebra