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Understanding Real Numbers and Their Classifications

Learn about rational and irrational numbers, the Density Property, and how to classify numbers. Practice identifying real numbers through examples and understanding the properties of each category.

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Understanding Real Numbers and Their Classifications

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  1. 4-7 The Real Numbers Course 3 Warm Up Problem of the Day Lesson Presentation

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

  3. 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. Learn to determine if a number is rational or irrational.

  5. Vocabulary irrational number real number Density Property

  6. 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

  7. 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

  8. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. 2 ≈1.4142135623730950488016… Irrational numberscan 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.

  9. Real Numbers Rational numbers Irrational numbers Integers Whole numbers The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

  10. 16 2 4 2 = = 2 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. 5 irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 C. whole, integer, rational, real

  11. 9 = 3 81 3 9 3 = = 3 Check It Out: Example 1 Write all names that apply to each number. 9 A. whole, integer, rational, real –35.9 –35.9 is a terminating decimal. B. rational, real 81 3 C. whole, integer, rational, real

  12. 0 3 = 0 Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. A. 21 irrational 0 3 B. rational

  13. 2 3 2 3 4 9 = Additional 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 4 9 D. rational

  14. Check It Out: Example 2 State if each number is rational, irrational, or not a real number. A. 23 is a whole number that is not a perfect square. 23 irrational 9 0 B. not a number, so not a real number

  15. 8 9 8 9 64 81 = Check It Out: Example 2 State if each number is rational, irrational, or not a real number. C. –7 not a real number 64 81 D. rational

  16. 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.

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

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

  19. 4 • 9 3 4 3 8 Find a real number between –2 and –2 . 5 8 Possible answer –2 . Lesson Quiz Write all names that apply to each number. 16 2 2. – 1. 2 real, irrational real, integer, rational State if each number is rational, irrational, or not a real number. 25 0 4. 3. rational not a real number 5.

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