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Understanding Real Numbers in Pre-Algebra

Learn to classify numbers as rational or irrational in pre-algebra. Discover the properties of real numbers, including rational and irrational subsets. Apply density properties and examples to deepen understanding.

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Understanding Real Numbers in Pre-Algebra

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  1. Learn to determine if a number is rational or irrational. 3-10 The Real Numbers Pre-Algebra

  2. 3-10 The Real Numbers Pre-Algebra

  3. Lemurs 3-10 The Real Numbers Primates Mammals Animals Pre-Algebra Biologists classify animals based on shared characteristics. The gray lesser mouse lemur is an animal, a mammal, a primate, and a lemur. 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.

  4. 3-10 The Real Numbers Pre-Algebra 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

  5. Helpful Hint 2 ≈1.4142135623730950488016… 3-10 The Real Numbers A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Pre-Algebra 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.

  6. Irrational numbers Rational numbers Real Numbers Whole numbers 3-10 Integers The Real Numbers Pre-Algebra The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

  7. 16 2 4 2 3-10 The Real Numbers = = 2 Pre-Algebra 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. 5 irrational, real B. –12.75 –12.75 is a terminating decimal. rational, real 16 2 C. whole, integer, rational, real

  8. 0 3 3-10 The Real Numbers = 0 Pre-Algebra Examples 2: Determining the Classification of All Numbers State if the number is rational, irrational, or not a real number. A. 15 is a whole number that is not a perfect square. 15 irrational 0 3 B. rational

  9. 2 3 2 3 4 9 3-10 The Real Numbers = Pre-Algebra 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 4 9 D. rational

  10. 3-10 The Real Numbers Pre-Algebra 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.

  11. 1 2 3 5 2 5 3 5 2 5 2 5 1 2 5 5 1 5 3 5 2 5 1 2 3 3 3 3 3-10 The Real Numbers A real number between 3 and 3is 3 . = 6 ÷ 2 3 + 3 ÷ 2 = 7 ÷ 2 = 3 Find a real number between 3 and 3 . 4 5 3 5 3 3 4 Pre-Algebra Examples 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.

  12. 3-10 The Real Numbers Pre-Algebra 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

  13. Helpful Hint 2 ≈1.4142135623730950488016… 3-10 The Real Numbers A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits. Pre-Algebra 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.

  14. Irrational numbers Rational numbers Real Numbers Whole numbers 3-10 Integers The Real Numbers Pre-Algebra The set of real numbers consists of the set of rational numbers and the set of irrational numbers.

  15. 81 3 9 = 3 9 3 3-10 The Real Numbers = = 3 Pre-Algebra Try This: 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

  16. 3-10 The Real Numbers Pre-Algebra Try This: Examples 2 State if the 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

  17. 8 9 8 9 64 81 3-10 The Real Numbers = Pre-Algebra Try This: Examples 2 State if the number is rational, irrational, or not a real number. C. –7 not a real number 64 81 D. rational

  18. 3-10 The Real Numbers Pre-Algebra 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.

  19. 1 2 4 7 3 7 4 7 3 7 3 7 1 2 7 7 4 7 1 2 4 3-10 The Real Numbers A real number between 4 and 4 is 4 . = 9 ÷ 2 = 4 4 + 4 ÷ 2 Find a real number between 4 and 4 . = 8 ÷ 2 5 7 1 7 6 7 2 7 3 7 4 7 4 4 4 4 4 4 Pre-Algebra Try This: Example 3 There are many solutions. One solution is halfway between the two numbers. To find it, add the numbers and divide by 2.

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