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Rational and Irrational Numbers

Learn to distinguish between rational and irrational numbers, including real numbers, integers, and whole numbers. Discover how rational numbers are fractions or decimals, while irrational numbers are non-repeating decimals. Explore the classification of numbers using Venn Diagrams and examples. Understand that square roots of non-perfect squares are irrational. A fraction with a denominator of 0 is undefined.

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Rational and Irrational Numbers

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  1. Rational and Irrational Numbers

  2. Rational and Irrational Numbers Essential Question How do I distinguish between rational and irrational numbers?

  3. Vocabulary real number irrational number

  4. Real Numbers Rational numbers Irrational numbers Integers Whole numbers The set of real numbers is all numbers that can be written on a number line. It consists of the set of rational numbers and the set of irrational numbers.

  5. Recall that rational numbers can be written as the quotient of two integers (a fraction) or as either terminating or repeating decimals. 4 5 23 3 = 3.8 = 0.6 1.44 = 1.2

  6. Irrational numberscan be written only as decimals that do not terminate or repeat. They cannot be written as the quotient of two integers. If a whole number is not a perfect square, then its square root is an irrational number. For example, 2 is not a perfect square, so 2 is irrational. Caution! A repeating decimal may not appear to repeat on a calculator, because calculators show a finite number of digits.

  7. Make a Venn Diagram that displays the following sets of numbers: Reals, Rationals, Irrationals, Integers, Wholes, and Naturals. Reals Rationals -2.65 Integers -3 -19 Wholes 0 Irrationals Naturals 1, 2, 3...

  8. 16 2 4 2 = = 2 Additional Example 1: Classifying Real Numbers Write all classifications 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

  9. 9 = 3 81 3 9 3 = = 3 Check It Out! Example 1 Write all classifications 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

  10. A fraction with a denominator of 0 is undefined because you cannot divide by zero. So it is not a number at all.

  11. 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

  12. Additional Example 2: Determining the Classification of All Numbers State if each number is rational, irrational, or not a real number. 4 0 C. not a real number

  13. 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. undefined, so not a real number

  14. 8 9 8 9 64 81 = Check It Out! Example 2 State if each number is rational, irrational, or not a real number. 64 81 C. rational

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