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Understanding and Comparing Real Numbers: Lesson on Irrational Numbers and Classification

Learn how to classify real numbers and distinguish between rational and irrational numbers. Compare and order real numbers and explore real-world examples involving mathematical expressions with real numbers.

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Understanding and Comparing Real Numbers: Lesson on Irrational Numbers and Classification

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  1. Lesson Menu Main Idea and New Vocabulary Key Concept: Irrational Numbers Example 1: Classify Numbers Example 2: Classify Numbers Example 3: Classify Numbers Example 4: Compare Real Numbers Example 5: Compare Real Numbers Example 6: Compare Real Numbers Example 7: Order Real Numbers Example 8: Real-World Example

  2. Compare mathematical expressions involving real numbers. • irrational number • real number Main Idea/Vocabulary

  3. Key Concept

  4. Answer: The decimal ends in a repeating pattern. It is a rational number because it is equivalent to . Classify Numbers Name all sets of numbers to which the real number 0.090909… belongs. Example 1

  5. Name all sets of numbers to which the real number 0.454545… belongs. A. whole B. rational C. whole, integer, rational D. irrational Example 1 CYP

  6. Name all sets of numbers to which the real number belongs. Answer: Since = 5, it is a whole number, an integer, and a rational number. Classify Numbers Example 2

  7. Name all sets of numbers to which the real number belongs. A. whole B. whole, integer C. whole, integer, rational D. irrational Example 2 CYP

  8. Name all sets of numbers to which the real number belongs. Answer: –3.464101615… Since the decimal does not terminate or repeat, it is an irrational number. Classify Numbers Example 3

  9. Name all sets of numbers to which the real number belongs. A. whole B. integer, rational C. whole, integer, rational D. irrational Example 3 CYP

  10. Replace with <, >, or = to make a true statement. Compare Real Numbers Write each number as a decimal. Example 4

  11. Since 3.872983346… is less than 3.875, . Compare Real Numbers Answer: Example 4

  12. Replace with <, >, or = to make 4 a true statement. A. < B. > C. = Example 4 CYP

  13. Replace with <, >, or = to make a true statement. Write as a decimal. ≈ 3.224903099… = 3.222222222… Compare Real Numbers Example 5

  14. Since 3.222222222… is less than 3.224903099…, Answer: Compare Real Numbers Example 5

  15. Replace with <, >, or = to make a true statement. A. < B. > C. = Example 5 CYP

  16. Replace with <, >, or = to make 12.3% ● a true statement. = 0.1 Compare Real Numbers Write each number as a decimal. 12.3% = 0.123 Example 6

  17. Since 0.123 is greater than 0.1, 12.3% > . Answer: 12.3%> Compare Real Numbers Example 6

  18. Replace with <, >, or = to make 90% a true statement. A. < B. > C. = Example 6 CYP

  19. Order from least to greatest. Verify your answer by graphing on a number line. Order Real Numbers Write each number as a decimal. Then order the decimals. 3 = 3.00 Example 7

  20. Answer: From least to greatest, the order is 3, , , and 4 . Order Real Numbers Example 7

  21. Order from least to greatest. A. B. C. D. Example 7 CYP

  22. BEACH VOLLEYBALLThe time in seconds that it takes an object to fall d feet is About how many seconds would it take for a volleyball thrown 32 feet up in the air to fall from its highest point to the sand? Example 8

  23. Use a calculator to approximate the time it would take for a volleyball to fall to the sand. Replace d with 32. ≈ 1.4 s Use a calculator. Answer: It would take a volleyball thrown 32 feet up in the air about 1.4 seconds to fall to the sand. Example 8

  24. PHYSICAL SCIENCEThe time in seconds that it takes an object to fall d feet is About how many seconds longer would it take for a ball dropped from 100 feet to fall to the ground than a ball dropped from 30 feet? A. about 70 s B. about 9.7 s C. about 2.5 s D. about 1.1 s Example 8 CYP

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