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Unformatted text preview: Section 1.1 Number Systems 11 Version: Fall 2007 1.1 Exercises In Exercises 1 8 , find the prime factor ization of the given natural number. 1. 80 2. 108 3. 180 4. 160 5. 128 6. 192 7. 32 8. 72 In Exercises 9 16 , convert the given dec imal to a fraction. 9. . 648 10. . 62 11. . 240 12. . 90 13. . 14 14. . 760 15. . 888 16. . 104 In Exercises 17 24 , convert the given repeating decimal to a fraction. 17. . 27 Copyrighted material. See: http://msenux.redwoods.edu/IntAlgText/ 1 18. . 171 19. . 24 20. . 882 21. . 84 22. . 384 23. . 63 24. . 60 25. Prove that √ 3 is irrational. 26. Prove that √ 5 is irrational. In Exercises 27 30 , copy the given ta ble onto your homework paper. In each row, place a check mark in each column that is appropriate. That is, if the num ber at the start of the row is rational, place a check mark in the rational col umn. Note: Most (but not all) rows will have more than one check mark. 27. N W Z Q R 2 2 / 3 0.15 . 2 √ 5 12 Chapter 1 Preliminaries Version: Fall 2007 28. N W Z Q R 10 / 2 π 6 . 9 √ 2 . 37 29. N W Z Q R 4 / 3 12 √ 11 1 . 3 6 / 2 30. N W Z Q R 3 / 5 √ 10 1 . 625 10 / 2 / 5 11 In Exercises 31 42 , consider the given statement and determine whether it is true or false. Write a sentence explaining your answer. In particular, if the state ment is false, try to give an example that contradicts the statement. 31. All natural numbers are whole num bers. 32. All whole numbers are rational num bers. 33. All rational numbers are integers. 34. All rational numbers are whole num bers. 35. Some natural numbers are irrational. 36. Some whole numbers are irrational. 37. Some real numbers are irrational. 38. All integers are real numbers. 39. All integers are rational numbers. 40. No rational numbers are natural num bers. 41. No real numbers are integers. 42. All whole numbers are natural num bers. Section 1.1 Number Systems 13 Version: Fall 2007 1.1 Answers 1. 2 · 2 · 2 · 2 · 5 3. 2 · 2 · 3 · 3 · 5 5. 2 · 2 · 2 · 2 · 2 · 2 · 2 7. 2 · 2 · 2 · 2 · 2 9. 81 125 11. 6 25 13. 7 50 15. 111 125 17. 3 11 19. 8 33 21. 28 33 23. 7 11 25. Suppose that √ 3 is rational. Then it can be expressed as the ratio of two integers p and q as follows: √ 3 = p q Square both sides, 3 = p 2 q 2 , then clear the equation of fractions by multiplying both sides by q 2 : p 2 = 3 q 2 (1) Now p and q each have their own unique prime factorizations. Both p 2 and q 2 have an even number of factors in their prime factorizations. But this contradicts equa tion ( 1 ), because the left side would have an even number of factors in its prime factorization, while the right side would have an odd number of factors in its prime factorization (there’s one extra 3 on the right side)....
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