Real Numbers and the Plane

Real Numbers and the Plane
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0.1

1. To understand Definition 1.1, you must believe that | x | = - x for negative x's. Using x = - 3 as an example, explain in words why multiplying x by - 1 produces the same result as taking the absolute value of x.

2. A common shortcut used to write inequalities is -4 < x < 4 in place of “ -4 < x and x < 4. ” Unfortunately, many people mistakenly write 4 < x < - 4 in place of “ 4 < x or x < - 4. ” Explain why the string 4 < x < - 4 could never be true. (Hint: What does this inequality string imply about the numbers on the far left and far right? Here, you must write “ 4 < x or x < - 4. ”)

3. Explain the result of Theorem 1.1 (ii) in your own words, assuming that all constants involved are positive.

4. Suppose a friend has dug holes for the corner posts of a rectangular deck. Explain how to use the Pythagorean Theorem to determine whether or not the holes truly form a rectangle (90° angles).

In exercises 5-32, solve the inequality.

5. 3x + 2 < 11 6. 4x + 1 < - 5

7. 2x -3 < - 7 8. 3x -1 < 9

9. 4-3x < 6 10. 5-2x < 9

11. 4 x + 1 < 7 12. -1 < 2 - x < 3

13. -2 < 2-2x < 3 14. 0 < 3 - x < 1

15. x² + 3x -4 > 0 16. x² + 4x + 3 < 0

17. x² - x -6 < 0 18. x² + 1 > 0

19. 3x² + 4 > 0 20. x² + 3x + 10 > 0

21. | x -3 | < 4 22. | 2x + 1 | < 1

23. | 3 - x | < 1 24. | 3 + x | > 1

25. | 2x + 1 | > 2 26. | 3x -1 | < 4

27. 28.

29. 30.

31. 32.

In exercises 33-38, find the distance between the pair of points.

33. (2,1), (4,4) 34. (2,1), ( -1,4)

35. ( -1, -2), (3, -2) 36. (1,2), (3,6)

37. (0,2), ( -2,6) 38. (4,1), (2,1)

In exercises 39-42, determine if the set of points form the vertices of a right triangle.

39. (1,1), (3,4), (0,6) 40. (0,2), (4,8), ( -2,12)

41. ( -2,3), (2,9), ( -4,13) 42. ( -2,3), (0,6), ( -3,8)

In exercises 43-46, the data represent populations at various times. Plot the points, discuss any patterns that are evident, and predict the population at the next step.

43. (0,1250), (1,1800), (2,2450), (3,3200)

44. (0,3160), (1,3250), (2,3360), (3,3490)

45. (0,4000), (1,3990), (2,3960), (3,3910)

46. (0,2100), (1,2200), (2,2100), (3,1700)

47. As discussed in the text, a number is rational if and only if its decimal representation terminates or repeats. Calculators and computers perform their calculations using a finite number of digits. Explain why such calculations can only produce rational numbers.

48. In Example 1.8, we discussed how the tendency of the data points to “curve up” corresponds to larger increases in consecutive y - values. Explain why this is true.

49. The ancient Greeks analyzed music mathematically. They found that if pipes of length L and are struck, they make tones that blend together nicely. We say that these tones are one octave apart. In general, nice harmonies are produced by pipes (or strings) with rational ratios of lengths. For example, pipes of length L and L form a fifth (e.g., middle C and the G above middle C). On a piano keyboard, 12 fifths are equal to 7 octaves. A glitch in piano tuning, known as the Pythagorean comma, results from the fact that 12 fifths with total length ratio do not equal 7 octaves with length ratio Show that the difference is about 1.3 %.

50. For the 12 keys of a piano octave to have exactly the same length ratios (see exercise 49), the ratio of consecutive lengths should be a number x such that x¹² = 2. Briefly explain why. This means that . There are two problems with this equal-tempered tuning. First, is irrational. Explain why it would be difficult to get the pipe or string exactly the right length. In any case, musicians say that equal-tempered pianos sound “dull. ”

51. The use of squares in the Pythagorean Theorem has found a surprising use in the analysis of baseball statistics. In Bill James' Historical Abstract, a rule is stated that a team's winning percentage P is approximately equal to where R is the number of runs scored by the team and G is the number of runs scored against the team. For example, in 1996 the Texas Rangers scored 928 runs and gave up 799 runs. The formula predicts a winning percentage of In fact, Texas won 90 games and lost 72 for a winning percentage of 0.556. Fill out the following table (data from the 1996 season). What are possible explanations for teams that win more (or less) games than the formula predicts?

Team R G P wins losses win %

Yankees 871 787 92 70

Braves 773 648 96 66

Phillies 650 790 67 95

Dodgers 703 652 90 72

Indians 952 769 99 62

52. It can be very difficult to prove that a given number is irrational. According to legend, the following proof that is irrational so upset the ancient Greek mathematicians that they drowned a mathematician who revealed the result to the general public. The proof is by contradiction; that is, we imagine that is rational and then show that this cannot be true. If were rational, we would have that for some integers p and q. Assume that is in simplified form (i.e., any common factors have been divided out). Square the equation to get . Explain why this can only be true if p is an even integer. Write p = 2r and substitute to get Then, rearrange this expression to get q² = 2r². Explain why this can only be true if q is an even integer. Something has gone wrong: explain why p and q can't both be even integers. Since this can't be true, we conclude that is irrational.

53. In the text, we stated that a number is rational if and only if its decimal representation repeats or terminates. In this exercise, we prove that the decimal representation of any rational number repeats or terminates. To start with a concrete example, use long division to show that . Note that when you get a remainder of 1, it's all over: you started with a 1 to divide into, so the sequence of digits must repeat. For a general rational number , there are q possible

remainders (0, 1, 2, , q - 1 ). Explain why when doing long division you must eventually get a remainder you have had before. Explain why the digits will then either terminate or start repeating.

54. The existence of irrational numbers may seem like a minor technical footnote to the rational numbers we normally use. This is far from true. Imagine that a number is to be chosen at random. Believe it or not, the number would almost certainly be an irrational number! To see why this would happen, suppose you are choosing a number between 0 and 1 by writing down its decimal representation one digit at a time. Pick the first digit, then the second digit (e.g., if you choose 4 then 6 your number starts out 0.46 ) and so on. If you continue choosing the digits randomly, what is the likelihood that you will repeat a pattern forever? If you don't, you have chosen an irrational number!