1. Call the Feds! We’ve Got Nested Radicals! Alan Craig

2. What are Nested Radicals? Examples:

3. We could keep this up forever!

4. If we did, what would we get? = ?

5. Let’s work up to it. What are the values of these expressions?

6. What are the Values?

10. What value is this sequence of numbers approaching?

11. Now what do you think the value of this infinite nested radical is?

12. You’re Right!

13. Let’s see an example of where an infinite nested radical could arise. Warning: Brief Excursion into Trigonometry! Trigonometry

14. Half-Angle Formula We will use the half-angle formula for cosine to take another look at this sequence and its limit.

15. Let’s use the formula to find.

16. Let’s rationalize the last expression by multiply numerator and denominator by 2.

17. Let’s use the formula to find.

19. Now multiply both sides by 2.

20. Let’s use the formula to find.

21. Repeatedly using the ½ angle formula:

22. As the angle  gets smaller and smaller, what value is the cos(  ) approaching?

23. Repeatedly using the ½ angle formula: Recall cos(0) = 1, so 2 cos(  ) is approaching 2 as  approaches 0.

24. That’s all the trigonometry for this session.

25. Now let’s ‘prove’ it.

26. Set x equal to the expression.

27. Square both sides.

28. Subtract the original equation from the squared equation.

30. Now solve the equation.

31. Solve the equation.

33. Why did we not use x = -1 ?

34. So

35. What about? = 3 ???

36. Using the same process as before, we get

37. Recall the Quadratic Formula We have So a = 1, b = -1, and c = -3 and

38. So, No, we do not get 3

39. Let’s ask a slightly different question. Is there a positive integer a, such that if we replace 3 under the nested radical with a, the nested radical will equal 3?

40. Let’s ask a slightly different question. That is, is there an a that makes the equation below true?

41. Let’s ask a slightly different question. That is, is there an a that makes the equation below true? Yes! And we are going to find it.

42. Subtract the original equation from the squared equation.

43. Finding a (Using the quadratic formula)

44. Finding a We want x = 3, so

45. Finding a

48. So we have shown that

49. Now let’s generalize our result. ‘Prove’ that for any integer k > 1, there is a positive integer a, such that Note: The following is not a true mathematical proof of this theorem (which would use limits of bounded, monotonically increasing sequences) but does suggest the core reasoning and result of such a proof.

50. Finding a

54. We have shown that For any integer k > 1, there is exactly one integer a = k (k - 1), such that

55. Examples

56. Alternatively, we might have noticed that we need to solve in such a way that we get two numbers that multiply to make a and subtract to make 1. Further, one of the numbers must be k. Thus, the other number must be k - 1 and a must be k ( k - 1). Another Way

57. That is

58. The END?

59. No! This is way too much fun!

60. Let’s Kick it Up a Notch!

61. Note that what we did before was a special case of this expression with b = 1.

62. Let’s Kick it Up a Notch! For each integer k > 1, there are exactly k - 1 pairs of integers a and b, 0 < b < k, that satisfy this equation. Further,

63. As before, square the equation. But before we subtract the original equation from the squared equation, we must isolate the radical (so that it will subtract away).

64. Now subtract.

66. Note one solution of this quadratic equation: Just keep this in mind for now. We will prove our assertion by factoring and come back to this later.

67. For integer solutions of we need two integers that multiply to make a and have a difference of b. One of the numbers must be k, so the other is k - b. Thus, Factor

68. There are exactly k – 1 such pairs a and b : (k – 1) Pairs (difference)

69. If k = 4, the k – 1 = 3 pairs a and b are : Example: k = 4

71. One Last Thought Consider this continued fraction:

72. Suppose it converges to x, then Does this look familiar?

73. So these are equal!

74. In particular, set a = b = 1. The Golden Ratio 

75. ?