Title: Call the Feds Weve Got Nested Radicals
1 Call the Feds!Weve Got
Nested Radicals!
- Alan Craig
- F. Lane Hardy Seminar
- 11-3-08
2What are Nested Radicals?
Examples
3We could keep this up forever!
4If we did, what would we get?
?
5Lets work up to it.What are the values of these
expressions?
6Lets work up to it.What are the values of these
expressions?
7What are the Values?
8What are the Values?
9What are the Values?
10What are the Values?
11What value is this sequence of numbers
approaching?
12Now what do you think the value of this infinite
nested radical is?
13Youre Right!
14Lets see an example of where an infinite nested
radical could arise.Warning Brief Excursion
into Trigonometry!
Trigonometry
15Half-Angle Formula
- We will use the half-angle formula for cosine to
take another look at this sequence and its limit.
16Lets use the formula to find .
17Lets use the formula to find .
Lets rationalize the last expression by
multiplying numerator and denominator by 2.
18Lets use the formula to find .
19Lets use the formula to find .
20Lets use the formula to find .
Now multiply both sides by 2.
21Lets use the formula to find .
Wow! A nested radical.
22Repeatedly using the ½ angle formula
23Repeatedly using the ½ angle formula
As the angle a gets smaller and smaller
approaching 0, what value is the cos(a)
approaching?
24Repeatedly using the ½ angle formula
Recall cos(0) 1, so 2 cos(a) is approaching 2
as a approaches 0.
25Repeatedly using the ½ angle formula
That is,
26Thats all the trigonometry for this session.
27We have shown in two different ways that the
equation ought to be true
To Recap
28Now lets prove it.
29Set x equal to the expression.
30Square both sides.
31Subtract the original equation from the squared
equation.
32Subtract the original equation from the squared
equation.
33Now solve the equation.
34Solve the equation.
35Solve the equation.
36Solve the equation.
Why did we not use x -1?
37So
38What about?
39Does
3 ???
40Using the same process as before, we get
41Recall the Quadratic Formula
- We have
- So a 1, b -1, and c -3 and
42So, No, we do not get 3
43Lets 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?
44Lets ask a slightly different question.
- That is, is there an a that makes the equation
below true?
45Lets 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.
46Subtract the original equation from the squared
equation.
47Finding a
(Using the quadratic formula)
48Finding a
49Finding a
50Finding a
51Finding a
52So we have shown that
53Now lets generalize our result.
- Prove that for any integer k gt 1, there is a
unique 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.
54Finding a
55Finding a
56Finding a
57Finding a
58We have shown that
- For any integer k gt 1, there is exactly one
integer a k (k - 1), such that
59We have shown that
- For any integer k gt 1, there is exactly one
integer a k (k - 1), such that
- That is, every integer can be represented as an
infinite nested radical!
60Example k 4
61Example k 5
62Another Way
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. (Why?)
Thus, the other number must be k - 1 and a must
be k (k - 1).
63That is
64The END?
65The END?
- No!
- This is way too much fun!
66Lets Kick it Up a Notch!
67Lets Kick it Up a Notch!
Note that what we did before was a special case
of this expression with b 1.
68Lets Kick it Up a Notch!
For each integer k gt 1, there are exactly k - 1
pairs of integers a and b, 0 lt b lt k, that
satisfy this equation. Further,
69As 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).
70Now subtract.
71Now subtract.
We will solve this by factoring now but keep it
in mind for later.
72Factor
- 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,
73(k 1) Pairs
- There are exactly k 1 such pairs a and b
(difference)
Recall that 0 lt b lt k
74Example k 4
- If k 4, the k 1 3 pairs a and b are
75Example k 4
76One Last Thought
Consider this continued fraction
77Suppose it converges to x, then
78Notice the shaded area is also x
79Rewriting the continued fraction
80See what we get!
Does this look familiar?
81Yes, these are equal!!!
82In particular, set a b 1.
83The Golden Ratio f
(But thats another F. Lane Hardy talk.)
84?
Reference Zimmerman, S., Ho, C. (2008). On
infinitely nested radicals. Mathematics Magazine,
81(1), 3-15.