Happy Saturday to you! We had our last regular BHMC meeting this morning. Our next and final BHMC event of the school year will be this Friday evening, May 4, at 6:30 PM in Jonas 204 (our usual meeting place). The meeting will be a celebration of the BHMC’s inaugural year, with certificates to be awarded and dessert to be eaten. Everyone who’s attended a BHMC meeting this year is invited to attend with their families. We hope to see you there!
At this morning’s meeting participants filled out a survey commenting on their experience this past year, after which Dr. Nag spoke a bit on Euler and reviewed his amazing formula (see last time’s post). Dr. Nag used the identity to answer one of the problems from last time (Problem 6).
Dr. Barrus introduced the problem of the day, inspired by a question contributed by Mr. Spear: Suppose we start with a negative number and a positive number, say -48 and 27, and we construct a list of numbers by finding the next number by adding the last two numbers from the list, as follows:
You can see that the first terms of the list were alternately negative and positive until the -15, after which the terms remained negative. We can do this with another choice of negative and positive numbers…
… and observe the same thing–the terms alternate in sign for a while and eventually end up all having the same sign (positive, in this case). You’ll notice that in this second example that only the first four terms changed signs every time, while in the first example there were six terms that alternated in sign before the signs settled to all being negative. This led to our first question of the day:
Question #1: Suppose the list started with -100. What should the second number be so as to make the alternating sign pattern last as long as possible?
In discussing the problem the group arrived at an answer (at least when the second number was required to be a whole number); can you? As a few students noticed, you can achieve even longer -/+ patterns if the second number is allowed to include some decimal places; is it possible to choose a (non-integer) number that should come after the -100 to make the alternating sign pattern last as long as you want?
A bit further into our discussion Dr. Barrus suggested looking at the terms of the list a bit more abstractly. If “-a” is our first term and “b” is our second term, then constructing the list in the same way as above, we get
If you want to make sure that the third term is negative (to preserve the alternating sign pattern), you can require that -a + b < 0, which means that b < a. On the other hand, to make the fourth term positive (so that the first four terms follow the pattern – + – +), we would require that -a + 2b > 0, which yields b > (1/2)a. So b has to be somewhere between (1/2)a and 1a… what if we continue down the sequence, requiring -2a+3b<0, -3a+5b>0, and so on–what more can we say about what size b has to have in relation to a?
Thinking along the lines of the last paragraph, you’ll run into an interesting sequence of fractions, ones where the numerator and denominator are consecutive Fibonacci numbers. This leads to the question posed by Mr. Spear:
Question #2: What number do the fractions
get closer and closer to the farther down the list you get?
(Note: Because he didn’t get a chance to ask him, Dr. Barrus wasn’t sure whether Mr. Spear meant the fractions above or their reciprocals–the list
If you like, you might try to figure out what number these fractions approach, and how the answer to this question is related to the answer to Question #2. Comparing the two numbers might even help in answering Question #2!)
Have thoughts, or better yet, questions for further thought on these problems? Write them in our comments section! Because we went a long way towards answering Questions #1 and Questions #2 in our meeting this morning, Dr. Barrus poses one final question on today’s topic:
Question #3: Is there a way to predict, just from the first two numbers in your list that you start your addition with, whether the numbers in the list will eventually be all negative or all positive? For instance, in the examples above, could we predict from the -48 and 27 that the first sequence would end up producing only negative numbers? Or, from the -631 and 442, that the numbers would eventually all be positive? If so, how?
Have a great week, everyone, write your thoughts here on the blog, and we’ll see you Friday!
(Edit: the time of Friday’s celebration was corrected above–see you at 6:30.)
. (We often write z to stand for a complex number a + ib, and we use either r or |z| to stand for the modulus.) If 
represents the angle formed by the positive real axis and the line segment joining z and the origin, then 
. Using a little trigonometry, we get 
and 
.
,
, which gives rise to the amazing identity 
.
.
