• Activity: can you write these numbers as sums of distinct Fibonacci numbers?
• Theorem: any natural number can be written as a sum of one or more non-consecutive Fibonacci numbers. Also, there is only one way to do this. (This is called Zeckendorf's Theorem)
• Looking at the ratios of consecutive Fibonacci numbers, we previously saw they approach a number we call "the Golden Ratio"
which will turn out to be irrational (more on this later).• A "Fibonacci-like" sequence starting with any two randomly selected natural numbers (in place of 1 and 1) will have its ratios approach the same value
• A continued fraction representation of
: Here is a video which shows the continued fraction being developed, by Michael McCafferty. (You could stop watching at about 2:00 when he has finished the continued fraction, or go on as he explains more advanced properties.)• From this we learn that
satisfies the equation
which we can solve to find the exact value
What is important to get from each of these: (some of this I mentioned last time)
• You should know how to get the next number in the Fibonacci sequence (or any Fibonacci-like sequence) by adding together the two previous numbers
• You should make sure to understand how the algebraic formula an = an-1 + an-2 represents that rule
• You should understand how we computed the ratios of successive Fibonacci numbers and how we concluded that those ratios were approaching a single number as we went farther and farther out
• Same thing for the "Fibonacci-like" example in the reading which started with 192 and 16
• Make sure that you see how the Fibonacci numbers and the spiral are being shown in the examples from nature: the sunflower seed head and the chambered nautilus shell in particular
• Understand how we got the continued fraction representation of
and that the fact that it contains only 1's means that is very special indeed• Know how to solve the equation to find the exact value of
(including why we choose the + sign in the quadratic formula).We will be going into more detail about rational and irrational numbers later in the course, and will prove that the square root of 5 is irrational, so that will prove that
is also irrational. For now it is enough to know that a rational number is a number that can be written in the form of a fraction, and that the decimal expansion of a rational number either terminates or eventually repeats.Homework:
Journal assignment:
Use your journal to start work on understanding each of the points above on the topic of Fibonacci numbers. Describe any part that seems unclear to you and try to work it out, possibly by discussion on the Piazza discussion board. (Don't worry if you don't get this totally understood by next time; it's just important to identify what you need to work on and to get started on it. The homework problems below may help with some things.)
Homework problems: (the first one was assigned last time, I'm just repeating it herre)
1) Choose any two natural numbers (not 1 and 1, and not 192 and 16, but some other pair, preferably not too big) and make the "Fibonacci-like" sequence starting with them. Write down at least the first 10 numbers in the sequence. Then compute the ratios of successive numbers in your sequence, as we did for the Fibonacci numbers. Do your ratios appear to be approaching
? Why or why not?2) We can verify that
Do the same to simplify
3) Now simplify
4) Solve for x:
5) Solve for x:
Reading for tomorrow: We will start a new topic, prime numbers and prime factorization of natural numbers (from Math is Fun). This will lead eventually to discussion of public-key cryptography, a very important (and lucrative!) application of number theory. Something to look forward to.
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