Thursday 1 September class

• Activity: Pigeonhole principle statements and mis-statements. Here is the handout. We did not finish all of these in class: you should look at the rest of them at home.

• Fibonacci numbers: beginning with the reading from Math is Fun.
 - How to get the next number in the sequence by adding together the two previous numbers
 - A spiral that is constructed on squares whose side lengths are the Fibonacci numbers
 - Subscript notation so we can write the above rule in algebraic form: $a_n = a_{n-1} + a_{n-2}$
 - Looking at the ratios of consecutive Fibonacci numbers, they approach a number we call "the Golden Ratio" $\varphi \approx 1.618$ which will turn out to be irrational (more on this later).
  - Fibonacci numbers in nature [the link I used: Math is Fun, also look at Fibonacci Numbers and Nature]
  - A beautiful video showing many real-world instances related to Fibonacci numbers, the Golden Ratio, and things that follow from them: Nature by Numbers, by Cristobal Vila


What is important to get from each of these:
• 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 $a_n = a_{n-1} + a_{n-2}$ 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
• 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

We will be going into more detail about rational and irrational numbers later in the course, and will prove that $\sqrt{5}$ is irrational, so that will prove that $\varphi$ 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: 
Record any new thing you learned from the homework discussion today.

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 problem:
 Choose any two natural numbers (not 1 and 1, but some other pair) and make the "Fibonacci-like" sequence starting with them and using the Fibonacci rule to get the next number. 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 $\varphi$? Why or why not?

Reading for next time: We will find a continued fraction expansion for the Golden Ratio, which leads to an equation we can solve to find its exact value (which is, alas, irrational).  I'm still looking for sources for this, so no links yet, sorry.