- I'd like to push our third exam to the 16th of April -- is that okay?
- New assignment:
Fri 4/2 More Infinity - Our poster presentations are coming up at the end of April (the 26th, 28th, and 30th). By next Wednesday, 4/7, you should provide me with a one-page, typed proposal for your project. You should try to find something that you're interested in, that also has a mathematical angle that you can share with your classmates.
- Reminder: Problems pp. 156-161, #2, 6, 7, 8, 13, 20, 37. Due Mon, 4/5.
- Last time we did some interesting problems concerning the Hotel
Infinity. Today we want to continue with our exploration of infinity,
and hopefully we'll see that there are two different sized infinities
by the end of the day (real numbers -- the set containing both the
rational and irrational -- both repeating and non-repeating decimals --
numbers are more numerous than the natural numbers).
- We'll start, however, with one more Hotel Infinity problem. This one involves the rational numbers. Are the rational numbers -- repeating decimals -- bigger than the natural numbers?
-
There's another way to show that the rational numbers are countable
("denumerable"). We'll do so using a one-to-one
correspondence, and make use of
- A funky tree....
- Here's the rule: starting with a node (n/d), make "children" of the
node by this process:
/ \ - So here we go: start with 1/1, and do it again, do it again, ....
/ \ - You can check that we have all rational numbers by just picking random rationals, and figuring out where they would appear in the tree. For example, try 7/11.
- What do ratios of successive Fibonaccis look like?
- Here's the rule: starting with a node (n/d), make "children" of the
node by this process:
- A funky tree....
- Now we'll show that the set of real numbers (decimals,
both rational and irrational) are too numerous for the natural
numbers.
- Motivation: irrationals are more actually more numerous then rationals -- as seen by throwing a ten-sided die....
- Cantor's diagonalization proof