Warmup problems:
- In an analog clock, how often small and big hands overlap during 24 hours?
- …how often they are opposite?
- …at 90 degrees?
1/13=0.0769(53841)... periodic decimal
fraction.
3/17=0.1765... - takes a while, but still decimal expansion is periodic.
We (sort of) proved: a number is
rational if and only if it has periodic decimal expansion.
Areas of
parallelograms: determinant: for a pair of vectors (a,b) and (c,d), the area of the parallelogram spanned by them is ad-bc (this expression is called determinant).
Sometimes long vectors generate parallelograms of small area, for example (13,21) and (21,34)... (Hmm.... Fibonacci numbers here for some reason...). Sometimes the area of the paralellogram is just 0! (For example, for (2,2), (5,5))
Game: each player starts with $6,
contribute any amount to the general fund. Then the general fund is distributed equally
among with exception of
those who contributed least.
Homework
- 0.12345678910111213141516... (we just write down all the natural numbers in their decimal form). Is this number rational?
- 22/23=0.95652... Can you find period here?
- Find a lot of
pairs of vectors with zero area.
