From the homework:
Q1. Exponential Equation. Suffice to say f(x)=ab^x works in different ways, all of which we will tackle on Thursday.
brief notes--we assume that each person contaminates two people who HAVEN'T yet had the disease. this creates overlaps that can make a problem like this feel weird.
notice that each generation infects 2^g (where g is the generation), but the total number infected is actually (2^(g+1) -1), as the total number infected include all of those who have already had this disease.
Q2. Creating a proof tree. Proof by exhaustion, getting rid of all 'corner' cases.
why do we only care about the ones place when solving these? what about the numbers allows us to not care about the 10s or the 100s?
which numbers are self limiting?
Q3. Here ... what does it mean to prove? What can we show?
Some cool proofs here: http://www-formal.stanford.edu/jmc/creative/node2.html
How do we understand these? take each one, and attempt to DO what they say. Check the ideas and make sure that they balance with what we know of the world.
"Assume a covering. The number of dominoes projecting from the top row to the second row is odd. Likewise the number from the second to the third is odd, etc. Therefore, the total number of vertical dominoes is the sum of seven odd numbers and hence odd. Likewise the the number of horizontal dominoes is odd. Odd + odd is even so the total is even, but the total is 31. There is an apparent mathematical induction here in the 'etc.'. We will see later that the idea itself does not include the induction." ...this is beautiful.
Some roads we might tackle:
if 'n' is odd, is 'n^2' odd?
if 'n' is even, is 'n^2' even?
some cool info on that: http://www.math.uiuc.edu/~hildebr/347.summer14/even-odd-proofs-sol.pdf
New Question: Pythagorean Theorem:
a^2 + b^2 = c^2 ... but why?
show one proof in class
there are more, here: http://www.cut-the-knot.org/pythagoras/
Garfield's Proof: http://denisegaskins.com/2008/09/24/mathematician-for-president/ (president, not cat)
Pythagorean Triples:
creation of them: http://www.wikiwand.com/en/Pythagorean_triple