Review, Part 1, MAT115

When I think about the big ideas, these are the primary ones:

1. Binary representations (powers of two; e.g. Fraudini, primitive counting, binary finger dancing!:); these come back later, when we got to the Egyptians. They're also how computers work!
2. Pascal's Triangle (seems to contain everything, in a systematic way):
   • Powers of 2 (sums of rows)
   • Counting numbers (second diagonals)
   • Triangular numbers (third diagonals)
   • Tetrahedral numbers (fourth diagonals)
   • Numbers of various Facebook graphs: for example, if we have four individuals in a Facebook, then there are six total different friendships possible (the third triangular number: $6=\frac{4*(4-1)}{2}$).

     So we go to the sixth row of Pascal's triangle

     

     (remembering that we number the rows from 0), and read off all the different ways of creating a Facebook:

     1 6 15 20 15 6 1

     (total of $2^6=64$), meaning that there is 1 Facebook containing no friendships, 6 Facebooks containing one friendship, 15 Facebooks containing two friendships, etc.

   • (Eventually Fibonacci numbers, in a systematic way)
   • (Eventually a fractal: Sierpinski's triangle)

3. The primary fractorizations/decompositions, which each go something like this: "Every (or almost every) counting number is either _______ or is a ______ of _______ numbers, in a unique way."
   1. primes - every counting number (greater than one) is either prime, or a product of primes in a unique way.
      • The Sieve of Eratosthenes was a tool for finding primes, in days of yore.
      • Twin primes

   2. binary representation -- every counting number is either a power of two, or can be written as a sum of distinct powers of two in just one way.
   3. (and Fibonacci numbers, eventually).
   A focus on uniqueness: unique representations.

4. Using historical information (e.g. Yanghui's triangle) and mathematics to learn something about cultures, systems of writing, etc.

5. Symmetry reared its lovely head, when we detected an error in Yanghui's triangle via a failure of symmetry.

6. One-to-one correspondence
   • It's how we count (even infinite things!).
   • It's an ancient idea, and has all sorts of interesting realizations across cultures: tallies, cairns, body numbers, etc.

7. Graphs (and trees are graphs)
   1. Definitions:
      • What is a graph?
      • Vertex/Node
      • Edge/Arc
      • degree of a vertex
      • Directed graph
      • Simple graphs (no loops or parallel edges)
      • Complete graphs (simple, every pair of nodes connected)
      • Planar graphs (no false intersections)
      • $K_5$, $K_{3,3}$, and Kuratowski's theorem
   2. Examples:
      • Facebook graphs (where nodes are labelled)
      • Directed graphs (e.g. mattresses)
      • The enemy of my enemy is my friend
      • Pascal's triangle applications
      • Special graphs (e.g. complete graphs, trees)

   3. Trees proved useful to carry out many of our calculations: in particular
      • primitive counting relied on dividing sheep, and we used a tree to keep track of the sheep in their pens.
      • We used trees to break down numbers in different bases and number representations (e.g. binary, Mayan, Babylonian).
      • We used a tree to break down a number by its prime factorization.
      • To represent the results of testing

   4. Dual graphs allowed us to use symmetry to solve problems. For example we can think of the dual of a Facebook graph as all the "unfriendships" between some people.
   5. Graphs ultimately figure into the Platonic solids, e.g. the Tetrahedron, and a number of other topics (e.g. Euler Graphs, planar graphs).
   6. Know your graph terminology (vertices/nodes, edges/arcs, regions/faces)

8. true vaccination rate: Two-toss sampling: there's a formula for figuring out the true rate of something, given that we know the number answering "yes" and the number answering "no" (with those lying if they get two heads, say -- roughly a quarter of those surveyed).

   Be prepared to use it.

9. Natural frequencies: Be familiar with Strogatz's Chapter 23: Chances Are

10. For the Covid modeling, just know the rough details (e.g. vocabulary)