Nathan Brunelle Department of Computer Science University of Virginia Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death
Problem: True or false: there arbitrary long blocks of consecutive composite integers. Extra Credit: find a short, induction-free proof.
Today: Numbers Build numbers from scratch Microcosm of course arc Going from basic, to more complex/general, to potentially absurd Gives a complete story of number systems Gain intuition on complexity of numbers Philosophical Q: do numbers exist?
Peano-Arithmetic Numbers from Scratch
X = 2 X X X X … Extra Credit Problem: Solve the following equation for X: where the stack of exponentiated x’s extends forever. What approaches fail? What techniques work and why? Lessons and generalizations
How do we get integers?
Peano-Arithmetic Numbers from Scratch
How do we get rationals?
How do we get reals?
Beyond Reals: Surreals Reals constructed from Dedekind Cuts
Beyond Reals: Surreals
Problem: Given any five points in/on the unit square, is there always a pair with distance ≤ ? 1 1 What approaches fail? What techniques work and why? Lessons and generalizations
Problem: Given any five points in/on the unit equilateral triangle, is there always a pair with distance ≤ ½ ? What approaches fail? What techniques work and why? Lessons and generalizations
What approaches fail? What techniques work and why? Lessons and generalizations x y Problem: For the given infinite ladder of resistors of resistance R each, what is the resistance measured between points x and y?
Other “numbers” of interest
Quaternions- Multiplication Table
Octonians- Multiplication Table
Sedenions- Multiplication Table