1. Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death

2. Problem: True or false: there arbitrary long blocks of consecutive composite integers. Extra Credit: find a short, induction-free proof.

3. 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?

4. Peano-Arithmetic Numbers from Scratch

8. 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

9. How do we get integers?

10. Peano-Arithmetic Numbers from Scratch

11. How do we get rationals?

12. How do we get reals?

13. Beyond Reals: Surreals Reals constructed from Dedekind Cuts

14. Beyond Reals: Surreals

16. 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

17. Problem: Given any five points in/on the unit equilateral triangle, is there always a pair with distance ≤ ½ ? 1 1 1 What approaches fail? What techniques work and why? Lessons and generalizations

18. 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?

19. Other “numbers” of interest

20. Quaternions- Multiplication Table

21. Octonians- Multiplication Table

22. Sedenions- Multiplication Table