Nathan Brunelle Department of Computer Science University of Virginia www.cs.virginia.edu/~njb2b/theory Theory of Computation CS3102 – Spring 2014 A tale.

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Theory of Computation CS3102 – Spring 2014 A tale of computers, math, problem solving, life, love and tragic death Nathan Brunelle Department of Computer.
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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