Numbers! - Winter Semester 2016/17 - Tübingen

Contact

Office: C5A38. Office hour: Mondays 14:00-15:00, or by appointment.

Overview

A lecture course about the construction and properties of different numbers systems. We start by reviewing the construction of the 'classical' number systems \( \mathbb{N} \), \( \mathbb{Z} \), \( \mathbb{Q} \), \( \mathbb{R} \), \( \mathbb{C} \). We see how to finally prove familiar properties of numbers, like addition is associative, multiplication is commutative.

From here we introduce Conway's system of surreal numbers, a system which supersedes the real numbers \( \mathbb{R} \), and includes infinite numbers \( \omega \), infinitesimal numbers \( \varepsilon = 1/\omega \), and more. In the surreal numbers arithmetic with infinity (\(\omega\)) make sense, so we can talk about \( \omega + 1 \), \( 2 \omega \), even \( \omega - 1 \) (an infinite number less than infinity?), and \( \frac{1}{2} \omega \). Even more weird and wonderful numbers like \( \sqrt[3]{\tfrac{1}{4} \omega - 1} + \tfrac{\pi}{\omega^2} \) also make sense.

After the winter break, we will focus more on the game theory side of surreal numbers. We will introduce and analyse the games of Hackenbush and Nim by using (and extending) our theory of surreal numbers.

Lecture notes and extra stuff

Notes are in the process of being written/cleaned up. A version of the lecture notes with slightly updated formatting, and better theorem numbering is available.

See the visualisation of the surreal number tree on on the wikipedia page for Surreal numbers.

A more precise proof for the construction of $ y = \frac{1}{x} $ is given in Section 3.4 (page 17--19) of An introduction to Conway's Games and Numbers (Schleicher, Stoll).

Handout: Justifying the meaning of relations \( <, >, =, \parallel \) for games.

Link: Undergraduate thesis of Qingyun Wu on the mathematics behind Go endgames. Shows how to study Go endgames using combinatorial game theory.

Lecture outline

Problem sheets/exercises

Exam/assessment information

For those talking the course for credit, a ~2 hour final exam will be available during the semester break.

The final exam will be on Wednesday, 1 March 2017 in N9 at 14:00 s.t.

Any students intending to take the exam, please email me at charlton(at)math.uni-tuebingen.de to confirm your attendence.

Exam guidelines

Some of the questions will test a general understanding of surreal numbers, and game theory. I will also you to reproduce/recreate some proofs from the lectures. These should be proofs that you could create easily on your own with if you understand the definitions involved. It would be a good idea to look at problem sheet 3.

English-German dictionaries ARE permitted.

For the exam I will focus mainly on the following topics

For the exam I will NOT ask about the following topics

Literature

Good overview of the construction of \( \mathbb{N} \), \( \mathbb{Z} \), \( \mathbb{Q} \), \( \mathbb{R} \), \( \mathbb{C} \) in Chapters 1, 2, and 3. Covers some aspects of surreal numbers briefly in Chapter 13 (specifically section 1, 2, and 6), with interleaved with the game theory side.

Conway's own book on surreal numbers. First part gives gives a good introduction to surreal numbers, and covers many more advanced aspects. Second part treats the game theory side.

Donald Knuth's 'mathematical novelette'. A dialogue between two characters who gradually explore and build Conway's surreal number system. A useful learning tool, which teaches one how to think mathematically; the book records the characters' false starts, frustrations and finally successes in proving their own theorems on surreal numbers.