Surreal number

The following pages in this resource may be of interest to you

Dyadic fractions subdivide an inch

■ Surreal number/The dyadics is still under construction. It will be a very simple introduction that might make surreal numbers seem easy to understand (if not a bit silly.) The figure to the right illustrates how rulers often subdivide the inch into dyadic fractions. The construction of surreal numbers begins with these dyadics (ranging from −∞ to +∞.) Each is assigned "birthday" and is associated with a pair of sets. Curiously, each set consists only of the null set $\{\,\},$ as well as sets that contain the null set in some way. Making matters even more complicated is each surreal can be associated with an endless variety of these pairs of sets.
■ Surreal number/Root 2 begins by discussing how to construct a pair of infinite series that converge to ${\sqrt {2}}$ (one from above and one from below). One derivation of this series is presented, but we also hint at how Newton's method might be used to construct the same pair of sequences. Later, we discuss how to generalize this method to include other roots $(x^{1/n})$. These pairs of sequences can be used to express surreal forms for irrational numbers that are the n-th root of an integer.
■ Surreal number/Counting introduces concepts involving infinity that are necessary for any student wishing to learn about surreal numbers. No claim is made that this prerequisite material is sufficient. This introduction begins with Cantor's diagonal argument

present informal introduction to countable and uncountable infinities that beginners can grasp. It makes not attempt to the leave the reader with the rigorous understanding of this discussion required to fully understand surreal numbers.

More contributions are welcome! To create an essay or article start with a title that not already on the list below. It is very easy to change a title.

• ChatBot assistance
• Counting
• Draftspace
• Root 2
• See also
• Simple hackenbush
• The dyadics

• If you need help, leave a message at User talk:Guy vandegrift.

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References

Wikiversity/Wikipedia/Wikibooks

• Introduction to Category Theory/Products and Coproducts of Sets
• Introduction to Category Theory/Sets and Functions
• Set theory
• w:Infinity#Real_analysis
• w:Transfinite number#Examples
• w:Hyperreal number
• w:Von Neumann universe

External links

• Review on amazon.com]
• staxeschange definition of all dyadics
• Scientific American badiou-and-science part-1 I might copy graph

• Steven Charlton - An Very Brief Introduction to Surreal Numbers (alternate link)

• what is a game? (mit)
• .9999
• wikipedia:Surreal number
• whitman.edu (Grimm.pdf) Includes games
• scietificamerican.com
• Roughan Directed Acyclic Graphs (DAG); 2*3 & 3/4 + 3/4/ (too complicated for me)

proof of conways simplicity rule

https://math.stackexchange.com/questions/816540/proof-of-conways-simplicity-rule-for-surreal-numbers

$0=\{|\}$

$n+1=\{n|\}$

$-n-1=\{|-n\}$

${\tfrac {2p+1}{2^{q+1}}}=\left\{{\tfrac {p}{2q}}|{\tfrac {p+1}{2q}}\right\}$

Limits and analysis

Infinity is easy to imagine, but difficult to incorporate}}

Infinity is easy to imagine, but difficult to incorporate into rigorous mathematics. The following calculation certainly violates the rules of mathematics:

$\lim _{\epsilon \to 0}{\frac {1}{\epsilon }}=\infty \;$ and $\;\lim _{\epsilon \to 0}{\frac {2}{\epsilon }}=\infty \;$ implies $\;{\frac {\infty }{\infty }}=2.$

This is why expressions like $\infty /\infty$ and $0/0$ of often called indeterminant. Most of the time, mistakes like this can be avoided by utilizing concepts taught in a course on mathematical analysis. Both the Wikipedia article and a query sent to an online chatbot suggest that no widely known problems in this field have been solved using surreal numbers. For that reason, any forrey into surreal numbers should probably be viewed as recreational.

Images

Infinitesimals (ε) and infinities (ω) on the hyperreal number line (1/ε = ω/1). This image appears in the Wikipedia articles w:Hyperreal number, w:Infinitesimal, and w:Infinity.

Latex Positive dyadic rationals: days 1-5

${\begin{matrix}&&&&&&&&&&&&&&&1&&&&&&&&&&&&&&&\\&&&&&&&{\tfrac {1}{2}}&&&&&&&&&&&&&&&2&&&&&&&\\&&&{\tfrac {1}{4}}&&&&&&&&{\tfrac {3}{4}}&&&&&&&&1{\tfrac {1}{2}}&&&&&&&&3&&&\\&{\tfrac {1}{8}}&&&&{\tfrac {3}{8}}&&&&{\tfrac {5}{8}}&&&&{\tfrac {7}{8}}&&&&1{\tfrac {1}{4}}&&&&1{\tfrac {3}{4}}&&&&2{\tfrac {1}{2}}&&&&4&\\{\tfrac {1}{16}}&&{\tfrac {3}{16}}&&{\tfrac {5}{16}}&&{\tfrac {7}{16}}&&{\tfrac {9}{16}}&&{\tfrac {11}{16}}&&{\tfrac {13}{16}}&&{\tfrac {15}{16}}&&1{\tfrac {1}{8}}&&1{\tfrac {3}{8}}&&1{\tfrac {5}{8}}&&1{\tfrac {7}{8}}&&2{\tfrac {1}{4}}&&2{\tfrac {3}{4}}&&3{\tfrac {1}{2}}&&5\end{matrix}}$

Categories

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