bogdan's (micro)blog

09:15 pm on Sep 10, 2012 | read the article | tags: nice2know

**the field of surreal numbers is the largest totally ordered field that can be constructed.** it’s nice2know because of the elegant construction and the neat ideas behind it.

let’s start with a totally ordered set and a rule:

“given two subsets, L and R of the initial set, with L strictly less than R,

{L|R}will denote the number strictly greater than L and strictly less than R”.

we can now elegantly construct:

**{|}** ~ 0 (where both L and R are empty sets);

**{0|}** ~ 1 (where L={|} and R is empty);

**{1|}** ~ 2 … {n|} ~ n+1, thus embedding the natural numbers.

making L empty and R one of the “naturals” we get negative integers: **{|n}** ~ -n-1

we can extend further the initial set closing it with respect to **{0|1}** ~ 1/2 and thus embedding the dyadic numbers set, which is a dense set in reals.

following the density of dyadic numbers, for any real number a we can construct infinite dyadic subsets L and R for which L < a < R, i.e. **{L|R}** ~ a.

the construction goes even further, building transfinite numbers like **{{1,2,3,…}|}** ~ ω and **{0|{1/2,1/4,1/8,…}}** ~ ε.

warning:

each surreal number has more than one representation, much like fractions. ex: 1/2 ~ **{0|1}** ~ **{1/4|3/4}**. this is why i avoided to use **{0|}** = 1.

definitions:

field, totally ordered set, dense set, transfinite number

*painting: joan miro – dancer*

- #artist 2
- #arts 4
- #away 3
- #bucharest 1
- #buggy 2
- #business 1
- #clothes 1
- #comics 1
- #contest 3
- #dragosvoicu 1
- #education 1
- #food 2
- #friends 14
- #hobby 23
- #howto 9
- #ideas 27
- #life lessons 3
- #me 59
- #mobile fun 4
- #music 51
- #muvis 15
- #muviz 13
- #myth buxter 1
- #nice2know 15
- #night out 1
- #openmind 2
- #outside 3
- #poems 4
- #quotes 1
- #raspberry 4
- #remote 56
- #replied 51
- #sci-tech 7
- #sciencenews 1
- #sexaid 6
- #subway 39
- #th!nk 3
- #theater 1
- #zen! 4

aceast sait folosește cookie-uri pentru a îmbunătăți experiența ta, ca vizitator. în același scop, acest sait utilizează modulul Facebook pentru integrarea cu rețeaua lor socială. poți accesa aici politica mea de confidențialitate.