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*

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