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I know how to obtain surreal numbers in n-day and I know about <= in surreal numbers axiom 2. We will also discover a lot of new representatives for well-known values. For example : $\{-1|1\}=0$ or $\{1/2|2\}=1$ , ... But I'm confused with some new representatives for example :$\{-2|1\}=0$. Why is not it equal to $-1$ or what is $\{-3|3/4\}=?$ or what is $\{1/4|3\}=?$

Do Anybody know all solution about them(equivalence representatives)?

Note:Please for answer ,don't example to surreal numbers origin representive {0|1}=1/2 or etc. I know about it . I want to know equivalence or new representatives for example $\{-2|1\}=?$ or $\{-4|3/4\}$ or etc...

farshad
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    What are you taking about ? You do as if everybody knows "surreal-numbers" and what '$n$-day" means in this context, but I fear that almost nobody knows that. You should at least give a thorough web reference ! – Jean Marie Jun 19 '17 at 11:56
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    @JeanMarie Everyone familiar with/following the tag "surreal-numbers" probably understands what farshad meant by "$n$-day". The Wikipedia page for the surreal numbers addresses what the surreal numbers are and the "day"s of their construction quite thoroughly. – Mark S. Jun 29 '17 at 22:28

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$\{a\mid b\}$ always represents the earliest surreal number $x$ (as defined by which day it first appears) such that $a<x<b$ (this also generalises quite naturally to when the left and right sets are not singletons). For instance, if $a$ is negative and $b$ is positive, $\{a\mid b\}$ always represents $0$. As I mentioned in my answer to your other question about surreal numbers, this can get very tedious to check manually for all but the simplest examples.

Arthur
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  • Thanks Arthur. What is {1/4|3}=? or etc. please tell me to this example... – farshad Jun 19 '17 at 12:36
  • @farshad The earliest number created that lies between $1/4$ and $3$ is $1$, so it represents $1$. You can check with the axiom that ${0\mid{}}\leq {1/4\mid 3}$ and ${1/4\mid 3} \leq {0\mid {}}$ if you want to confirm it. In order to do that, you will, of course, have to choose representatives of $1/4$ and $3$, and representatives of the numbers appearing in the left and right sets of those numbers, and so on. I think the proof is doable in less than a page if you choose the shortest (i.e. original) forms for each number. – Arthur Jun 19 '17 at 12:42
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    Thanks a lot Arthur. Then {3/2|4}=2 because 2 is earliest number created that lies between 3/2 and 3. Do I say true? – farshad Jun 19 '17 at 13:17
  • @farshad Exactly. – Arthur Jun 19 '17 at 13:19
  • Thank you very much Arthur. – farshad Jun 19 '17 at 13:39