I can obtain surreal numbers in n-day then I don't want only list of surreal numbers example in 2-day: -2<-1<-1/2<0<1/2<1<2 but I want list of equivalence surreal numbers exemplar 2-day: equivalence surreal numbers 2day Do one know it for 3-day and 4-day?! Note: I know {-1/2|-1}=-3/4 . I want more example {-2|3/4}?
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3I can't really understand what you've written, but it seems like you want something exactly like your picture, but with every possible fifth day number and fifth day representation instead of second day. Note that in the fifth day, there are $32$ numbers, and many of them are going to have many different forms, so the list is going to be huge. Just of the top of my head, with very quick rudimentary calculations, I'd guess there are somewhere around $2^{20}$ forms. That's a long list to ask someone on the internet to just do for you. – Arthur Jun 18 '17 at 08:39
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I said there are $32$ numbers on the fifth day. I was wrong. There are $63$. And $0$ alone has $2^{30}$ representations. So for day five, the list of possible forms is probably somewhere between twenty and fifty billion entries long. – Arthur Jun 18 '17 at 09:17
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Do you know the notion of sign sequences for / as surreal numbers? They make computing a surreal number more straight forward (though the result is a sign sequence that might be hard to decipher!). For instance your list of inequalities can be written as $(--) < (-) < (-+) < () < (+-) < (+) <(++)$. Then if I want to know what ${-1 \ | \ -\frac{1}{2}} = {(-) \ | \ (-+)}$ is, I know it's $(-+-)$ which I can convert back to its dyadic form $(-+-) = -\frac{3}{4}$. – nombre Jun 18 '17 at 09:30
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Arthur yes. but I want {xL|xR} that xL and xR is only a number of 3-day and not some number.... – farshad Jun 18 '17 at 10:04
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@farshad I know that, and have taken that into account. Otherwise the list would be infinite, as $0$ is represented by, among other things, $\langle{}\mid n\rangle$ for any positive $n$. See my answer below for calculation details. – Arthur Jun 18 '17 at 10:21
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Thanks nombre. I want source about sign sequence. and I want more example equivalence sureeal number for instance :{-2|3/4}? – farshad Jun 18 '17 at 10:38
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This one is the simplest surreal number between -2 and 3/4. If 0 is between those then it is it. Else if -1 xor +1 is between those it is it. Else etc...Gonshor introduced sign sequences in An introduction to the theory of surreal numbers. – nombre Jun 18 '17 at 12:13
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Ok, calculation time, just to make it clear what I'm talking about in my comment above: on day $3$, the number $3$ appears. It appears in many different forms, namely all the forms that has empty right set, and where the left set contains $2$. How many such are there? Well, there are $7$ numbers from day $2$ that are available, and the only restriction is that $2$ appears. That means there are $64$ representations of $3$.
Similarly, there are $64$ third-day representations of $0$, because as long as everything in the left set is negative, and everything in the right set is positive, and $0$ doesn't appear, the form represents $0$. Then there are the remaining $14$ numbers, and they all have a similar number of representations (maybe half as many representations? I haven't done the math, add the combinatorics gets a bit more hairy, but it's close). The list will just be too long past day $2$ to ever bother.
To answer your question: yes, it is known what representations are available, but it takes a special kind of dedication to actually write them down. Now, I love the concept of surreal numbers. I wrote an answer here a while back, basically declaring my love for them. But they get very tedious, very fast, and this question is a perfect example of that.
Arthur
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I want more equivalence surreal numbers for instance:{-3|3/2}? that xL and yL are one number. – farshad Jun 18 '17 at 10:40
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@farshad But in your example image, there is $2 = {-1, 0,1\mid }$, where $x_L$ has three numbers. You weren't looking for something exactly like your example image, then? – Arthur Jun 18 '17 at 10:42
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Yes Arthur. but you told me They are many numbers and I know in numbers example {-1,0,1|} max or min xLs or xRs are important. Then I decide to restrict them to one number for xL or xR example {-3|3/2} or {1/4|4} or {-2|3} ?! for instance: {1/2|2}=1 – farshad Jun 18 '17 at 14:55
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The number of new surreals born on a given day is 2^days.
The sequence of number for a given day has the same spread pattern as the Van Der Corput Sequence. https://en.wikipedia.org/wiki/Van_der_Corput_sequence
The numbers on a given day will be the numbers in that sequence minus 1/2 times 2^(n-1). I believe, but they are related and you can surely use it to find the surreal numbers labels and ordering of naming labels of the numbers as they are born according to the process described by Conway.
I wrote a python program on that gives birth to the surreal numbers over the first 7 days of creation: https://github.com/peawormsworth/PySurreal/blob/master/Creating%20Surreal%20Numbers%20in%20Python3.ipynb
peawormsworth
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The first 32 surreal numbers in reduced (L|R) form...
0 => ( None|None )
-1 => ( None|0 )
1 => ( 0|None )
-2 => ( None|-1 )
-1/2 => ( -1|0 )
1/2 => ( 0|1 )
2 => ( 1|None )
-3 => ( None|-2 )
-3/2 => ( -2|-1 )
-3/4 => ( -1|-1/2 )
-1/4 => ( -1/2|0 )
1/4 => ( 0|1/2 )
3/4 => ( 1/2|1 )
3/2 => ( 1|2 )
3 => ( 2|None )
-4 => ( None|-3 )
-5/2 => ( -3|-2 )
-7/4 => ( -2|-3/2 )
-5/4 => ( -3/2|-1 )
-7/8 => ( -1|-3/4 )
-5/8 => ( -3/4|-1/2 )
-3/8 => ( -1/2|-1/4 )
-1/8 => ( -1/4|0 )
1/8 => ( 0|1/4 )
3/8 => ( 1/4|1/2 )
5/8 => ( 1/2|3/4 )
7/8 => ( 3/4|1 )
5/4 => ( 1|3/2 )
7/4 => ( 3/2|2 )
5/2 => ( 2|3 )
4 => ( 3|None )
Generated here: https://github.com/peawormsworth/tools/blob/master/Surreal%20Golf.ipynb
peawormsworth
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Put all of the sureals created each day into a numeric ascending ordered list and prepend it with null. The first new number will have a left side of the first number in your list and a right side of the last number in your list. Now prepend your new number to the beginning of the list and roll the last number in your list to the beginning. When null has returned to the beginning of the list, you will have completed a full day cycle and your list will contain all the numbers created up to that next day. In order to prepare the list for another cycle, evaluate the numeric value of (L,R) pairs for the new numbers using (L+R)/2. Now your list will be ready to produce a new day of numbers using the same process. This process can be repeated indefinitely.
For example, your initial ordered list would be: (),2,1,1/2,0,-1/2,-1,-2 The first new number is (first,last), which is ((),-2). Now put the new number on the beginning of the list: ((),-2),(),2,1,1/2,0,-1/2,-1,-2. "roll" the list so the last entry moves to the first: -2,((),-2),(),2,1,1/2,0,-1/2,-1
Repeat the process to find the second new number: (-2,-1). Put it on the front of list and roll back to front: -1,(-2,-1),-2,((),-2),(),2,1,1/2,0,-1/2
The next repetition makes: -1/2,(-1/2,-1)-1,(-2,-1),-2,((),-2),(),2,1,1/2,0
Now repeat this process until () is at the beginning. You will have all your new number for the new day in (L,R) form. Convert this form back to value labels (numbers) and repeat the process to find the next day and so on.
In general, on each day, the new numbers will be between all the existing numbers on previous days. So if you order you numbers from low to high, the new numbers will be all the pairs that you can make of two consecutive number (with empty sets imagined to be on the ends.
peawormsworth
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Or you could just list your numbers in ascending order, double each of them, add empty sets to the ends and then read off all the new numbers as pairs:
Ascending order:
-2,-1,-1/2,0,1/2,1,2
Double each:
-2,-2,-1,-1,-1/2,-1/2,0,0,1/2,1/2,1,1,2,2
Add empty sets to ends:
(),-2,-2,-1,-1,-1/2,-1/2,0,0,1/2,1/2,1,1,2,2,()
Read off the pairs as the new numbers:
((),-2) (-2,-1) (-1,-1/2) (-1/2,0) (0,1/2) (1/2,1) (1,2) (2,())
Being:
-3, -3/2, -3/4, - 1/4, 1/4, 3/4, 3/2, 3
Combine these with previous number to repeat for the process of the next day
– peawormsworth Jun 21 '21 at 11:21