List of equivalence surreal numbers to 4 day?

Question

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}?

Answer 1

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.

Answer 2

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.