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One way to represent surreal numbers is the sign expansion. Now Wikipedia describes how to compare them, how to convert them to the standard representation of left/right sets, how to negate them, and how to add and multiply them (however those operations are described basically by converting to left/right sets, adding them in that form, and converting back).

However I'm interested in "reading" them. My question is now twofold: First, I want to know if what I think I've already found out is correct, and second, I'd like to know how to "read" more of them (ideally, a scheme that at least in principle allows to "read" all of them). With "reading", I mean finding out a number/formula representation that represents the same number, without going through the recursion formula defining the surreal numbers.

Here's what I think I've already found out:

First, consider the numbers with finite domain. Those are effectively represented by a finite string of $+$ and $-$. Since a number gets negated by changing all signs at once, it suffices to consider those numbers which do not start with a $-$ (i.e. the nonnegative numbers).

The special case that the string contains no $-$ is simple: The number of $+$ in the string is the number represented.

Otherwise, since the string begins with a $+$, it contains somewhere the substring $+-$. The first such substring can be interpreted as "decimal point", separating the integer part before it and the fractional part after it. The integer part before it is interpreted as before: The number of $+$ is directly the integer part of the number. The fractional part is then decoded as follows: Whenever the same sign follows, a $0$ is appended to the number, and whenever a different sign follows, an $1$ is appended to the number. Finally, when the end of the string is reached, a final $1$ is appended. The digit string such obtained is interpreted in base 2.

For example, the digit string $+++--+--$ contains a $-$, therefore we identify the first $+-$ as "decimal point". That is, we get $++\mathbf{+-}-+--$. Before the $+-$ we have two $+$, so the integer part is $2$. The fractional part starts with $-$ which is the same as the previous $-$ from the "decimal point", so we start with $0$. Then we get a sign change to $+$, giving an $1$, followed by a second sign change to $-$ again, giving another $1$, and no sign change giving another $0$. Finally, another $1$ is appended, giving a total fractional part of $.01101_2 = .40625_10$. Therefore the number is $2.40625$.

Next, consider the numbers with domain $\omega$. I think the same interpretation can be applied here, except that you of course never reach a final digit, and thus never reach an end.

However there are two special cases, namely the ones which would result in an infinite number of $0$s (i.e. the string ending either in $++++\dots$ or $----\dots$) and an infinite number of $1$s (i.e. a string ending in $+-+-\dots$); those, interpreted as real numbers, would give the finite binary fractions again. I think those correspond to $x+\epsilon$ and $x-\epsilon$ with $x$ a dyadic fraction (because those are exactly the generation $\omega$ numbers which are missing, and the ordering would also fit).

Now I want to know:

  • Is my interpretation of the sign expansion correct?
  • And how can the sign expansions with domain $>\omega$ be interpreted?
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Your interpretation of finite sign expansions is slightly off. There are essentially two equivalent approaches to interpreting these finite expansions: One makes conversion between binary representations easy, and one hinting at the general structure even for domains greater than $\omega$.

The easy-conversion rule begins the way you started, but after the first instance of "$+-$" you can simply read off the remaining symbols as bits with $+$ being 1 and $-$ being zero, and then append a final $1$. Therefore, $+++−−+−−$ would be $2+\left(.01001_{2}\right)$. The other way to think about it is that the first three $+$ signs take you up to $3$, and then the first (and only the first, for a finite expansion) sign change starts a sequence of diminishing returns: $+++−−+−−=1+1+1-\frac12-\frac14+\frac18-\frac1{16}-\frac1{32}=2+\frac14+\frac1{32}$.

In any case, your guess about expansions with domain $\omega$ is essentially correct. Every domain-$\omega$ expansion that doesn't end in a tail of $-$ or $+$ corresponds to the real number with the corresponding binary expansion. If it's all $+$, then it's $\omega$. Among the positives, if it contains a $-$ but ends in a tail of $+$, then that would be a tail of $1$s in the binary expansion for some real number $x$, but the sign expansion actually corresponds to $x-\varepsilon=x-\frac1{\omega}$. Similarly, a tail of $-$ would be $x+\varepsilon$.

A short answer to your second question would be just to give you some examples, using $\cdots$ to indicate a pattern of "length" $\omega$. $+++\cdots+=\omega+1$, $+++\cdots-=\omega-1$, $+++\cdots+++\cdots=\omega2$, $+++\cdots---\cdots=\frac{\omega}2$ $+--+-+-+-+-+\cdots+=\frac13+\varepsilon$, $+----\cdots+=2\varepsilon$, $+----\cdots-=\varepsilon/2$, $+----\cdots---\cdots=\varepsilon^2$.

A thorough answer is given by Philip Ehrlich's "Conway names, the simplicity hierarchy and the surreal number tree", which is currently available at http://www.ohio.edu/people/ehrlich/ConwayNames.pdf

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