How can $\pi$ be defined as a surreal number. I want to express $\pi$ in the Surreal Number notation $\{L|R\}$. What is the most natural or intuitive way of doing so, seeing as there are many (possibly infinite) ways of expressing the same surreal number.
 A: A simple explanation can be found in this wikipedia article, from which I take the following quotation:

... any real number $a$ can be represented by $\{L_a \mid R_a\}$, where $L_a$ is the set of all dyadic rationals less than $a$ and $R_a$ is the set of all dyadic rationals greater than $a$ (reminiscent of a Dedekind cut). Thus the real numbers are also embedded within the surreals.

A: If you want to have a more explicit expression, start with the largest integer below $\pi$, that is $a_0=3$, and then iterate the following algorithm:


*

*If $a_n<\pi$, it goes into the left set, and $a_{n+1} = a_n + 1/2^{n+1}$.

*If $a_n>\pi$, it goes to the right set, and $a_{n+1} = a_n - 1/2^{n+1}$.
This gives
$$\pi = \{3, 3.125, 3.140625, \ldots  |\, 3.5, 3.25, 3.1875, 3.15625, \ldots\}$$
A: Perhaps a more natural construction would be to consider the sequences $I_n$ and $C_n$ of $n$-gons (a) inscribed in and (b) circumscribing a circle of unit diameter for $n\in\mathbb{N}$. Now takes $i_n$ to be the circumference of $I_n$ and $c_n$ the circumference of $C_n$. $i_n$ and $c_n$ are algebraic, so can certainly be constructed without first constructing $\pi$, so it makes sense to set
$$\pi = \{ i_n \mid c_n \}$$
