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.

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
$$\pi = \{3, 3.125, 3.140625, \ldots  |\, 3.5, 3.25, 3.1875, 3.15625, \ldots\}$$
