How can $\pi$ be defined as a surreal number.

Question

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.

Answer 1

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.

Answer 2

Being infinite there is no finite representation of π in Surreal numbers. I think there are two options

1. Just declare your variable to be a surreal number with a numeric label of π, and then continue on with your equations. ie: don't calculate it, just say: "x is a surreal and its value is π"

2. Express π as a written estimate.

If you want to hold a π estimate in a computer, then I would say: π = + 0 + 1 + 1 + 1 + 1 − 1÷2 − 1÷4 − 1÷8 + 1÷16 − 1÷32 − 1÷64 + 1÷128 − 1÷256 − 1÷512 + ... for as long as you want to continue. And then extract the signs: +++++---+--+--, and convert to a bit format: 11111000100100. Then I would call this format sinary and declare the left side to be this value with the right side trimmed back to the first 0 and the left side is this value trimmed back to the first 1. ie:

original = 11111000100100
.   left = 11111000100
.  right = 1111100010010

These estimate expressions for pi are:

π = + 0 + 1 + 1 + 1 + 1 − 1÷2 − 1÷4 − 1÷8 + 1÷16 − 1÷32 − 1÷64 + 1÷128 − 1÷256 − 1÷512 = 3.142578125

πₗ = + 0 + 1 + 1 + 1 + 1 − 1÷2 − 1÷4 − 1÷8 + 1÷16 − 1÷32 − 1÷64 = 3.140625

πᵣ = + 0 + 1 + 1 + 1 + 1 − 1÷2 − 1÷4 − 1÷8 + 1÷16 − 1÷32 − 1÷64 + 1÷128 − 1÷256 = 3.14453125

which says:

π ≈ { 201/64 | 805/256 } = numeric label 1609/512

I suggest that expressing any infinite surreal in a finite format should only contain the central estimate of the number. Because the left and right set will be contained within this representation already and storing them separately provides no new information to our estimated representation.

Answer 3

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

Answer 4

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 $$\pi = \{ i_n \mid c_n \}$$