Surreal Numbers

Dan’s Surreal Number Definitions
Definition of x:

x = (x^L|x^R), where every element of x^L is strictly less than every element of x^R and x is strictly between the two.

Some specific definitions of x are:

( | ) = 0

(n | ) = n + 1

( | n) = n – 1

(p/2^q | (p+1)/2^q) = (2p+1)/2^q+1

Definition of "Simplest":

Given that x = (x^L | x^R), x is the "simplest" number between x^L and x^R.

The order of simplicity is as follows:

0 is the simplest;

then the ± integers in order (± 1 are simpler than ± 2, ± 2 are simpler than ± 3, ...);

if there is no integer between x^L and x^R, then the simplest number is the fraction between them with the lowest power of 2 in the denominator.

Definition of "Between":

x is between L and R iff max(L) < x < min(R)

max(L) = x^L = largest element of L

min(R) = x^R = smallest element of R

Some specific examples are:

(1 | 3) = 2

(1 | 5) = 2

(Ľ | ľ) = ˝

(˝ | 2˝) = 1

Definition of x > y, x < y:

x > y iff (there is no element of x^R less than y, and x is not less than any element of y^R)

x < y iff y > x

Definition of x = y, x > y, x < y

x = y iff (x < y and x > y)

x > y iff (x > y and y ¹ x)

x < y iff y < x

Definition of x + y:

x + y = (x^L + y, x + y^L | x^R + y, x + y^R)

Definition of –x:

–x = (–x^L | –x^R)

Definition of x · y:

x · y = (x^L·y + x· y^L – x^L·y^L, x^R·y + x· y^R – x^R·y^R | x^L·y + x· y^R – x^L·y^R, x^R·y + x· y^L – x^R·y^L)

Further Reading