Talk:Surreal number

Drafts: 1

Table

Birthday	Surreal number	Label
0	{|}	0
1	{|0}	-1
	{0|}	1
2	{|-1}	-2
	{-1|0}	-1/2
	{0|1}	1/2
	{1|}	2
3	{|-2}	-3
	{-2|-1}	-3/2
	{-1|-1/2}	-3/4
	{-1/2|0}	-1/4
	{0|1/2}	1/4
	{1/2|1}	3/4
	{1|2}	3/2
	{2|}	3
4	{|-3}	-4
	{-3|-2}	-5/2
	{-2|-3/2}	-7/4
	{-3/2|-1}	-5/4
	{-1|-3/4}	-7/8

Proving that 1>0

The recursive definition of surreal numbers is completed by defining comparison:

Definition

Given numeric forms x = { X_L | X_R } and y = { Y_L | Y_R }, x ≤ y if and only if both:

• There is no x_L ∈ X_L such that y ≤ x_L: every element in the left part of x is strictly smaller than y.
• There is no y_R ∈ Y_R such that y_R ≤ x: every element in the right part of y is strictly larger than x.

Substitutions

$x=0=\{X_{L}|X_{R}\}=\{|\}$ and $y=1=\{Y_{L}|Y_{R}\}=\{0|\}$ We must prove two things:

1. There is nothing in $X_{L}$ that is larger than y. This is trivial because there is nothing in $X_{L}$.

2. There is nothing in $Y_{R}$ that is less than or equal to x. This is no problem because there is nothing in $Y_{R}.$Therefore, $1>0$

X_L={} and y={

Surreal numbers are real numbers, with a twist

Surreal numbers are defined to include positive and negative rational numbers (p/q where p and q are integers), as well as irrational numbers (π and 2^1/2), but with a twist: Also included are not one, but an infinite number of infinities, along with a similar collection of "zeros", often called ε (in the limit that ε→0.)

What can you do with surreal numbers? Among other things, they allow you to declare that:

$$.{\overline {9}}\;=\;\;.9999\ldots \;<1,$$

where, $(\ldots )$ denotes an infinite number of nines (see disclaimer.)This article will make no attempt to explain how surreal numbers might or might not be used in applied mathematics, except to point out that some of their properties are commonly used when dealing with real numbers.