Surreal Number Tianruo Chen

Introduction In mathematics system, the surreal number system is an arithmetic continuum containing the real number as infinite and infinitesimal numbers.

Construction of surreal number Surreal number is a pair of sets of previously created surreal number. If L and R are two sets of numbers, and no member of L is ≥ any member of R, then we get a number {L|R} We can construct all numbers in this way For example, { 0 | } = 1 { 1 | } = 2 { 0 | 1 } = 1/2 { 0 | 1/2 } = 1/4

Convention If x={L|R}, we write x L for the typical member of L and x R for the typical member of R. So we can write {x L |x R } to represent x itself.

Definition Definition 1 We say x ≥ y iff no x R ≤ y and x≤ no y L Definition 2 x=y iff x ≥ y and y ≥ x x>y iff x≥y and y is not more than or equal to x

Let’s construct the surreal numbers Every number has the for {L|R} based on the construction. But what do we have at the beginning? Since initially there will be no earlier constructed number. The answer is that there is a certain set of number named the empty set Ø. So the earliest number can only be {L|R} where L=R=Ø. In the simplist notation { | }. We call this number 0.

Is the surreal number well- formed We have mentioned that no member of L is ≥ any member of R. We call the number well-formed if it satisfies this requirement. So are any members of the right set less than or equal to any members of the left set? Since both the sets are empty for { | }. It doesn’t matter here.

The construction of -1 and 1 We can create 3 new numbers now. {0| }, { |0} and {0|0} Since the last number {0|0} is not well-formed, because 0≤0. We only have 2 appropriate surreal number {0| } and { |0}. Here we call 1={0| } and -1={ |0}. We can prove that -1= -1, -1<0, -1<1, 0<1, 1=1 For example, Is -1≥ 1? -1≥1 iff no -1 R ≤ 1 and -1≤ no 1 L But 0≤1 and -1≤0, So we don’t have -1≥1

The Construction of 2,½,-2,-½ As we find before, -1<0<1 And we have particular set { }, {-1},{0},{1},{-1,0},{-1,1},{0,1},{-1,0,1} We use it for constructing surreal number with L and R { |R}. {L| }, {-1|0}, {-1|0,1}, {-1|1}, {0|1},{-1,0|1} We define {1| }=2, {0 |1}=½ And For number x={ 0,1| }, 0<x and 1<x, since 1<x already tells us 0<1<x, the entry 0 didn’t tell us anything indeed. So x={0,1| }={1| }=2

The Construction of 2,½,-2,-½ 0={−1 | }={ | 1} ={-1| 1} 1={−1, 0 | } 2={0, 1 | } = {−1, 1 | } ={−1, 0, 1 | } -1={ | 0, 1} −2={ | − 1, 0} = { | − 1, 1} ={ | − 1, 0, 1} ½={−1, 0 | 1} -½={−1 | 0, 1}

When the first number were born

Arithmetical operation Definition of x ＋ y x+y = {x L + y,x + y L | x R + y, x + y R } Definition of –x -x = { -x R | -x L } Definition of xy. xy = {x L y + xy L – x L y L, x R y + xy r –x R y R |x L y +xy R –x L y R,x R y + xy L -x R y L }

The number {Z| } Since Because 0 is in Z, 1={0| } and -1={ |0} are also in Z. Therefore, all numbers born from these previous number set are in Z. Then we can create a new surreal number {Z| } What is the value of it? It is a number that greater than all integers. It’s value is infinity. We use Greek letter ω to denote it

Red-Blue Hackenbush Game Rule: There are two players named “Red” and “Blue” Two players alternate moves, Red moves by cutting a red segment and Blue, by cutting a blue one When a player is unable to move, he loses. A move consists of hacking away one of the segments, and removing that segment and all segments above it that are not connected to the ground.

A sample Game

Analyzing Games Every game has to end with a winner or a loser and where there are finite number of possible moves and the game must end in finite time. Let’s consider about the following Hackenbush Game And we assume that Blue makes the first move ， there are seven possible moves we can reach.

The tree for the game We can now draw the following complete tree for the game In this case, if Red play correctly, he can always win if Blue has the first move.

Some Fractional Games Let’s assume that components of positions are made of entirely n blue segments, it will have a value of +n, and if there are n red segments, it will have the value of –n. In the picture (A), the blue has exactly 1 move, so it can be assigned the value of +1. However, in all other four diagrams, blue can win whether he starts first or not and Red has more and more options. So what is the value of the other four pictures?

Some Fractional Games Let’s consider the picture (F), it has a value of 0, since whoever moves first will lose. And the red segments has the value of -1. This meant that two copies of picture(B) has the sum value of +1. So it the picture (B) has the value ½. And consider about the picture (G), we can get the picture(C) has the value ¼.

Finding a Game’s Value Let’s consider the following game. We can find that the value of the picture is +1. (three blue moves for +3 and 2 red moves for -2) If blue need to move, the remaining picture will have values of 0,-1 and -2. If the red move first the remaining picture will have value of +2 and +3. V ={ B 1,B 2,…,B n |R 1,R 2,…,R m }

Finding a Game’s Value For the sample game above, we can write the value as: {−2, −1, 0|2, 3}. We can ignore the “bad” moves and all that really concerns us are the largest value on the left and the smallest value on the right: {−2, −1, 0|2, 3} = {0|2} = 1

Calculating a Game Value Let’s work out the value of the following Hackenbush Game. The pair of games on the left show all the possibility that can be obtained with a blue move and the ones on the right are from a red move.

Calculating a Game’s Value From previous work, we know the game values of all the components except for the picture (B) Repeating the previous steps, we get: We know that the value of (C),(D),(E) and (F) are +½,+¼,0 and +1. So we get the value of B={+½, 0|+1,+1}=+¾ Value of A ={+¼,0|+¾,+½}=+ ⅜

Thank you for listening Reference: [Conway, 1976] Conway, J. H. (1976). On Numbers and Games. Academic Press, London, New York, San Francisco. [Elwyn R. Berlekamp, 1982] Elwyn R. Berlekamp, John H. Conway, R. K. G. (1982). Winning Ways, Volume 1: Games in General. Academic Press, London, New York, Paris, San Diego, San Francisco, Sa ̃o Paulo, Sydney, Tokyo, Toronto. [Knuth, 1974] Knuth, D. E. (1974). Surreal Numbers. Addison- Wesley, Reading, Massachusetts, Menlo Park, California, London, Amsterdam, Don Mills, Ontario, Sydney. Hackenbush.Tom Davis December 15, 2011