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 xL for the typical member of L and xR for

the typical member of R.• So we can write {xL|xR} to represent x itself.

Definition• Definition 1• We say x ≥ y iff no xR ≤ y and x≤ no yL

• 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=1For example, Is -1≥ 1?-1≥1 iff no -1R ≤ 1 and -1≤ no 1L

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,-½0={−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 = {xL + y,x + yL| xR + y, x + yR }

• Definition of –x• -x = { -xR | -xL }

• Definition of xy.• xy = {xLy + xyL – xLyL, xRy + xyr –xRyR |xLy +xyR –xLyR,xRy + xyL-xRyL}

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

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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 ={ B1,B2,…,Bn|R1,R2,…,Rm}

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 http://www.geometer.org/mathcircles December 15, 2011