Surreal Number - PowerPoint Presentation

Slide1

Surreal Number

Tianruo

ChenSlide2

Introduction

In mathematics system, the surreal number system is an arithmetic continuum containing the real number as infinite and infinitesimal numbers.Slide3

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 } = 1/4Slide4

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.Slide5

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 xSlide6

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.Slide7

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.Slide8

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≥1Slide9

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| }=2Slide10

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} Slide11

When the first number were bornSlide12

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

}Slide13

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 itSlide14

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. Slide15

A sample

G

ameSlide16

Analyzing Games

Every

game

has

to

end

with

a

winner

or

a

loser

and

where

there

a

re

finite

number of possible moves and the game must end in finite time.Let’s consider about the following Hackenbush GameAnd we assume that Blue makes the first move，there are seven possible moves we can reach.Slide17

The

t

ree

for

the

game

W

e

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.Slide18

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?Slide19

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 .Slide20

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

}Slide21

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

} = 1Slide22

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.Slide23

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:

W

e

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}=+¾Value of A ={+¼,0|+¾,+½}=+⅜Slide24

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