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Conway John H.

A K Peters, Ltd. / CRC Press

242

1-56881-127-6

2nd edition

Rest in Peace John H. Conway.

This book grants an intuitive introduction to the class of surreal numbers, which enables us to distinguish between different kinds of infintity.

Starting with the form {L | R} the core idea of surreal numbers has great philosophical value. Any number has a left and right side, which is distinguished. The only rule is that there must not be any number L>=R. Starting with this basis, the numbers grow like a tree.

The number 0 is { | }. On both sides we only find the empty set.

Therefore, Conway founded number 0 on day 0.

Now we have created the possibilites for

{ | } and {0 | } and { | 0} and {0 | 0}.

{0 | 0} is not a number, as 0 >= 0.

Number 1 is {0 | }.

Numer -1 is { | 0}.

On day 1 numbers -1 and 1 were created.

With the condition that there must not be L >= R we now have

{ | R}, {L | }, {-1 | 0}, {-1 | 0, 1}, {-1 | 1}, {0 | 1} {-1, 0 | 1}

Number 2 is {1 | }.

Number 1/2 is {0 | 1}.

Number -2 is { | -1}.

Number -1/2 is {-1 | 0}.

On day 2 we have created these new numbers.

New numbers are born on each new consecutive day, etc. creating an infinite number tree. Conway now plays with the idea, which numbers are born on day "infinite". He shows which numbers are created on this day. The game with infinity begins.