Generating the Surreal Numbers

n

plot type

list

number line

tree

layered graph

new surreal numbers of generation 2

2 ↔ {1|} ↔ {{0|}|} ↔ {{{|}|}|}

1

2

↔ {0|1} ↔ {{|}|{0|}} ↔ {{|}|{{|}|}}

-

1

2

↔ {-1|0} ↔ {{|0}|{|}} ↔ {{|{|}}|{|}}

-2 ↔ {|-1} ↔ {|{|0}} ↔ {|{|{|}}}

A surreal

{L|R}

is recursively defined as an ordered pair of sets of surreals; the first surreal is

0≡{∅|∅}

, with both left and right sets empty; this is written as

{|}

for compactness. Once any surreal has been defined, it can also be included in the left or right sets, iteratively producing further surreals. The slider selects the first few generations of surreal numbers, which can be viewed as a list, a plot on the number line, or in tree or graph form, showing which surreals were used in the construction of others.