◄ WORLD V · SONNY 5DART 576 · a helldive across the board

THE SURREAL NUMBERS numbers born from games

Conway found an entire number system inside game theory. Every number is a pair of sets {L | R} — everything less than it, everything greater. From { | } = 0 the whole line unfolds by “birthday”: day 1 gives 1 and −1, day 2 gives ½ and 2, and onward to every real, every infinite and infinitesimal number — the largest ordered field there is. The simplest number strictly between two others is always the one born first.

THE TECHNIQUE x = {L | R}; simplest number between

The demo verifies the ordering 0 < ½ < 1 and that ½ = {0 | 1} is born on day 2: live demo







  
HISTORY & CREDIT Conway · Knuth 1974

  
“Surreal numbers are just the reals rewritten.” — they properly contain the reals plus infinities (ω) and infinitesimals (1/ω), all in one ordered field. cited

  
the cut · a number is {L | R} — born between what is below and what is above.
the birthday · 0 on day 0; each day fills every new gap with its simplest number.
Conway · discovered analyzing Go endgames; Knuth named them in a 1974 novella.

  
A field wide enough for the infinite and the infinitesimal. construction






  
RECOMMEND FOR I-13 the ordering, on the compiler

  
On i-13, cross-multiplying dyadic values confirms −1 < 0 < ½ < 1, with ½ born on day 2:

  
$ i13 run gm_surreal-numbers.i13
RUN OK · 56 step(s)
zero_lt_half = 1  half_lt_one = 1  neg1_lt_zero = 1
ordered = 1
half_birthday = 2   -- {0|1}, simplest between 0 and 1

  
Recommend as a NULL — a construction/definition (B39). The surreal ordering and birthdays are fixed by the definition; every correct evaluation agrees. A number system, not a same-function DOF. NULL — numbers born from games.