Surreal Numbers

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The Surreal Numbers, or just Surreals, are an Esoteric (or should I say surreal? No. I shouldn't) numbering system (like the complexes or the naturals) that are pretty fucking weird (though they were invented to be).

In surreal numbers, the number { | } is called zero. { 0 | } is the number after 0 (some people like to call it 1). Likewise { 1 | } is the number after 1 (2), and so on. { | 0 } is -1, and so on in that direction as well.

{ 0, 1 | } is the number after 0 and 1, which is equivalent to { 1 | }. { 0 | 1 } is the number greater than 0 but less than 1, which is 1/2, and { 0 | 1, 1/2 } is 1/4.

We are allowed to have as many numbers on each side as we want, even an infinite number. { 0 | 1, 1/2, 1/4, … } is the infinitesimal, and { 0, 1, 2, … | } is infinity.

A more technical description can be found at Wikipedia:surreal number.

Attempts have been made to incorporate surreals into an esolang, but so far none have fully succeeded.