Editing Ontology:Q5,38 (section)

== Background ==

{{TTS|tts=star|<nowiki>*</nowiki>}} (or {{TTS|tts=star one|<nowiki>*</nowiki>1}}) is a number-like object used to describe games in combinatorial game theory. It represents a game state where there is at least one space which can be taken by either player and count toward either player winning or losing. A positive number counts a space which will be taken by one player, while a negative number counts a space which will be taken by the other player. A rational number is a combination of spaces that count toward one player averaged against spaces that count toward the other. However, a star number is an un-taken space which genuinely does not count toward either player until a particular player is selected to move first.

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=== {{TTS|tts=star|<nowiki>*</nowiki>}} as matrix ===

Thinking of {{TTS|tts=star|<nowiki>*</nowiki>}} as a number similar to an integer or decimal may be less useful than thinking of it as a composite object containing multiple elements that assemble into a [[E:algebra (data structure)|mathematical data structure]]. Matrices in particular are somewhat similar to game states: any particular vector laid out vertically can be concatenated onto a matrix as a new column through an operation called augmentation. If {{TTS|tts=star|<nowiki>*</nowiki>}} gains its contradictory properties from the internal contents of a vector, then it would make sense that augmenting matrices would combine two sets of contradictory elements into one set of contradictory elements that could resolve into zero. Matrices have interesting capabilities, such as multiplying against themselves to produce -1 or 1 — this has allowed complex numbers to be turned into matrices for some applications. The same thing could apply to {{TTS|tts=star|<nowiki>*</nowiki>}}.