The Number Systems Poster

A poster of the 52 number systems in the book "The Number Systems" by Mark Kortink. The boxes show the number systems and each arrow runs from a number system to one constructed from it. The colours show the number systems family, and the badge the basic algebraid type. The concept of number has been extended and generalised many times. Successive constructions enlarge the natural numbers to the integers, rationals, reals, and complex numbers, introducing additive inverses, multiplicative inverses, completeness, and algebraic closure in turn. Other developments go in quite different directions: the p-adic numbers replace the usual metric on the rationals; quaternions and octonions sacrifice commutativity and then associativity; dual and split-complex numbers introduce nilpotents or zero divisors; hyperreal, Hahn, and surreal numbers introduce infinitesimal and infinite quantities; ordinal and cardinal numbers extend finite counting into the transfinite; and tropical and related systems alter the arithmetic operations themselves. This book develops a comparative atlas of these and other number systems. Instead of requiring every number system to satisfy one algebraic axiom system, it describes each system by a common structural profile: its construction, algebraic laws, order, topology, valuation or norm, completeness, distinguished elements, cardinality, embeddings, and relationships to other systems. The comparison separates properties essential to particular notions of number from those peculiar to familiar systems such as the real numbers. A final synthesis classifies the main mechanisms for constructing number systems and the properties gained or lost in passing between them. The comparison also aims to identify recurring structural patterns that may suggest further generalisations of the concept of number.