Numbers
Numbers [ edit ] Many number systems, such as the integers and the rationals , enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules , vector spaces and algebras also form groups. Integers [ edit ] The group of integers � under addition, denoted ( � , + ) , has been described above. The integers, with the operation of multiplication instead of addition, ( � , ⋅ ) do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, � = 2 is an integer, but the only solution to the equation � ⋅ � = 1 in this case is � = 1 2 , which is a rational number, but not an in...