User talk:Jason Potkanski/Locke Math
The study of quantity starts with numbers, first the familiar natural numbers and integers ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, whence such popular results as Fermat's last theorem. Two famous unsolved problems in number theory are the twin prime conjecture and Goldbach's conjecture.
As the number system is further developed, the integers are recognised as a subset of the rational numbers ("fractions"). These, in turn, are contained within the real numbers, which are used to represent continuous quantities. Real numbers are generalised to complex numbers. These are the first steps of a hierarchy of number systems that include the quarternions and octonions. Consideration of numbers larger than all finite natural numbers leads to the concept of transfinite numbers. In this formalism, infinite cardinal numbers, the aleph numbers, allow meaningful comparison of the size of infinitely large sets.