Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e.
arithmetic,
algebra,
geometry, and
analysis).
[citation needed] In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to
logic, to
set theory (
foundations), to the empirical mathematics of the various sciences (
applied mathematics), and more recently to the rigorous study of
uncertainty.
Quantity
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, from which come such popular results as
Fermat's Last Theorem. Number theory also holds two problems widely considered to be unsolved: the
twin prime conjecture and
Goldbach's conjecture.
As the number system is further developed, the integers are recognized 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 generalized to
complex numbers. These are the first steps of a hierarchy of numbers that goes on to include
quarternions and
octonions. Consideration of the natural numbers also leads to the
transfinite numbers, which formalize the concept of "
infinity". Another area of study is size, which leads to the
cardinal numbers and then to another conception of infinity: the
aleph numbers, which allow meaningful comparison of the size of infinitely large sets.
Structure
Many mathematical objects, such as
sets of numbers and
functions, exhibit internal structure as a consequence of
operations or
relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure; for instance
number theory studies properties of the set of
integers that can be expressed in terms of
arithmetic operations. Moreover, it frequently happens that different such structured sets (or
structures) exhibit similar properties, which makes it possible, by a further step of
abstraction, to state
axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study
groups,
rings,
fields and other abstract systems; together such studies (for structures defined by algebraic operations) constitute the domain of
abstract algebra. By its great generality, abstract algebra can often be applied to seemingly unrelated problems; for instance a number of ancient problems concerning
compass and straightedge constructions were finally solved using
Galois theory, which involves field theory and group theory. Another example of an algebraic theory is
linear algebra, which is the general study of
vector spaces, whose elements called
vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of
geometry and
algebra have very strong interactions in modern mathematics.
Combinatorics studies ways of enumerating the number of objects that fit a given structure.
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