Pure mathematics
Quantity
Main article: Arithmetic
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. The twin
prime conjecture
and Goldbach's conjecture are two unsolved problems in number
theory.
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 quaternionsand octonions. Consideration of the natural
numbers also leads to the transfinite numbers, which formalize the concept
of "infinity". According to the fundamental theorem of algebra all solutions of equations in one unknown
with complex coefficients are complex numbers, regardless of degree. Another
area of study is the size of sets, which is described with thecardinal
numbers. These include 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 ofoperations 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 andalgebra have very strong interactions in modern
mathematics. Combinatorics studies ways of enumerating the number
of objects that fit a given structure.
Space
The study of space originates
with geometry –
in particular, Euclidean geometry, which combines space and
numbers, and encompasses the well-known Pythagorean theorem. Trigonometry is the branch of mathematics that deals
with relationships between the sides and the angles of triangles and with the
trigonometric functions. The modern study of space generalizes these ideas to
include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology.
Quantity and space both play a role in analytic
geometry, differential geometry, and algebraic geometry. Convex and discrete
geometry were
developed to solve problems in number theory and functional analysis but now
are pursued with an eye on applications in optimization and computer science. Within differential geometry
are the concepts of fiber
bundles and
calculus on manifolds, in
particular, vector and tensor
calculus. Within algebraic geometry is the description of geometric
objects as solution sets of polynomial equations, combining the concepts of
quantity and space, and also the study of topological groups, which combine structure and
space. Lie
groups are used
to study space, structure, and change. Topology in all its many ramifications may have been
the greatest growth area in 20th-century mathematics; it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of
modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse
theory. Topology also includes the now solved Poincaré conjecture,
and the still unsolved areas of the Hodge
conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proved only with
the help of computers.
Change
Understanding and describing
change is a common theme in the natural
sciences, and calculus was developed as a tool to investigate it. Functions arise
here, as a central concept describing a changing quantity. The rigorous study
of real
numbersand functions of a real variable is known as real
analysis, with complex
analysis the
equivalent field for the complex
numbers. Functional analysis focuses
attention on (typically infinite-dimensional) spaces of functions. One of many applications of
functional analysis is quantum
mechanics. Many problems lead naturally to relationships between a
quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can
be described bydynamical systems; chaos
theory makes
precise the ways in which many of these systems exhibit unpredictable yet stilldeterministic behavior.
