Mathematics

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From Wikipedia, the free encyclopedia
"Math" and "Maths" redirect here. For other uses, see Mathematics (disambiguation)
and Math (disambiguation).
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Mathematics is a field of study that discovers and organizes methods, theories and
theorems that are developed and proved for the needs of empirical sciences and
mathematics itself. There are many areas of mathematics, which include number
theory (the study of numbers), algebra (the study of formulas and related
structures), geometry (the study of shapes and spaces that contain them), analysis
(the study of continuous changes), and set theory (presently used as a foundation
for all mathematics).

Mathematics involves the description and manipulation of abstract objects that

consist of either abstractions from nature or—in modern mathematics—purely abstract
entities that are stipulated to have certain properties, called axioms. Mathematics
uses pure reason to prove properties of objects, a proof consisting of a succession
of applications of deductive rules to already established results. These results
include previously proved theorems, axioms, and—in case of abstraction from nature—
some basic properties that are considered true starting points of the theory under
consideration.[1]

Mathematics is essential in the natural sciences, engineering, medicine, finance,

computer science, and the social sciences. Although mathematics is extensively used
for modeling phenomena, the fundamental truths of mathematics are independent of
any scientific experimentation. Some areas of mathematics, such as statistics and
game theory, are developed in close correlation with their applications and are
often grouped under applied mathematics. Other areas are developed independently
from any application (and are therefore called pure mathematics) but often later
find practical applications.[2][3]

Historically, the concept of a proof and its associated mathematical rigour first
appeared in Greek mathematics, most notably in Euclid's Elements.[4] Since its
beginning, mathematics was primarily divided into geometry and arithmetic (the
manipulation of natural numbers and fractions), until the 16th and 17th centuries,
when algebra[a] and infinitesimal calculus were introduced as new fields. Since
then, the interaction between mathematical innovations and scientific discoveries
has led to a correlated increase in the development of both.[5] At the end of the
19th century, the foundational crisis of mathematics led to the systematization of
the axiomatic method,[6] which heralded a dramatic increase in the number of
mathematical areas and their fields of application. The contemporary Mathematics
Subject Classification lists more than sixty first-level areas of mathematics.

Areas of mathematics
Before the Renaissance, mathematics was divided into two main areas: arithmetic,
regarding the manipulation of numbers, and geometry, regarding the study of shapes.
[7] Some types of pseudoscience, such as numerology and astrology, were not then
clearly distinguished from mathematics.[8]

During the Renaissance, two more areas appeared. Mathematical notation led to
algebra which, roughly speaking, consists of the study and the manipulation of
formulas. Calculus, consisting of the two subfields differential calculus and
integral calculus, is the study of continuous functions, which model the typically
nonlinear relationships between varying quantities, as represented by variables.
This division into four main areas—arithmetic, geometry, algebra, and calculus[9]—
endured until the end of the 19th century. Areas such as celestial mechanics and
solid mechanics were then studied by mathematicians, but now are considered as
belonging to physics.[10] The subject of combinatorics has been studied for much of
recorded history, yet did not become a separate branch of mathematics until the
seventeenth century.[11]

At the end of the 19th century, the foundational crisis in mathematics and the
resulting systematization of the axiomatic method led to an explosion of new areas
of mathematics.[12][6] The 2020 Mathematics Subject Classification contains no less
than sixty-three first-level areas.[13] Some of these areas correspond to the older
division, as is true regarding number theory (the modern name for higher
arithmetic) and geometry. Several other first-level areas have "geometry" in their
names or are otherwise commonly considered part of geometry. Algebra and calculus
do not appear as first-level areas but are respectively split into several first-
level areas. Other first-level areas emerged during the 20th century or had not
previously been considered as mathematics, such as mathematical logic and
foundations.[14]

Number theory
Main article: Number theory

This is the Ulam spiral, which illustrates the distribution of prime numbers. The
dark diagonal lines in the spiral hint at the hypothesized approximate independence
between being prime and being a value of a quadratic polynomial, a conjecture now
known as Hardy and Littlewood's Conjecture F.
Number theory began with the manipulation of numbers, that is, natural numbers
(
N
)
,
{\displaystyle (\mathbb {N} ),} and later expanded to integers
(
Z
)
{\displaystyle (\mathbb {Z} )} and rational numbers
(
Q
)
.
{\displaystyle (\mathbb {Q} ).} Number theory was once called arithmetic, but
nowadays this term is mostly used for numerical calculations.[15] Number theory
dates back to ancient Babylon and probably China. Two prominent early number
theorists were Euclid of ancient Greece and Diophantus of Alexandria.[16] The
modern study of number theory in its abstract form is largely attributed to Pierre
de Fermat and Leonhard Euler. The field came to full fruition with the
contributions of Adrien-Marie Legendre and Carl Friedrich Gauss.[17]

Many easily stated number problems have solutions that require sophisticated
methods, often from across mathematics. A prominent example is Fermat's Last
Theorem. This conjecture was stated in 1637 by Pierre de Fermat, but it was proved
only in 1994 by Andrew Wiles, who used tools including scheme theory from algebraic
geometry, category theory, and homological algebra.[18] Another example is
Goldbach's conjecture, which asserts that every even integer greater than 2 is the
sum of two prime numbers. Stated in 1742 by Christian Goldbach, it remains unproven
despite considerable effort.[19]

Number theory includes several subareas, including analytic number theory,

algebraic number theory, geometry of numbers (method oriented), diophantine
equations, and transcendence theory (problem oriented).[14]

Geometry
Main article: Geometry

On the surface of a sphere, Euclidean geometry only applies as a local

approximation. For larger scales the sum of the angles of a triangle is not equal
to 180°.
Geometry is one of the oldest branches of mathematics. It started with empirical
recipes concerning shapes, such as lines, angles and circles, which were developed
mainly for the needs of surveying and architecture, but has since blossomed out
into many other subfields.[20]

A fundamental innovation was the ancient Greeks' introduction of the concept of

proofs, which require that every assertion must be proved. For example, it is not
sufficient to verify by measurement that, say, two lengths are equal; their
equality must be proven via reasoning from previously accepted results (theorems)
and a few basic statements. The basic statements are not subject to proof because
they are self-evident (postulates), or are part of the definition of the subject of
study (axioms). This principle, foundational for all mathematics, was first
elaborated for geometry, and was systematized by Euclid around 300 BC in his book
Elements.[21][22]
The resulting Euclidean geometry is the study of shapes and their arrangements
constructed from lines, planes and circles in the Euclidean plane (plane geometry)
and the three-dimensional Euclidean space.[b][20]