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thematics before the renaissance mathematics was divided into two main areas arithmetic regarding the study and manipulation of numbers and geometry regarding the study of shapes 11 some types of pseudoscience such as numerology and astrology were not then clearly distinguished from mathematics 12 beginning with the renaissance two more areas became predominant new mathematical notation led to modern algebra which roughly speaking begins with the study and manipulation of algebraic expressions calculus consisting of the two subfields differential calculus and integral calculus originated with geometry but evolved into 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 13 endured until the end of the 19th century other areas that were previously studied by mathematicians such as celestial mechanics and solid mechanics are now considered as belonging to physics 14 the subject of combinatorics has been studied for much of recorded history yet did not become a separate branch of mathematics until the 17th century 15 at the end of the 19th century the foundational crisis in mathematics and the systematic use of the axiomatic method led to an explosion of new areas of mathematics 16 8 the 2020 mathematics subject classification contains no less than sixty three first level areas 10 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 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 9 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 evolved from 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 17 the study of numbers arguably dates back to ancient babylon and probably china but developed into a distinct discipline in ancient greece two prominent early number theorists were euclid and diophantus of alexandria 18 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 19 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 20 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 21 number theory includes several subareas including analytic number theory algebraic number theory geometry of numbers method oriented diophantine analysis and transcendence theory problem oriented 9 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 22 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 23 24 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 22 euclidean geometry was developed without change of methods or scope until the 17th century when rené descartes introduced what is now called cartesian coordinates this constituted a major change of paradigm instead of defining real numbers as lengths of line segments see number line it allowed the representation of points using their coordinates which are numbers algebra and later calculus can thus be used to solve geometrical problems geometry was split into two new subfields synthetic geometry which uses purely geometrical methods and analytic geometry which uses coordinates systemically 25 analytic geometry allows the study of curves unrelated to circles and lines such curves can be defined as the graph of functions the study of which led to differential geometry they can also be defined as implicit equations often polynomial equations which spawned algebraic geometry analytic geometry also makes it possible to consider euclidean spaces of higher than three dimensions 22 in the 19th century mathematicians discovered non euclidean geometries which do not follow the parallel postulate by questioning that postulate s truth this discovery has been viewed as joining russell s paradox in revealing the foundational crisis of mathematics this aspect of the crisis was solved by systematizing the axiomatic method and adopting that the truth of the chosen axioms is not a mathematical problem 26 8 in turn the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that do not change under specific transformations of the space 27 today s subareas of geometry include 9 projective geometry introduced in the 16th century by girard desargues extends euclidean geometry by adding points at infinity at which parallel lines intersect this simplifies many aspects of classical geometry by unifying the treatments for intersecting and parallel lines affine geometry the study of properties relative to parallelism and independent from the concept of length differential geometry the study of curves surfaces and their generalizations which are defined using differentiable functions manifold theory the study of shapes that are not necessarily embedded in a larger space riemannian geometry the study of distance properties in curved spaces algebraic geometry the study of curves surfaces and their generalizations which are defined using polynomials topology the study of properties that are kept under continuous deformations algebraic topology the use in topology of algebraic methods mainly homological algebra discrete geometry the study of finite configurations in geometry convex geometry the study of convex sets which takes its importance from its applications in optimization complex geometry the geometry obtained by replacing real numbers with complex numbers algebra main article algebra the quadratic formula which concisely expresses the solutions of all quadratic equations the rubik s cube group is a concrete application of group theory 28 algebra is the art of manipulating equations and formulas diophantus 3rd century and al khwarizmi 9th century were the two main precursors of algebra 29 30 diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution 31 al khwarizmi introduced systematic methods for transforming equations such as moving a term from one side of an equation into the other side 32 the term algebra is derived from the arabic word al jabr meaning the reunion of broken parts that he used for naming one of these methods in the title of his main treatise 33 34 algebra became an area in its own right only with françois viète 1540 1603 who introduced the use of variables for representing unknown or unspecified numbers 35 variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas 36 until the 19th century algebra consisted mainly of the study of linear equations presently linear algebra and polynomial equations in a single unknown which were called algebraic equations a term still in use although it may be ambiguous during the 19th century mathematicians began to use variables to represent things other than numbers such as matrices modular integers and geometric transformations on which generalizations of arithmetic operations are often valid 37 the concept of algebraic structure addresses this consisting of a set whose elements are unspecified of operations acting on the elements of the set and rules that these operations must follow the scope of algebra thus grew to include the study of algebraic structures this object of algebra was called modern algebra or abstract algebra as established by the influence and works of emmy noether 38 and popularized by van der waerden s book moderne algebra some types of algebraic structures have useful and often fundamental properties in many areas of mathematics their study became autonomous parts of algebra and include 9 group theory field theory vector spaces whose study is essentially the same as linear algebra ring theory commutative algebra which is the study of commutative rings includes the study of polynomials and is a foundational part of algebraic geometry homological algebra lie algebra and lie group theory boolean algebra which is widely used for the study of the logical structure of computers the study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory 39 the latter applies to every mathematical structure not only algebraic ones at its origin it was introduced together with homological algebra for allowing the algebraic study of non algebraic objects such as topological spaces this particular area of application is called algebraic topology 40 calculus and analysis main articles calculus and mathematical analysis a cauchy sequence consists of elements such that all subsequent terms of a term become arbitrarily close to each other as the sequence progresses from left to right calculus formerly called infinitesimal calculus was introduced independently and simultaneously by 17th century mathematicians newton and leibniz 41 it is fundamentally the study of the relationship between variables that depend continuously on each other calculus was expanded in the 18th century by euler with the introduction of the concept of a function and many other results 42 presently calculus refers mainly to the elementary part of this theory and analysis is commonly used for advanced parts 43 analysis is further subdivided into real analysis where variables represent real numbers and complex analysis where variables represent complex numbers analysis includes many subareas shared by other areas of mathematics which include 9 multivariable calculus functional analysis where variables represent varying functions integration measure theory and potential theory all strongly related with probability theory on a continuum ordinary differential equations partial differential equations numerical analysis mainly devoted to the computation of solutions of ordinary and partial differential equations that arise in many applications discrete mathematics main article discrete mathematics a diagram representing a two state markov chain the states are represented by a and e the numbers are the probability of flipping the state discrete mathematics broadly speaking is the study of individual countable mathematical objects an example is the set of all integers 44 because the objects of study here are discrete the methods of calculus and mathematical analysis do not directly apply c algorithms especially their implementation and computational complexity play a major role in discrete mathematics 45 the four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century 46 the p versus np problem which remains open to this day is also important for discrete mathematics since its solution would potentially impact a large number of computationally difficult problems 47 discrete mathematics includes 9 combinatorics the art of enumerating mathematical objects that satisfy some given constraints originally these objects were elements or subsets of a given set this has been extended to various objects which establishes a strong link between combinatorics and other parts of discrete mathematics for example discrete geometry includes counting configurations of geometric shapes graph theory and hypergraphs coding theory including error correcting codes and a part of cryptography matroid theory discrete geometry discrete probability distributions game theory although continuous games are also studied most common games such as chess and poker are discrete discrete optimization including combinatorial optimization integer programming constraint programming mathematical logic and set theory main articles mathematical logic and set theory the venn diagram is a commonly used method to illustrate the relations between sets the two subjects of mathematical logic and set theory have belonged to mathematics since the end of the 19th century 48 49 before this period sets were not considered to be mathematical objects and logic although used for mathematical proofs belonged to philosophy and was not specifically studied by mathematicians 50 before cantor s study of infinite sets mathematicians were reluctant to consider actually infinite collections and considered infinity to be the result of endless enumeration cantor s work offended many mathematicians not only by considering actually infinite sets 51 but by showing that this implies different sizes of infinity per cantor s diagonal argument this led to the controversy over cantor s set theory 52 in the same per...
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