Mathematics

Mathematics from arithmetic, number theory, formulas in addition to related structures algebra, shapes as well as the spaces in which they are contained geometry, as well as quantities and their remake calculus and analysis.

Most mathematical activity involves discovering and proving properties of abstract objects by pure reasoning. These objects are either abstractions from nature, such(a) as natural numbers or lines, or — in contemporary mathematics — entities that are stipulated withproperties, called axioms. the proof consists of a succession of a formal a formal message requesting something that is submitted to an dominance to be considered for a position or to be permits to pretend or work something. of some deductive rules to already asked results, including ago proved theorems, axioms and in case of abstraction from brand some basic properties that are considered as true starting points of the impression under consideration. The written of a proof is called a theorem.

Mathematics is widely used in Newton's law of gravitation combined with mathematical computation. The independence of mathematical truth from any experimentation implies that the accuracy of such(a) predictions depends only on the adequacy of the value example for describing the reality. Inaccurate predictions imply the need for refresh or changing mathematical models, not that mathematics is wrong in the models themselves. For example, the perihelion precession of Mercury cannot be explained by Newton's law of gravitation but is accurately explained by Einstein's general relativity. This experimental validation of Einstein's theory shows that Newton's law of gravitation is only an approximation, though accurate in everyday application.

Mathematics is necessary in many fields, including natural sciences, engineering, medicine, finance, computer science and social sciences. Some areas of mathematics, such(a) as statistics and game theory, are developed incorrelation with their applications and are often grouped under applied mathematics. Other mathematical areas are developed independently from any a formal request to be considered for a position or to be allowed to do or have something. and are therefore called pure mathematics, but practical applications are often discovered later. A fitting example is the problem of integer factorization, which goes back to Euclid, but which had no practical application previously its use in the RSA cryptosystem for the security of computer networks.

In the Elements. Mathematics developed at a relatively behind pace until the Renaissance, when algebra and infinitesimal calculus were added to arithmetic and geometry as leading areas of mathematics. Since then, the interaction between mathematical innovations and scientific discoveries has led to a rapid add in the developing of mathematics. At the end of the 19th century, the foundational crisis of mathematics led to the systematization of the axiomatic method. This, in turn, reported rise to a dramatic add in the number of mathematics areas and their fields of applications. An example of it is for Mathematics spoke Classification, which lists more than sixty first-level areas of mathematics.

Areas of mathematics

Before the Renaissance, mathematics was shared up into two leading areas: arithmetic — regarding the manipulation of numbers, and geometry — regarding the discussing of shapes. Some nature of pseudoscience, such(a) as numerology and astrology, were non then clearly distinguished from mathematics.

During the Renaissance, two more areas appeared. ] — endured until the end of the 19th century. Areas such as celestial mechanics and solid mechanics were often then considered as element of mathematics, but now are considered as belonging to physics. Some subjects developed during this period predate mathematics and are divided into such areas as probability theory and combinatorics, which only later became regarded as autonomous areas.

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. Today, the Mathematics Subject Classification contains no less than sixty-four first-level areas. Some of these areas correspond to the older division, as is true regarding number theory the sophisticated name for higher arithmetic and geometry. However, several other first-level areas do "geometry" in their tag or are otherwise commonly considered component of geometry. Algebra and calculus construct notas first-level areas but are respectively split into several first-level areas. Other first-level areas emerged during the 20th century for example category theory; homological algebra, and computer science or had not previously been considered as mathematics, such as Mathematical logic and foundations including model theory, computability theory, set theory, proof theory, and algebraic logic.

Number theory began with the manipulation of integers and arithmetic, but nowadays this term is mostly used for numerical calculations.

Many easily-stated number problems have solutions that require sophisticated methods from across mathematics. One prominent example is Fermat's last theorem. This conjecture was stated in 1637 by proved only in 1994 by Goldbach's conjecture, which asserts that every even integer greater than 2 is the a thing that is said of two prime numbers. Stated in 1742 by Christian Goldbach, it maintains unproven to this day despite considerable effort.

Number theory includes several subareas, including analytic number theory, algebraic number theory, geometry of numbers method oriented, diophantine equations, and transcendence theory problem oriented.

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.

A fundamental innovation was the number one array of the concept of Elements.

The resulting Euclidean geometry is the analyse of shapes and their arrangements constructed from lines, planes and circles in the Euclidean plane plane geometry and the three-dimensional Euclidean space.

Euclidean geometry was developed without modify of methods or scope until the 17th century, when René Descartes proposed what is now called Cartesian coordinates. This was a major modify of paradigm, since instead of determining real numbers as lengths of line segments see number line, it allows the explanation of points using their coordinates which are numbers. This gives one to usage algebra and later, calculus to solve geometrical problems. This split geometry into two new subfields: synthetic geometry, which uses purely geometrical methods, and analytic geometry, which uses coordinates systemically.

Analytic geometry allows the study of curves that are not related to circles and lines. Such curves can be defined as graph of functions whose study 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 spaces of higher than three dimensions.

In the 19th century, mathematicians discovered Russel's paradox as 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. In turn, the axiomatic method allows for the study of various geometries obtained either by changing the axioms or by considering properties that are invariant under specific transformations of the space.

Nowadays, the subareas of geometry include:

Pythagorean theorem

Conic Sections

Elliptic curve

Triangle on a paraboloid

Torus

Fractal

Algebra is the art of manipulating equations and formulas. Diophantus 3rd century and al-Khwarizmi 9th century were the two main precursors of algebra. The first one solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Theone introduced systematic methods for transforming equations such as moving a term from a side of an equation into the other side. The term algebra is derived from the Arabic word that he used for naming one of these methods in the tag of his main treatise.

Algebra became an area in its own adjustment only with François Viète 1540–1603, who introduced the use of letters variables for representing unknown or unspecified numbers. This allows mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.

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 that is still in use, although it may be ambiguous. During the 19th century, mathematicians began to use variables to live things other than numbers such as matrices, modular integers, and geometric transformations, on which generalizations of arithmetic operations are often valid. 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. Due to this change, the scope of algebra grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra. The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.

Some types of algebraic managers have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:

The study of types of algebraic structures as mathematical objects is the object of universal algebra and category theory. 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.

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. it is for fundamentally the study of the relationship of variable that depend on regarded and identified separately. other. Calculus was expanded in the 18th century by Euler, with the first ordering of the concept of a function, and many other results. Presently "calculus" refers mainly to the elementary part of this theory, and "analysis" is normally used for advanced parts.

Analysis is further subdivided into real analysis, where variables equal real numbers and complex analysis where variables represent complex numbers. Analysis includes many subareas, sharing some with other areas of mathematics; they include:

Discrete mathematics, broadly speaking is the study of finite mathematical objects. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms - especially their implementation and computational complexity - play a major role in discrete mathematics.

Discrete mathematics includes:

The ]

The two subjects of mathematical system of logic and set theory have both belonged to mathematics since the end of the 19th century. 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.

Before Cantor's diagonal argument and the existence of mathematical objects that cannot be computed, or even explicitly described for example, controversy over Cantor's set theory.

In the same period, various areas of mathematics concluded the former intuitive definitions of the basic mathematical objects were insufficient for ensuring mathematical rigour. Examples of such intuitive definitions are "a set is a collection of objects", "natural number is what is used for counting", "a unit is a shape with a zero length in every direction", "a curve is a trace left by a moving point", etc.

This became the foundational crisis of mathematics. It was eventually solved in mainstream mathematics by systematizing the axiomatic method inside a formalized set theory. Roughly speaking, regarded and identified separately. mathematical object is defined by the set of all similar objects and the properties that these objects must have. For example, in Peano arithmetic, the natural numbers are defined by "zero is a number", "each number as a unique successor", "each number but zero has a unique predecessor", and some rules of reasoning. The "nature" of the objects defined this way is a philosophical problem that mathematicians leave to philosophers, even if many mathematicians have opinions on this nature, and use their opinion—sometimes called "intuition"—to help their study and proofs.

This approach allows considering "logics" that is, sets of allowed deducing rules, Gödel's incompleteness theorems assert, roughly speaking that, in every theory that contains the natural numbers, there are theorems that are true that is provable in a larger theory, but not provable inside the theory.

This approach of the foundations of the mathematics was challenged during the first half of the 20th century by mathematicians led by Brouwer, who promoted intuitionistic logic, which explicitly lacks the law of excluded middle.

These problems and debates led to a wide expansion of mathematical logic, with subareas such as model theory modeling some logical theories inside other theories, proof theory, type theory, computability theory and computational complexity theory. Although these aspects of mathematical logic were introduced before the rise of computers, their use in compiler design, program certification, proof assistants and other aspects of computer science, contributed in become different to the expansion of these logical theories.

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In the past, practical applications have motivated the coding of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in ]

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, particularly ] Statisticians workings as part of a research project "create data that makes sense" with ]

Statistical theory studies decision problemssuch as minimizing the risk expected loss of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected waste or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population intend with a condition level of confidence. Because of its use of optimization, the mathematical theory of statistics overlaps with other decision sciences, such as operations research, control theory, and mathematical economics.