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Algebra

Analysis

Applied Mathematics

Computational Mathematics

Geometry and Topology

Logic and Foundations of Mathematics

Number Theory

Probability Theory

Statistics

The mathematics is the science of logical reasoning and abstract. It studies quantity, measurements, areas, structures and variations. A mathematical work is to look for patterns, make conjectures and through rigorous deductions from axioms and definitions, establish new results.

The math has been built over many years. Results and millennial theories remain valid and useful and yet it continues to grow permanently.

The math has been built over many years. Results and millennial theories remain valid and useful and yet it continues to grow permanently.

Areas of study

Mathematics is used as a crucial device in numerous fields of understanding such as engineering, medicine, physics, chemistry, biology, and social sciences. Applied mathematics, the branch of that deals with application of mathematical knowledge in other areas of knowledge, sometimes leads to the development of a new branch, as happened with statistics or game theory. The study of pure mathematics, without concern for their applicability often proved useful years or centuries ahead. It helped in the development of conical or number theory made by the Greeks and astronomy discoveries made by Kepler in the seventeenth century, or to the development of computer security today.

The teaching may also designate learning basic or elementary basics that learning and research initiation (upper education). At different times and places, the choice of subjects taught and teaching methods change.In some countries, the choice of programs schools in public education is done by official institutions.

The teaching may also designate learning basic or elementary basics that learning and research initiation (upper education). At different times and places, the choice of subjects taught and teaching methods change.In some countries, the choice of programs schools in public education is done by official institutions.

History

Archeological records show that mathematics has always been part of human activity. It evolved from counting, measurement, calculation and systematic study of geometric shapes and motions of physical objects. More abstract reasoning involving logical argument arose with the Greek mathematicians in about 300 BC, notably with the work The Elements of Euclid. The need for greater rigor was perceived and established by the nineteenth century.

The mathematics developed mainly in Mesopotamia, Egypt, Greece, India, and the Middle East. From the Renaissance, development intensified in Europe, when new scientific discoveries led to a rapid growth that lasts until today.

Long search underwent for a consensus regarding the definition of what is mathematics. However, in the former decades of the twentieth century, it got a definition that has wide approval among mathematicians. It is the science of regularities (patterns ). According to this definition, the work of the mathematician is to examine abstract patterns, real and imagined, visual or mental. The mathematicians seek regularities in numbers, space, science and imagination and formulate theories which try to explain the observed relationships. Another definition would be that it is the study of abstract structures defined axiomatically, using formal logic as a common structure. The specific structures usually have their origin in the natural sciences, most commonly in physics, but mathematicians also define and investigate structures, for example, to realize that the structures provide a unifying generalization for several subfields or a useful tool for common calculations.

The first known object that certifies the ability to calculate is the bone. Ishango, a fibula of baboon risks indicate a score dating from 20,000 years ago. Many numbering systems existed. The Rhind Papyrus is a document that has withstood the time andhows the numerals written in ancient Egypt.

At the time of the Renaissance, a part of the Arabic texts were studied and translated into Latin. The mathematical research focused then in Europe. The algebraic calculation developed rapidly with the work of the French François Viète and Renãcopy; Descartes. At that time logarithms tables were also created, which were extremely important to the scientific advancement of the sixteenth century.

They were replaced only after the creation of computers. The perception that the real numbers are not sufficient for solving certain equations also dates from the sixteenth century. At this time began the development of so-called complex numbers, with only a definition and four operations. A deeper understanding of complex numbers was only achieved in the eighteenth century with Euler. In the early seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz discovered the notion of infinitesimal calculus and introduced the notion of flexor (word later abandoned). Throughout the eighteenth and nineteenth centuries, it has developed strongly with the introduction of new abstract structures, notably the groups (thanks to the work of EvaristeGalois ) on the solvability of polynomial equations, and rings defined in the work of Richard Dedekind .

The mathematics developed mainly in Mesopotamia, Egypt, Greece, India, and the Middle East. From the Renaissance, development intensified in Europe, when new scientific discoveries led to a rapid growth that lasts until today.

Long search underwent for a consensus regarding the definition of what is mathematics. However, in the former decades of the twentieth century, it got a definition that has wide approval among mathematicians. It is the science of regularities (patterns ). According to this definition, the work of the mathematician is to examine abstract patterns, real and imagined, visual or mental. The mathematicians seek regularities in numbers, space, science and imagination and formulate theories which try to explain the observed relationships. Another definition would be that it is the study of abstract structures defined axiomatically, using formal logic as a common structure. The specific structures usually have their origin in the natural sciences, most commonly in physics, but mathematicians also define and investigate structures, for example, to realize that the structures provide a unifying generalization for several subfields or a useful tool for common calculations.

The first known object that certifies the ability to calculate is the bone. Ishango, a fibula of baboon risks indicate a score dating from 20,000 years ago. Many numbering systems existed. The Rhind Papyrus is a document that has withstood the time andhows the numerals written in ancient Egypt.

At the time of the Renaissance, a part of the Arabic texts were studied and translated into Latin. The mathematical research focused then in Europe. The algebraic calculation developed rapidly with the work of the French François Viète and Renãcopy; Descartes. At that time logarithms tables were also created, which were extremely important to the scientific advancement of the sixteenth century.

They were replaced only after the creation of computers. The perception that the real numbers are not sufficient for solving certain equations also dates from the sixteenth century. At this time began the development of so-called complex numbers, with only a definition and four operations. A deeper understanding of complex numbers was only achieved in the eighteenth century with Euler. In the early seventeenth century, Isaac Newton and Gottfried Wilhelm Leibniz discovered the notion of infinitesimal calculus and introduced the notion of flexor (word later abandoned). Throughout the eighteenth and nineteenth centuries, it has developed strongly with the introduction of new abstract structures, notably the groups (thanks to the work of EvaristeGalois ) on the solvability of polynomial equations, and rings defined in the work of Richard Dedekind .

Compartments

- Probability theory and statistics:Probability theory attempts to describe and study mathematical models of random phenomena from a theoretical perspective. Statistics is the area that wants to create methods, principles, criteria, etc. to discuss data from random phenomena or data from experiments and observations from reality. Knowledge and probability theory could be used to formulate such methods, principles and criteria, something that shows that probability theory and statistical theory are closely linked. Models are used in many academic sciences, these are usually deterministic. This means that given some initial known values, we can predict a future event. Isaac Newton showed that his laws of motion are deterministic because they can predict the time it takes for the earth to make one revolution around the sun. In probability theory, random phenomena are studied, where future outcome cannot be predicted exactly. For example, coin throw is a random phenomenon: even though we have complete knowledge about the coins design, such as that it is symmetric, we cannot predict in which case it will be heads or tails. Instead of a deterministic model requires a probabilistic theory. The relevant difference between probability theory and statistical theory is that in probability theory, (a) we have a given random model and try to predict the outcome of a random trial, while in statistical theory it is the opposite, and we have (b) an outcome of a random trial and to describe the underlying random model. A biochemist can use statistical methods to develop medication that relieves headaches. He offered medicine to different people, the variation between people mean that they experience very different change in their headaches. A statistical analysis of data from such an experiment can answer how much relief can be expected on average.
- Arithmetic:The science of numbers and operations on quantities of numbers is called arithmetic. Arithmetic operations include addition, subtraction, multiplication and division (the four operations) and congruence relation, factorization and potencies. Arithmetic was part of quadrivium at medieval University.
- Geometry:Geometry is the science of spatial structures. During the 1600s, Renãcopy; Descartes further defined geometry with the help of algebraic formulations, a topic that came to be known as analytic geometry. Some implications of Descartes discoveries are that different conic customers represented in the form of short equations, and that the plane geometric figures were depicted in a Cartesian coordinate system. The science that studies the angles and their relationships between each other is called trigonometry. The relationships between geometric and trigonometric rates are strong. In modern times, the topology has become an important area, where spatial structures are studied.
- Algebra:Algebra is a science of quantitative balance. Elementary algebra, linear algebra and abstract algebra are examples of areas that deal with algebraic structures.
- Mathematical Analysis:Calculus is about change. A large part of the analysis consists of theories of values, from which the theory of differential, a measure of change, and integrals, and the limit of a sum are formed. Sometimes people talk about vector analysis, which uses mathematical analysis and linear algebra to solve problems.
- Discrete Mathematics:It is about integers. It is an important branch of combinatory theory which discusses the combinations and permutations of a selection.

The pioneer names

The pioneer names are RenÃ© Descartes, Leonardo Fibonacci, Carl Friedrich Gauss, Leonhard Euler, Isaac Newton, David Hilbert, Emmy Noether and many more.

Research and discoveries

The math still continues to develop intensively throughout the world today.

Considerate and relating modification in computable extents is the common theme in the natural sciences and calculus was developed as the most useful tool for this task. The description of the variation of a quantity value is obtained by means of the concept of function. The real numbers are used to represent continuous quantities and their properties. The functional analysis deals with functions defined on spaces of infinite dimensions which typically forms the basis for the formulation of quantum mechanics, among many other things.

To clarify and investigate the foundations of mathematics, the fields were developed in set theory, mathematical logic and model theory.

When computers were designed, various theoretical issues led to the development of theories of computability, computational complexity, information and algorithmic information, which are investigated in computer science.

An important theory developed by winner of Nobel Prize, John Nash, is game theory, which currently has applications in various fields such as the study of trade disputes.

Computers also contributed to the development of chaos theory, which deals with the fact that many nonlinear dynamical systems have a behavior that, in practice, is unpredictable. Chaos theory has close relations with fractal geometry, as the set of Mandelbrot and Mary, discovered by Lorenz, known by the attractor that bears his name.

An important field in applied mathematics is statistics, which allows the description, analysis and prediction of random phenomena and is used in all sciences. The numerical analysis investigates the methods to solve numerically and efficiently several problems using computers and taking into account the rounding errors. Discrete mathematics is the common name for these fields useful in computer science.

Mathematical research is not limited to the demonstration of theorems. One of the most successful methods of mathematical research is the development of reconciliation fields a priori remote highlighting similar phenomena (e.g., Euclidean geometry and linear differential equations). Similar phenomena can occur in trying to adapt the results of a field of mathematics to another, to reformulate elements demonstration equivalent terms, to attempt to axiomatization of an object (for example, this could be the notion of vector space) that would combine the two fields ... In the latter case, then the new object would become an object of study in itself. In some cases, identification of objects apriori different becomes necessary: the language of categories used to make this stuff.

Another research method is the confrontation examples and special cases. This confrontation may refute properties we thought or hoped to be true (conjecture). On the contrary, it can help to check properties or result in the offense. For example, in Riemannian geometry, the study of surfaces (i.e. objects in dimension 2) and their geodesic eventually led Anosov formalize the Anosovdiffeomorphism of a transformation with interesting dynamic properties.

Mathematics bears special relationship with all sciences, in the broadest sense. Data analysis (graphical interpretation, statistics ) uses a variety of mathematical skills. But advanced mathematical tools involved in the modeling.

All so-called hard, tend to an understanding of the real world. This understanding requires the establishment of a model taking into account a number of parameters considered causes of a phenomenon. This model is a mathematical object; the study provides a better understanding of the phenomenon, possibly a qualitative or quantitative predictions about its future.

The model uses skills that are essentially within the analysis and probability, but the algebraic and geometric methods are useful. Its relationship with the humanities is mainly by statistics and probabilities, but also by differential equations, stochastic or not, economics and finance (sociology, psychology , economics ,finance , management , language).

In particular, financial mathematics is a branch of applied mathematics to the understanding of the evolution of financial markets and the risk estimation. This branch develops at the border of probabilities and the analysis and use of statistics.

Considerate and relating modification in computable extents is the common theme in the natural sciences and calculus was developed as the most useful tool for this task. The description of the variation of a quantity value is obtained by means of the concept of function. The real numbers are used to represent continuous quantities and their properties. The functional analysis deals with functions defined on spaces of infinite dimensions which typically forms the basis for the formulation of quantum mechanics, among many other things.

To clarify and investigate the foundations of mathematics, the fields were developed in set theory, mathematical logic and model theory.

When computers were designed, various theoretical issues led to the development of theories of computability, computational complexity, information and algorithmic information, which are investigated in computer science.

An important theory developed by winner of Nobel Prize, John Nash, is game theory, which currently has applications in various fields such as the study of trade disputes.

Computers also contributed to the development of chaos theory, which deals with the fact that many nonlinear dynamical systems have a behavior that, in practice, is unpredictable. Chaos theory has close relations with fractal geometry, as the set of Mandelbrot and Mary, discovered by Lorenz, known by the attractor that bears his name.

An important field in applied mathematics is statistics, which allows the description, analysis and prediction of random phenomena and is used in all sciences. The numerical analysis investigates the methods to solve numerically and efficiently several problems using computers and taking into account the rounding errors. Discrete mathematics is the common name for these fields useful in computer science.

Mathematical research is not limited to the demonstration of theorems. One of the most successful methods of mathematical research is the development of reconciliation fields a priori remote highlighting similar phenomena (e.g., Euclidean geometry and linear differential equations). Similar phenomena can occur in trying to adapt the results of a field of mathematics to another, to reformulate elements demonstration equivalent terms, to attempt to axiomatization of an object (for example, this could be the notion of vector space) that would combine the two fields ... In the latter case, then the new object would become an object of study in itself. In some cases, identification of objects apriori different becomes necessary: the language of categories used to make this stuff.

Another research method is the confrontation examples and special cases. This confrontation may refute properties we thought or hoped to be true (conjecture). On the contrary, it can help to check properties or result in the offense. For example, in Riemannian geometry, the study of surfaces (i.e. objects in dimension 2) and their geodesic eventually led Anosov formalize the Anosovdiffeomorphism of a transformation with interesting dynamic properties.

Mathematics bears special relationship with all sciences, in the broadest sense. Data analysis (graphical interpretation, statistics ) uses a variety of mathematical skills. But advanced mathematical tools involved in the modeling.

All so-called hard, tend to an understanding of the real world. This understanding requires the establishment of a model taking into account a number of parameters considered causes of a phenomenon. This model is a mathematical object; the study provides a better understanding of the phenomenon, possibly a qualitative or quantitative predictions about its future.

The model uses skills that are essentially within the analysis and probability, but the algebraic and geometric methods are useful. Its relationship with the humanities is mainly by statistics and probabilities, but also by differential equations, stochastic or not, economics and finance (sociology, psychology , economics ,finance , management , language).

In particular, financial mathematics is a branch of applied mathematics to the understanding of the evolution of financial markets and the risk estimation. This branch develops at the border of probabilities and the analysis and use of statistics.

Famous schools of mathematics

- Faculty of Mathematics, University of Cambridge
- Berlin Mathematical School
- Moscow State University of Economics, Statistics, and Informatics
- University of Sydney School of Mathematics and Statistics
- Department of Statistics of Rajshahi University
- School of Mathematics, University of Manchester
- University of Copenhagen Institute for Mathematical Sciences
- University of Houston College of Natural Sciences and Mathematics

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