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General Mathematics: Revision and Practice

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Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature or—in modern mathematics—entities that are stipulated to have certain properties, called axioms. A proof consists 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. [5] Until the 19th century, the development of mathematics in the West was mainly motivated by the needs of technology and science, and there was no clear distinction between pure and applied mathematics. [109] For example, the natural numbers and arithmetic were introduced for the need of counting, and geometry was motivated by surveying, architecture and astronomy. Later, Isaac Newton introduced infinitesimal calculus for explaining the movement of the planets with his law of gravitation. Moreover, most mathematicians were also scientists, and many scientists were also mathematicians. [110] However, a notable exception occurred with the tradition of pure mathematics in Ancient Greece. [111] Mathematics has a remarkable ability to cross cultural boundaries and time periods. As a human activity, the practice of mathematics has a social side, which includes education, careers, recognition, popularization, and so on. In education, mathematics is a core part of the curriculum and forms an important element of the STEM academic disciplines. Prominent careers for professional mathematicians include math teacher or professor, statistician, actuary, financial analyst, economist, accountant, commodity trader, or computer consultant. [167] The validity of a mathematical theorem relies only on the rigor of its proof, which could theoretically be done automatically by a computer program. This does not mean that there is no place for creativity in a mathematical work. On the contrary, many important mathematical results (theorems) are solutions of problems that other mathematicians failed to solve, and the invention of a way for solving them may be a fundamental way of the solving process. [178] [179] An extreme example is Apery's theorem: Roger Apery provided only the ideas for a proof, and the formal proof was given only several months later by three other mathematicians. [180]

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. 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.

What Is The Definition of Mathematics?

List of prime numbers—not just a table, but a list of various kinds of prime numbers (each with an accompanying table) The text uses language and terms consistently throughout. The sections follow a recognizable pattern from one section to another. 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°. There are a few word problems ending each chapter. They are generic and hard to date, but there is one question referencing a VHS tape. This is an easy update.

Applied mathematics [ edit ] Dynamical systems and differential equations [ edit ] Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator.

Branches of Mathematics

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. [38] [39] Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning 'the reunion of broken parts' [40] that he used for naming one of these methods in the title of his main treatise. Like other mathematical sciences such as physics and computer science, statistics is an autonomous discipline rather than a branch of applied mathematics. Like research physicists and computer scientists, research statisticians are mathematical scientists. Many statisticians have a degree in mathematics, and some statisticians are also mathematicians. There is no general consensus about a definition of mathematics or its epistemological status—that is, its place among other human activities. [156] [157] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. [156] There is not even consensus on whether mathematics is an art or a science. [157] Some just say, "mathematics is what mathematicians do". [156] This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study. [158] a b c d e f Kleiner, Israel (December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". Mathematics Magazine. Taylor & Francis, Ltd. 64 (5): 291–314. doi: 10.1080/0025570X.1991.11977625. JSTOR 2690647.

Ramana, B. V. (2007). Applied Mathematics. Tata McGraw–Hill Education. p.2.10. ISBN 978-0-07-066753-2 . Retrieved July 30, 2022. The mathematical study of change, motion, growth or decay is calculus. Before the Renaissance, mathematics was divided into two main areas: arithmetic, regarding the manipulation of numbers, and geometry, regarding the study of shapes. [18] Some types of pseudoscience, such as numerology and astrology, were not then clearly distinguished from mathematics. [19] Once written formally, a proof can be verified using a program called a proof assistant. [135] These programs are useful in situations where one is uncertain about a proof's correctness. [135] A bottle contains 45 milliliters of sugar and 67 milliliters of water. What fraction of sugar does the bottle contain? Round the result to two decimal places (then express as a percent)." The question is about fractions, and poorly worded, then instructions say to round to two decimal places.

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In the 20th century, the mathematician L. E. J. Brouwer even initiated a philosophical perspective known as intuitionism, which primarily identifies mathematics with certain creative processes in the mind. [59] Intuitionism is in turn one flavor of a stance known as constructivism, which only considers a mathematical object valid if it can be directly constructed, not merely guaranteed by logic indirectly. This leads committed constructivists to reject certain results, particularly arguments like existential proofs based on the law of excluded middle. [185] Lists of mathematics topics cover a variety of topics related to mathematics. Some of these lists link to hundreds of articles; some link only to a few. The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. The fundamental postulate of mathematical economics is that of the rational individual actor – Homo economicus ( lit. 'economic man'). [145] In this model, each individual seeks to maximize their self-interest, [145] and always makes optimal choices using perfect information. [146] [ bettersourceneeded] This atomistic view of economics allows it to relatively easily mathematize its thinking, because individual calculations are transposed into mathematical calculations. Such mathematical modeling allows one to probe economic mechanisms which would be very difficult to discover by a "literary" analysis. [ citation needed] For example, explanations of economic cycles are not trivial. Without mathematical modeling, it is hard to go beyond simple statistical observations or unproven speculation. [ citation needed]

The book is divisible into many subsets of the skills needed to master a certain mathematical skill. Tiwari, Sarju (1992). Mathematics in History, Culture, Philosophy, and Science. Mittal Publications. p.27. ISBN 978-81-7099-404-6 . Retrieved March 19, 2023.Main articles: Mathematical logic and Set theory The Venn diagram is a commonly used method to illustrate the relations between sets.

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