General Mathematics: Revision and Practice

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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. 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]

Ada Lovelace, in the 1840s, is known for having written the first computer program ever in collaboration with Charles Babbage Main articles: Mathematical logic and Set theory The Venn diagram is a commonly used method to illustrate the relations between sets. Bell, E. T. (2012). The Development of Mathematics. Dover Books on Mathematics (reprint, reviseded.). Courier Corporation. p.3. ISBN 978-0-486-15228-8 . Retrieved November 11, 2022.Archaeological evidence shows that instruction in mathematics occurred as early as the second millennium BCE in ancient Babylonia. [168] Comparable evidence has been unearthed for scribal mathematics training in the ancient Near East and then for the Greco-Roman world starting around 300 BCE. [169] The oldest known mathematics textbook is the Rhind papyrus, dated from c. 1650 BCE in Egypt. [170] Due to a scarcity of books, mathematical teachings in ancient India were communicated using memorized oral tradition since the Vedic period ( c. 1500– c. 500 BCE). [171] In Imperial China during the Tang dynasty (618–907 CE), a mathematics curriculum was adopted for the civil service exam to join the state bureaucracy. [172]

There are only 2 numbers that are twice the sum of their individual digits; one of them is zero (0). What is the other one? The history of mathematics is an ever-growing series of abstractions. Evolutionarily speaking, the first abstraction to ever be discovered, one shared by many animals, [71] was probably that of numbers: the realization that, for example, a collection of two apples and a collection of two oranges (say) have something in common, namely that there are two of them. As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also known how to count abstract quantities, like time—days, seasons, or years. [72] [73] The Babylonian mathematical tablet Plimpton 322, dated to 1800BC What do you get if you divide the number of hours in a week by the sum of the sides of a triangle, and the number of natural satellites of the earth? The fields of mathematics and computing intersect both in computer science, the study of algorithms and data structures, and in scientific computing, the study of algorithmic methods for solving problems in mathematics, science, and engineering.Which of the following number is an odd integer contains the digit 5, is divisible by 3 and lies between the square of 12 and 13?

The unreasonable effectiveness of mathematics is a phenomenon that was named and first made explicit by physicist Eugene Wigner. [7] It is the fact that many mathematical theories (even the "purest") have applications outside their initial object. These applications may be completely outside their initial area of mathematics, and may concern physical phenomena that were completely unknown when the mathematical theory was introduced. [123] Examples of unexpected applications of mathematical theories can be found in many areas of mathematics. Information theory is a branch of applied mathematics and Social science involving the quantification of information. Historically, information theory was developed to find fundamental limits on compressing and reliably communicating data.

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Mathematics has developed a rich terminology covering a broad range of fields that study the properties of various abstract, idealized objects and how they interact. It is based on rigorous definitions that provide a standard foundation for communication. An axiom or postulate is a mathematical statement that is taken to be true without need of proof. If a mathematical statement has yet to be proven (or disproven), it is termed a conjecture. Through a series of rigorous arguments employing deductive reasoning, a statement that is proven to be true becomes a theorem. A specialized theorem that is mainly used to prove another theorem is called a lemma. A proven instance that forms part of a more general finding is termed a corollary. [96] Also, each chapter began with the objectives, followed by the concepts (overview, any necessary terminology, explanations, examples, practice problems), a chapter summary, additional exercises, and chapter exam. Mathematics is used in most sciences for modeling phenomena, which then allows predictions to be made from experimental laws. [99] The independence of mathematical truth from any experimentation implies that the accuracy of such predictions depends only on the adequacy of the model. [100] Inaccurate predictions, rather than being caused by invalid mathematical concepts, imply the need to change the mathematical model used. [101] For example, the perihelion precession of Mercury could only be explained after the emergence of Einstein's general relativity, which replaced Newton's law of gravitation as a better mathematical model. [102]

Logic [ edit ] Venn diagrams are illustrations of set theoretical, mathematical or logical relationships. Applied mathematics [ edit ] Dynamical systems and differential equations [ edit ] Phase portrait of a continuous-time dynamical system, the Van der Pol oscillator. Which of these could be either a circumference divided by π or a song written and recorded by David Bouie?

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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. In the 19th century, the internal development of geometry (pure mathematics) led to definition and study of non-Euclidean geometries, spaces of dimension higher than three and manifolds. At this time, these concepts seemed totally disconnected from the physical reality, but at the beginning of the 20th century, Albert Einstein developed the theory of relativity that uses fundamentally these concepts. In particular, spacetime of special relativity is a non-Euclidean space of dimension four, and spacetime of general relativity is a (curved) manifold of dimension four. [126] [127] 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 new list of seven important problems, titled the " Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. [211] To date, only one of these problems, the Poincaré conjecture, has been solved. [212] See also 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. [31]