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Foundations Of Geometry
Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play. Axiomatic systems Based on ancient Greek methods, an ''axiomatic system'' is a formal description of a way to establish the ''mathematical truth'' that flows from a fixed set of assumptions. Although applicable to any area of mathematics, geometry is the branch of elementary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometries
This is a list of geometry topics. Types, methodologies, and terminologies of geometry. * Absolute geometry * Affine geometry * Algebraic geometry * Analytic geometry * Archimedes' use of infinitesimals * Birational geometry * Complex geometry * Combinatorial geometry * Computational geometry * Conformal geometry * Constructive solid geometry * Contact geometry * Convex geometry * Descriptive geometry * Differential geometry * Digital geometry * Discrete geometry * Distance geometry * Elliptic geometry * Enumerative geometry * Epipolar geometry * Finite geometry * Fractal geometry * Geometry of numbers * Hyperbolic geometry * Incidence geometry * Information geometry * Integral geometry * Inversive geometry * Inversive ring geometry * Klein geometry * Lie sphere geometry * Non-Euclidean geometry * Noncommutative algebraic geometry * Noncommutative geometry * Computational geometry#Numerical computational geometry, Numerical geometry * Ordered geometry * Parabolic geometry (diffe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexandria
Alexandria ( or ; ar, ٱلْإِسْكَنْدَرِيَّةُ ; grc-gre, Αλεξάνδρεια, Alexándria) is the second largest city in Egypt, and the largest city on the Mediterranean coast. Founded in by Alexander the Great, Alexandria grew rapidly and became a major centre of Hellenic civilisation, eventually replacing Memphis, in present-day Greater Cairo, as Egypt's capital. During the Hellenistic period, it was home to the Lighthouse of Alexandria, which ranked among the Seven Wonders of the Ancient World, as well as the storied Library of Alexandria. Today, the library is reincarnated in the disc-shaped, ultramodern Bibliotheca Alexandrina. Its 15th-century seafront Qaitbay Citadel is now a museum. Called the "Bride of the Mediterranean" by locals, Alexandria is a popular tourist destination and an important industrial centre due to its natural gas and oil pipelines from Suez. The city extends about along the northern coast of Egypt, and is the largest city on t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields (the term is no more in common use outside educational context). Linear algebra, which deals with linear equations and linear mappings, is used for modern presentations of geometry, and has many practical applications (in weather forecasting, for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as commutative algebra, and some not, such as Galois theory. The word ''algebra'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Three Dimensions
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal meaning of the term dimension. In mathematics, a tuple of numbers can be understood as the Cartesian coordinates of a location in a -dimensional Euclidean space. The set of these -tuples is commonly denoted \R^n, and can be identified to the -dimensional Euclidean space. When , this space is called three-dimensional Euclidean space (or simply Euclidean space when the context is clear). It serves as a model of the physical universe (when relativity theory is not considered), in which all known matter exists. While this space remains the most compelling and useful way to model the world as it is experienced, it is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Geometry
In mathematics, solid geometry or stereometry is the traditional name for the geometry of Three-dimensional space, three-dimensional, Euclidean spaces (i.e., 3D geometry). Stereometry deals with the measurements of volumes of various solid figures (or 3D figures), including Pyramid (geometry), pyramids, Prism (geometry), prisms and other polyhedrons; Cylinder (geometry), cylinders; cone (geometry), cones; Frustum, truncated cones; and ball (mathematics), balls bounded by spheres. History The Pythagoreanism, Pythagoreans dealt with the regular solids, but the pyramid, prism, cone and cylinder were not studied until the Platonism, Platonists. Eudoxus of Cnidus, Eudoxus established their measurement, proving the pyramid and cone to have one-third the volume of a prism and cylinder on the same base and of the same height. He was probably also the discoverer of a proof that the volume enclosed by a sphere is proportional to the cube of its radius.Paraphrased and taken in part from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is an inferential argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for further mathematical work. Proofs employ logic expressed in mathematical symbols ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Secondary School
A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' secondary education, lower secondary education'' (ages 11 to 14) and ''upper secondary education'' (ages 14 to 18), i.e., both levels 2 and 3 of the International Standard Classification of Education, ISCED scale, but these can also be provided in separate schools. In the United States, US, the secondary education system has separate Middle school#United States, middle schools and High school in the United States, high schools. In the United Kingdom, UK, most state schools and Independent school, privately-funded schools accommodate pupils between the ages of 11–16 or 11–18; some UK Independent school, private schools, i.e. Public school (United Kingdom), public schools, admit pupils between the ages of 13 and 18. Secondary schools follow on from primary school, primary schools and prepare for voc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical System
A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A formal system is essentially an "axiomatic system". In 1921, David Hilbert proposed to use such a system as the foundation for the knowledge in mathematics. A formal system may represent a well-defined system of abstract thought. The term ''formalism'' is sometimes a rough synonym for ''formal system'', but it also refers to a given style of notation, for example, Paul Dirac's bra–ket notation. Background Each formal system is described by primitive symbols (which collectively form an alphabet) to finitely construct a formal language from a set of axioms through inferential rules of formation. The system thus consists of valid formulas built up through finite combinations of the primitive symbols—combinations that are formed from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proposition
In logic and linguistics, a proposition is the meaning of a declarative sentence. In philosophy, " meaning" is understood to be a non-linguistic entity which is shared by all sentences with the same meaning. Equivalently, a proposition is the non-linguistic bearer of truth or falsity which makes any sentence that expresses it either true or false. While the term "proposition" may sometimes be used in everyday language to refer to a linguistic statement which can be either true or false, the technical philosophical term, which differs from the mathematical usage, refers exclusively to the non-linguistic meaning behind the statement. The term is often used very broadly and can also refer to various related concepts, both in the history of philosophy and in contemporary analytic philosophy. It can generally be used to refer to some or all of the following: The primary bearers of truth values (such as "true" and "false"); the objects of belief and other propositional attitudes (i.e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'. The term has subtle differences in definition when used in the context of different fields of study. As defined in classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question. As used in modern logic, an axiom is a premise or starting point for reasoning. As used in mathematics, the term ''axiom'' is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (''A'' and ''B'') implies ''A''), while non-logical axioms (e.g., ) are actually ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Elements
The ''Elements'' ( grc, Στοιχεῖα ''Stoikheîa'') is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry, elementary number theory, and incommensurable lines. ''Elements'' is the oldest extant large-scale deductive treatment of mathematics. It has proven instrumental in the development of logic and modern science, and its logical rigor was not surpassed until the 19th century. Euclid's ''Elements'' has been referred to as the most successful and influential textbook ever written. It was one of the very earliest mathematical works to be printed after the invention of the printing press and has been estimated to be second only to the Bible in the number of editions published since the first printing i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
