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Pythagorean Theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. This theorem can be written as an equation relating the lengths of the sides ''a'', ''b'' and the hypotenuse ''c'', often called the Pythagorean equation: :a^2 + b^2 = c^2 , The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ancient Greece
Ancient Greece ( el, Ἑλλάς, Hellás) was a northeastern Mediterranean civilization, existing from the Greek Dark Ages of the 12th–9th centuries BC to the end of classical antiquity ( AD 600), that comprised a loose collection of culturally and linguistically related city-states and other territories. Most of these regions were officially unified only once, for 13 years, under Alexander the Great's empire from 336 to 323 BC (though this excludes a number of Greek city-states free from Alexander's jurisdiction in the western Mediterranean, around the Black Sea, Cyprus, and Cyrenaica). In Western history, the era of classical antiquity was immediately followed by the Early Middle Ages and the Byzantine period. Roughly three centuries after the Late Bronze Age collapse of Mycenaean Greece, Greek urban poleis began to form in the 8th century BC, ushering in the Archaic period and the colonization of the Mediterranean Basin. This was followed by the age of Classica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states :c^2 = a^2 + b^2 - 2ab\cos\gamma, where denotes the angle contained between sides of lengths and and opposite the side of length . For the same figure, the other two relations are analogous: :a^2=b^2+c^2-2bc\cos\alpha, :b^2=a^2+c^2-2ac\cos\beta. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90 degrees, or radians), then , and thus the law of cosines reduces to the Pythagorean theorem: :c^2 = a^2 + b^2. The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known. History Though the notion of the cosine was not yet developed in his time, Euclid's ''Elements'', dating back to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Length
Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the International System of Units (SI) system the base unit for length is the metre. Length is commonly understood to mean the most extended dimension of a fixed object. However, this is not always the case and may depend on the position the object is in. Various terms for the length of a fixed object are used, and these include height, which is vertical length or vertical extent, and width, breadth or depth. Height is used when there is a base from which vertical measurements can be taken. Width or breadth usually refer to a shorter dimension when length is the longest one. Depth is used for the third dimension of a three dimensional object. Length is the measure of one spatial dimension, whereas area is a measure of two dimensions (length squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charisma
Charisma () is a personal quality of presence or charm that compels its subjects. Scholars in sociology, political science, psychology, and management reserve the term for a type of leadership seen as extraordinary; in these fields, the term "charisma" is used to describe a particular type of leader who uses "values-based, symbolic, and emotion-laden leader signaling". In Christian theology, the term appears as '' charism'', an endowment or extraordinary power given by the Holy Spirit."Spiritual gifts". ''A Dictionary of the Bible'' by W. R. F. Browning. Oxford University Press Inc. ''Oxford Reference Online''. Oxford University Press. Accessed 22 June 2011. Etymology The English term ''charisma'' is from the Greek (''khárisma''), which means "favor freely given" or "gift of grace". The term and its plural (''charismata'') derive from (''charis''), which means "grace" or indeed "charm" with which it shares the root. Some derivatives from that root (including "grace") have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstruseness
In philosophy, the terms obscurantism and obscurationism describe the anti-intellectual practices of deliberately presenting information in an abstruse and imprecise manner that limits further inquiry and understanding of a subject. There are two historical and intellectual denotations of ''obscurantism'': (1) the deliberate restriction of knowledge—opposition to the dissemination of knowledge; and (2) deliberate obscurity—a recondite style of writing characterized by deliberate vagueness. The term ''obscurantism'' derives from the title of the 16th-century satire (''Letters of Obscure Men'', 1515–1519), which was based upon the intellectual dispute between the German Catholic humanist Johann Reuchlin and the monk Johannes Pfefferkorn of the Dominican Order, about whether or not all Jewish books should be burned as un-Christian heresy. Earlier, in 1509, the monk Pfefferkorn had obtained permission from Maximilian I, Holy Roman Emperor (1486–1519), to burn all copies of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-dimensional
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found nec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-Euclidean Geometry
In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry. The essential difference between the metric geometries is the nature of parallel lines. Euclid's fifth postulate, the parallel postulate, is equivalent to Playfair's postulate, which states that, within a two-dimensional plane, for any given line and a point ''A'', which is not on , there is exactly one line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-dimensional Space
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characteristics shared by those elements (thus creating a conceptual model). As such, they are the essential basis of all valid deductive inferences (particularly in logic, mathematics and science), where the process of verification is necessary to determine whether a generalization holds true for any given situation. Generalization can also be used to refer to the process of identifying the parts of a whole, as belonging to the whole. The parts, which might be unrelated when left on their own, may be brought together as a group, hence belonging to the whole by establishing a common relation between them. However, the parts cannot be generalized into a whole—until a common relation is established among ''all'' parts. This does not mean that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system. This contrasts with synthetic geometry. Analytic geometry is used in physics and engineering, and also in aviation, Aerospace engineering, rocketry, space science, and spaceflight. It is the foundation of most modern fields of geometry, including Algebraic geometry, algebraic, Differential geometry, differential, Discrete geometry, discrete and computational geometry. Usually the Cartesian coordinate system is applied to manipulate equations for planes, straight lines, and circles, often in two and sometimes three dimensions. Geometrically, one studies the Euclidean plane (two dimensions) and Euclidean space. As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometric shapes in a numerical way and extracting numerical information from shapes' numerical defin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate System
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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