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Heron's Formula
In geometry, Heron's formula
Heron's formula
(sometimes called Hero's formula), named after Hero of Alexandria,[1] gives the area of a triangle by requiring no arbitrary choice of side as base or vertex as origin, contrary to other formulas for the area of a triangle, such as half the base times the height or half the norm of a cross product of two sides.Contents1 Formulation 2 Example 3 History 4 Proofs4.1 Trigonometric proof using the law of cosines 4.2 Algebraic proof using the Pythagorean theorem 4.3 Trigonometric proof using the law of cotangent
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Difference Of Squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = ( a + b ) ( a − b ) displaystyle a^ 2 -b^ 2 =(a+b)(a-b) in elementary algebra.Contents1 Proof 2 Geometrical demonstrations 3 Uses3.1 Factorization
Factorization
of polynomials 3.2 Complex number
Complex number
case: sum of two squares 3.3 Rationalising denominators 3.4 Mental arithmetic 3.5 Difference of two perfect squares4 Generalizations4.1 Difference of two nth powers5 See also 6 Notes 7 References 8 External linksProof[edit] The proof of the factorization identity is straightforward
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Distance
Distance
Distance
is a numerical measurement of how far apart objects are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). In most cases, "distance from A to B" is interchangeable with "distance from B to A". In mathematics, a distance function or metric is a generalization of the concept of physical distance
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Quadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side". Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is ∠ A + ∠ B + ∠ C + ∠ D = 360 ∘ . displaystyle angle A+angle B+angle C+angle D=360^ circ
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Volume
Volume
Volume
is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.[1] Volume
Volume
is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container; i. e., the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Three dimensional mathematical shapes are also assigned volumes. Volumes of some simple shapes, such as regular, straight-edged, and circular shapes can be easily calculated using arithmetic formulas
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Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or initially small fluctuations in initial data which might cause a large deviation of final answer from the exact solution[citation needed]. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called numerically stable
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Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. Specifically, a k-simplex is a k-dimensional polytope which is the convex hull of its k + 1 vertices. More formally, suppose the k + 1 points u 0 , … , u k ∈ R k displaystyle u_ 0 ,dots ,u_ k in mathbb R ^ k are affinely independent, which means u 1 − u 0 , … , u k − u 0 displaystyle u_ 1 -u_ 0 ,dots ,u_ k -u_ 0 are linearly independent
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Pythagorean Theorem
In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry
Euclidean geometry
among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the "Pythagorean equation":[1] a 2 + b 2 = c 2 , displaystyle a^ 2 +b^ 2 =c^ 2 , where c represents the length of the hypotenuse and a and b the lengths of the triangle's other two sides. Although it is often argued that knowledge of the theorem predates him,[2][3] the theorem is named after the ancient Greek mathematician Pythagoras
Pythagoras
(c
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Difference Of Two Squares
In mathematics, the difference of two squares is a squared (multiplied by itself) number subtracted from another squared number. Every difference of squares may be factored according to the identity a 2 − b 2 = ( a + b ) ( a − b ) displaystyle a^ 2 -b^ 2 =(a+b)(a-b) in elementary algebra.Contents1 Proof 2 Geometrical demonstrations 3 Uses3.1 Factorization
Factorization
of polynomials 3.2 Complex number
Complex number
case: sum of two squares 3.3 Rationalising denominators 3.4 Mental arithmetic 3.5 Difference of two perfect squares4 Generalizations4.1 Difference of two nth powers5 See also 6 Notes 7 References 8 External linksProof[edit] The proof of the factorization identity is straightforward
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Shoelace Formula
The shoelace formula or shoelace algorithm (also known as Gauss's area formula and the surveyor's formula[1]) is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates
Cartesian coordinates
in the plane.[2] The user cross-multiplies corresponding coordinates to find the area encompassing the polygon, and subtracts it from the surrounding polygon to find the area of the polygon within. It is called the shoelace formula because of the constant cross-multiplying for the coordinates making up the polygon, like tying shoelaces.[2] It is also sometimes called the shoelace method
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Law Of Cosines
In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) 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 − 2 a b cos ⁡ γ , displaystyle c^ 2 =a^ 2 +b^ 2 -2abcos gamma , where γ denotes the angle contained between sides of lengths a and b and opposite the side of length c. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle γ is a right angle (of measure 90°, or π/2 radians), then cos γ = 0, and thus the law of cosines reduces to the Pythagorean theorem: c 2 = a 2 + b 2 . displaystyle c^ 2 =a^ 2 +b^ 2
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Angle
2D anglesRight Interior Exterior2D angle pairsAdjacent Vertical Complementary Supplementary Transversal3D anglesDihedralAn angle formed by two rays emanating from a vertex.In planar geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.[1] Angles formed by two rays lie in a plane, but this plane does not have to be a Euclidean plane. Angles are also formed by the intersection of two planes in Euclidean and other spaces. These are called dihedral angles. Angles formed by the intersection of two curves in a plane are defined as the angle determined by the tangent rays at the point of intersection. Similar statements hold in space, for example, the spherical angle formed by two great circles on a sphere is the dihedral angle between the planes determined by the great circles. Angle
Angle
is also used to designate the measure of an angle or of a rotation
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Algebra
Algebra
Algebra
(from Arabic
Arabic
"al-jabr" literally meaning "reunion of broken parts"[1]) is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols;[2] it is a unifying thread of almost all of mathematics.[3] As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra; the more abstract parts are called abstract algebra or modern algebra. Elementary algebra
Elementary algebra
is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics
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Digital Object Identifier
In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to uniquely identify objects, standardized by the International Organization for Standardization
International Organization for Standardization
(ISO).[1] An implementation of the Handle System,[2][3] DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos. A DOI aims to be "resolvable", usually to some form of access to the information object to which the DOI refers. This is achieved by binding the DOI to metadata about the object, such as a URL, indicating where the object can be found. Thus, by being actionable and interoperable, a DOI differs from identifiers such as ISBNs and ISRCs which aim only to uniquely identify their referents
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Incenter
In geometry, the incenter of a triangle is a triangle center, a point defined for any triangle in a way that is independent of the triangle's placement or scale. The incenter may be equivalently defined as the point where the internal angle bisectors of the triangle cross, as the point equidistant from the triangle's sides, as the junction point of the medial axis and innermost point of the grassfire transform of the triangle, and as the center point of the inscribed circle of the triangle. Together with the centroid, circumcenter, and orthocenter, it is one of the four triangle centers known to the ancient Greeks, and the only one that does not in general lie on the Euler line
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Eric W. Weisstein
Eric Wolfgang Weisstein (born March 18, 1969) is an encyclopedist who created and maintains MathWorld and Eric Weisstein's World of Science (ScienceWorld). He is the author of the CRC Concise Encyclopedia of Mathematics. He currently works for Wolfram Research, Inc.Contents1 Education 2 Career2.1 Academic research 2.2 MathWorld, ScienceWorld
ScienceWorld
and Wolfram Research 2.3 Further scientific activities3 Footnotes 4 References 5 External linksEducation[edit] Weisstein holds a Ph.D. in planetary astronomy which he obtained from the California Institute of Technology's (Caltech) Division of Geological and Planetary Sciences in 1996 as well as an M.S. in planetary astronomy in 1993 also from Caltech. Weisstein graduated Cum Laude from Cornell University
Cornell University
with a B.A. in physics and a minor in astronomy in 1990
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