Euclidean (or, less commonly, Euclidian) is an adjective derived from the name of
Euclid
Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
, an ancient Greek mathematician.
Geometry
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Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
, the two-dimensional plane and three-dimensional space of Euclidean geometry as well as their higher dimensional generalizations
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Euclidean geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematics, Greek mathematician Euclid, which he described in his textbook on geometry, ''Euclid's Elements, Elements''. Euclid's approach consists in assuming a small set ...
, the study of the properties of Euclidean spaces
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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 ge ...
, systems of points, lines, and planes analogous to Euclidean geometry but without uniquely determined parallel lines
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Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of the line segment between them. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, and therefore is o ...
, the distance between pairs of points in Euclidean spaces
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Euclidean ball, the set of points within some fixed distance from a center point
Number theory
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Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
, the division which produces a quotient and a remainder
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Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is a ...
, a method for finding greatest common divisors
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Extended Euclidean algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's id ...
, a method for solving the Diophantine equation ''ax'' + ''by'' = ''d'' where ''d'' is the greatest common divisor of ''a'' and ''b''
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Euclid's lemma
In algebra and number theory, Euclid's lemma is a lemma that captures a fundamental property of prime numbers:
For example, if , , , then , and since this is divisible by 19, the lemma implies that one or both of 133 or 143 must be as well. In ...
: if a prime number divides a product of two numbers, then it divides at least one of those two numbers
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Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of Euclidean division of integers. Th ...
, a ring in which Euclidean division may be defined, which allows Euclid's lemma to be true and the Euclidean algorithm and the extended Euclidean algorithm to work
Other
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Euclidean relation In mathematics, Euclidean relations are a class of binary relations that formalize ":wikisource:Page:First six books of the elements of Euclid 1847 Byrne.djvu/26, Axiom 1" in Euclid's Elements, Euclid's ''Elements'': "Magnitudes which are equal to t ...
, a property of binary relations related to transitivity
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Euclidean distance map, a digital image in which each pixel value represents the Euclidean distance to an obstacle
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Euclidean rhythm, a method of distributing beats across musical steps based on Euclid's algorithm
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Euclidean zoning, a system of land use management modeled after the zoning code of Euclid, Ohio
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Euclidean division
In arithmetic, Euclidean division – or division with remainder – is the process of dividing one integer (the dividend) by another (the divisor), in a way that produces an integer quotient and a natural number remainder strictly smaller than ...
of the Intermediate Math League of Eastern Massachusetts
See also
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Euclid (disambiguation)
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Euclid's ''Elements'', a 13-book mathematical treatise written by Euclid, that includes both geometry and number theory
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Euclideon, an Australian computer graphics company
{{disambiguation
Mathematics disambiguation pages