In mathematics, the Borsuk – Ulam theorem, named after Stanisław Ulam and Karol Borsuk, states that every continuous function from an n-sphere into Euclidean n-space maps some pair of antipodal points to the same point.

The invention of Cartesian coordinates in the 17th century by René Descartes ( Latinized name: Cartesius ) revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

Because of Euclidean geometry's fundamental status in mathematics, it would be impossible to give more than a representative sampling of applications here.

According to Husserl, this view of logic and mathematics accounted for the objectivity of a series of mathematical developments of his time, such as n-dimensional manifolds ( both Euclidean and non-Euclidean ), Hermann Grassmann's theory of extensions, William Rowan Hamilton's Hamiltonians, Sophus Lie's theory of transformation groups, and Cantor's set theory.

In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions.

In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry.

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects.

In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain ( also called a Euclidean ring ) is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean division of the integers.

The three-dimensional Euclidean space R < sup > 3 </ sup > is a vector space, and lines and planes passing through the origin ( mathematics ) | origin are vector subspaces in R < sup > 3 </ sup >.

In mathematics, a careful distinction is made between the sphere ( a two-dimensional surface embedded in three-dimensional Euclidean space ) and the ball ( the interior of the three-dimensional sphere ).

In mathematics, specifically in real analysis, the Bolzano – Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R < sup > n </ sup >.

In mathematics, physics, and engineering, a Euclidean vector ( sometimes called a geometric or spatial vector, or — as here — simply a vector ) is a geometric object that has a magnitude ( or length ) and direction and can be added to other vectors according to vector algebra.

When it becomes necessary to distinguish these special vectors from vectors as defined in pure mathematics, they are sometimes referred to as geometric, spatial, or Euclidean vectors.

Vector calculus ( or vector analysis ) is a branch of mathematics concerned with differentiation and integration of vector fields, primarily in 3 dimensional Euclidean space The term " vector calculus " is sometimes used as a synonym for the broader subject of multivariable calculus, which includes vector calculus as well as partial differentiation and multiple integration.

In mathematics, the convex hull or convex envelope of a set X of points in the Euclidean plane or Euclidean space is the smallest convex set that contains X.

The ancient Greek deiknymi, or thought experiment, " was the most ancient pattern of mathematical proof ", and existed before Euclidean mathematics, where the emphasis was on the conceptual, rather than on the experimental part of a thought-experiment.

In mathematics, the Euclidean distance or Euclidean metric is the " ordinary " distance between two points that one would measure with a ruler, and is given by the Pythagorean formula.

In mathematics and computer science, an algorithm ( originating from al-Khwārizmī, the famous Persian mathematician Muḥammad ibn Mūsā al-Khwārizmī ) is a step-by-step procedure for calculations.

In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm.

For instance, instead of a hardware multiplier, a calculator might implement floating point mathematics with code in ROM, and compute trigonometric functions with the CORDIC algorithm because CORDIC does not require hardware floating-point.

Graph ( mathematics ) | Graphs like this are among the objects studied by discrete mathematics, for their interesting graph property | mathematical properties, their usefulness as models of real-world problems, and their importance in developing computer algorithm s.

Early in the 1970s, the asymmetric key algorithm was invented by staff member Clifford Cocks, a mathematics graduate: this fact was kept secret until 1997.

In mathematics, Horner's method ( also known as Horner scheme in the UK or Horner's rule in the U. S .) is either of two things: ( i ) an algorithm for calculating polynomials, which consists in transforming the monomial form into a computationally efficient form ; or ( ii ) a method for approximating the roots of a polynomial.

Such an algorithm contradicts fundamental laws of mathematics because, if it existed, it could be applied repeatedly to losslessly reduce any file to length 0.

A strong grasp of mathematics is not required to understand or implement this algorithm.

In theoretical computer science and mathematics, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm.

The thesis states that Turing machines indeed capture the informal notion of effective method in logic and mathematics, and provide a precise definition of an algorithm or ' mechanical procedure '.

Peter Williston Shor ( born August 14, 1959 ) is an American professor of applied mathematics at MIT, most famous for his work on quantum computation, in particular for devising Shor's algorithm, a quantum algorithm for factoring exponentially faster than the best currently-known algorithm running on a classical computer.

In mathematics, the sieve of Eratosthenes (), one of a number of prime number sieves, is a simple, ancient algorithm for finding all prime numbers up to any given limit.

It is interdisciplinary in nature, borrowing, adapting and enhancing method and theory from numerous other disciplines such as computer science ( e. g. algorithm and software design, database design and theory ), geoinformation science ( spatial statistics and modeling, geographic information systems ), artificial intelligence research ( supervised classification, fuzzy logic ), ecology ( point pattern analysis ), applied mathematics ( graph theory, probability theory ) and statistics.

In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1 / π.

Other methods are the Faugère F4 algorithm, based on the same mathematics as the Buchberger algorithm, and involutive approaches, based on ideas from differential algebra.

Because it is based on different mathematics ( lattice-based cryptography ) from RSA and ECC, the NTRU algorithm has different cryptographic properties.

Solutions of the initial value problem were computed by using the LSODE algorithm, as implemented in the mathematics package GNU Octave.

* bisection method, in mathematics, a root-finding algorithm

This discovery was a major paradigm shift in mathematics, as it freed mathematicians from the mistaken belief that Euclid's axioms were the only way to make geometry consistent and non-contradictory.

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

Illustration at the beginning of a medieval translation of Euclid's Element ( mathematics ) | Elements, ( c. 1310 )

A good stepping stone leads to many others, so some of the most powerful results in mathematics are known as lemmata, such as Bézout's lemma, Urysohn's lemma, Dehn's lemma, Euclid's lemma, Farkas ' lemma, Fatou's lemma, Gauss's lemma, Nakayama's lemma, Poincaré's lemma, Riesz's lemma, Schwarz's lemma, Itō's lemma and Zorn's lemma.

While Greek astronomy — thanks to Alexander's conquests — probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition ; in particular, there is no evidence that Euclid's Elements reached India before the 18th century.

Roger Ascham thought that his pupil Robert had an uncommon talent for languages and writing, " exceed almost all other by nature ", and regretted that he had done himself harm by preferring " Euclid's pricks and lines " ( mathematics ).

The Italy | Italian Jesuit Matteo Ricci ( left ) and the Chinese mathematics | Chinese mathematician Xu Guangqi ( right ) published the Chinese language | Chinese edition of Euclid's Elements ( 幾何原本 ) in 1607.

# Ἡ καθόλου πραγματεία, a work on the general principles of mathematics that perhaps included Apollonius's criticisms and suggestions for the improvement of Euclid's Elements

The views of Ramus on mathematics implied a limitation to the practical: he considered Euclid's theory on irrational numbers to be useless.

* Classical education-the teaching of mathematics within the quadrivium, part of the classical education curriculum of the Middle Ages, which was typically based on Euclid's Elements taught as a paradigm of deductive reasoning.

As a mathematics educator, Dodgson defended the use of Euclid's Elements as a geometry textbook ; Euclid and his Modern Rivals is a criticism of a reform movement in geometry education lead by the Association for the Improvement of Geometrical Teaching.

E. T. Bell in his book Men of Mathematics wrote about Lobachevsky's influence on the following development of mathematics: The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other ' axioms ' or accepted ' truths ', for example the ' law ' of causality which, for centuries, have seemed as necessary to straight thinking as Euclid's postulate appeared till Lobatchewsky discarded it.

In mathematics, the Sturm's sequence of a polynomial p is a sequence of polynomials associated to p and its derivative by a variant of Euclid's algorithm for polynomials.

* A commentary on Euclid's Elements, a fundamental mathematics text.