In mathematics, a Hurwitz quaternion ( or Hurwitz integer ) is a quaternion whose components are either all integers or all half-integers ( halves of an odd integer ; a mixture of integers and half-integers is not allowed ).

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry.

In mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R. Thus A is endowed with binary operations of addition and multiplication satisfying a number of axioms, including associativity of multiplication and distributivity, as well as compatible multiplication by the elements of the field K or the ring R.

In classical mathematics, analytic geometry, also known as coordinate geometry, or Cartesian geometry, is the study of geometry using a coordinate system and the principles of algebra and analysis.

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed.

Binary relations are used in many branches of mathematics to model concepts like " is greater than ", " is equal to ", and " divides " in arithmetic, " is congruent to " in geometry, " is adjacent to " in graph theory, " is orthogonal to " in linear algebra and many more.

In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space.

The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics ( for example Venn diagrams and symbolic reasoning about their Boolean algebra ), and the everyday usage of set theory concepts in most contemporary mathematics.

Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics.

General category theory, an extension of universal algebra having many new features allowing for semantic flexibility and higher-order logic, came later ; it is now applied throughout mathematics.

It is also a tool used in branches of mathematics including combinatorics, abstract algebra, and mathematical analysis.

Mathematics used in rendering includes: linear algebra, calculus, numerical mathematics, signal processing, and Monte Carlo methods.

In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group.

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.

Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory, and more.

In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.

The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics.

Diophantus ' work created a foundation for work on algebra and in fact much of advanced mathematics is based on algebra.

In mathematics and abstract algebra, a group is the algebraic structure, where is a non-empty set and denotes a binary operation called the group operation.

Elementary algebra introduces the basic rules and operations of algebra, one of the main branches of mathematics.

Elementary algebra is typically taught to secondary school students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic as " algebra ".

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

His influence spilled over into many other branches of mathematics, for example the contemporary theory of D-modules.

In Doppler's time in Prague as a professor he published over 50 articles on mathematics, physics and astronomy.

So he taught Spanish, physics, and mathematics at the New Albany High School in New Albany, Indiana for a year before he resolved to start over, at the age of 25, to become a professional astronomer.

In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated.

The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.

If an isomorphism can be found from a relatively unknown part of mathematics into some well studied division of mathematics, where many theorems are already proved, and many methods are already available to find answers, then the function can be used to map whole problems out of unfamiliar territory over to " solid ground " where the problem is easier to understand and work with.

* Encyclopaedia of Mathematics online encyclopaedia from Springer, Graduate-level reference work with over 8, 000 entries, illuminating nearly 50, 000 notions in mathematics.

There is some dispute over priority of various ideas: Newton's Principia is certainly the seminal work and has been tremendously influential, and the systematic mathematics therein did not and could not have been stated earlier because calculus had not been developed.

The first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics.

* Module ( mathematics ) over a ring, a generalization of vector spaces

In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers.

Platonism is considered to be, in mathematics departments the world over, the predominant philosophy of mathematics, especially regarding the foundations of mathematics.

So she pored over every book on mathematics in her father's library, even teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler.

In mathematics, specifically in ring theory, the simple modules over a ring R are the ( left or right ) modules over R which have no non-zero proper submodules.

According to Steinhaus, while he was strolling through the gardens he was surprised to over hear the term " Lebesgue measure " ( Lebesgue integration was at the time still a fairly new idea in mathematics ) and walked over to investigate.

In pure mathematics, a vector is any element of a vector space over some field and is often represented as a coordinate vector.

In addition to mathematics and statistics, the arithmetic mean is used frequently in fields such as economics, sociology, and history, though it is used in almost every academic field to some extent.

Combinatorics is an example of a field of mathematics which does not, in general, follow the axiomatic method.

These connections shed the boundaries between combinatorics and parts of mathematics and theoretical computer science, but at the same time led to a partial fragmentation of the field.

This was a major discovery in an important field of mathematics ; construction problems had occupied mathematicians since the days of the Ancient Greeks, and the discovery ultimately led Gauss to choose mathematics instead of philology as a career.

The survey of Hanover fueled Gauss's interest in differential geometry, a field of mathematics dealing with curves and surfaces.

* Computational complexity theory, a field in theoretical computer science and mathematics

The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business accounts.

The history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field.

In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.

In modern mathematics, the theory of fields ( or field theory ) plays an essential role in number theory and algebraic geometry.

Financial mathematics is a field of applied mathematics, concerned with financial markets.

The field is largely focused on the modelling of derivatives, although other important subfields include insurance mathematics and quantitative portfolio problems.

An example of such a finite field is the ring Z / pZ, which is essentially the set of integers from 0 to p − 1 with integer addition and multiplication modulo p. It is also sometimes denoted Z < sub > p </ sub >, but within some areas of mathematics, particularly number theory, this may cause confusion because the same notation Z < sub > p </ sub > is used for the ring of p-adic integers.

In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension.

In 1925 Peano switched Chairs unofficially from Infinitesimal Calculus to Complementary Mathematics, a field which better suited his current style of mathematics.

Al-Kindi wrote De Gradibus, in which he demonstrated the application of mathematics to medicine, particularly in the field of pharmacology.

The field is at the intersection of mathematics, statistics, computer science, physics, neurobiology, and electrical engineering.

Much of the mathematics behind information theory with events of different probabilities was developed for the field of thermodynamics by Ludwig Boltzmann and J. Willard Gibbs.