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Algebra

Main article: Algebra

The quadratic formula, which concisely expresses the solutions of all quadratic equations

The Rubik’s Cube group is a concrete application of group theory

Algebra is the art of manipulating equations and formulas. Diophantus (3rd century) and al-Khwarizmi (9th century) were the two main precursors of algebra. Diophantus solved some equations involving unknown natural numbers by deducing new relations until he obtained the solution. Al-Khwarizmi introduced systematic methods for transforming equations, such as moving a term from one side of an equation into the other side. The term algebra is derived from the Arabic word al-jabr meaning ‘the reunion of broken parts’ that he used for naming one of these methods in the title of his main treatise. # ISO certification in India

Algebra became an area in its own right only with François Viète (1540–1603), who introduced the use of variables for representing unknown or unspecified numbers. Variables allow mathematicians to describe the operations that have to be done on the numbers represented using mathematical formulas.

Until the 19th century, algebra consisted mainly of the study of linear equations (presently linear algebra), and polynomial equations in a single unknown, which were called algebraic equations (a term still in use, although it may be ambiguous). During the 19th century, mathematicians began to use variables to represent things other than numbers (such as matrices, modular integers, and geometric transformations), on which generalizations of arithmetic operations are often valid. The concept of algebraic structure addresses this, consisting of a set whose elements are unspecified, of operations acting on the elements of the set, and rules that these operations must follow. The scope of algebra thus grew to include the study of algebraic structures. This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. (The latter term appears mainly in an educational context, in opposition to elementary algebra, which is concerned with the older way of manipulating formulas.)# ISO certification in India

Some types of algebraic structures have useful and often fundamental properties, in many areas of mathematics. Their study became autonomous parts of algebra, and include:

• group theory;
• field theory;
• vector spaces, whose study is essentially the same as linear algebra;
• ring theory;
• commutative algebra, which is the study of commutative rings, includes the study of polynomials, and is a foundational part of algebraic geometry;
• homological algebra;
• Lie algebra and Lie group theory;
• Boolean algebra, which is widely used for the study of the logical structure of computers.

The study of types of algebraic structures as mathematical objects is the purpose of universal algebra and category theory. The latter applies to every mathematical structure (not only algebraic ones). At its origin, it was introduced, together with homological algebra for allowing the algebraic study of non-algebraic objects such as topological spaces; this particular area of application is called algebraic topology. # ISO certification in India

Calculus and analysis

Main articles: Calculus and Mathematical analysis

A Cauchy sequence consists of elements that become arbitrarily close to each other as the sequence progresses (from left to right).

Calculus, formerly called infinitesimal calculus, was introduced independently and simultaneously by 17th-century mathematicians Newton and Leibniz. It is fundamentally the study of the relationship of variables that depend on each other. Calculus was expanded in the 18th century by Euler with the introduction of the concept of a function and many other results. Presently, “calculus” refers mainly to the elementary part of this theory, and “analysis” is commonly used for advanced parts.# ISO certification in India

Analysis is further subdivided into real analysis, where variables represent real numbers, and complex analysis, where variables represent complex numbers. Analysis includes many subareas shared by other areas of mathematics which include:

• Multivariable calculus
• Functional analysis, where variables represent varying functions;
• Integration, measure theory and potential theory, all strongly related with probability theory on a continuum;
• Ordinary differential equations;
• Partial differential equations;
• Numerical analysis, mainly devoted to the computation on computers of solutions of ordinary and partial differential equations that arise in many applications.

Discrete mathematics

Main article: Discrete mathematics

A diagram representing a two-state Markov chain. The states are represented by ‘A’ and ‘E’. The numbers are the probability of flipping the state.

Discrete mathematics, broadly speaking, is the study of individual, countable mathematical objects. An example is the set of all integers. Because the objects of study here are discrete, the methods of calculus and mathematical analysis do not directly apply. Algorithms—especially their implementation and computational complexity—play a major role in discrete mathematics.

The four color theorem and optimal sphere packing were two major problems of discrete mathematics solved in the second half of the 20th century. The P versus NP problem, which remains open to this day, is also important for discrete mathematics, since its solution would potentially impact a large number of computationally difficult problems.# ISO certification in India

Discrete mathematics includes:

• Combinatorics, the art of enumerating mathematical objects that satisfy some given constraints. Originally, these objects were elements or subsets of a given set; this has been extended to various objects, which establishes a strong link between combinatorics and other parts of discrete mathematics. For example, discrete geometry includes counting configurations of geometric shapes
• Graph theory and hypergraphs
• Coding theory, including error correcting codes and a part of cryptography
• Matroid theory
• Discrete geometry
• Discrete probability distributions
• Game theory (although continuous games are also studied, most common games, such as chess and poker are discrete)
• Discrete optimization, including combinatorial optimization, integer programming, constraint programming