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Algebra is the branch of mathematics concerning the study of the rules of operations and
relations, and the constructions and concepts arising from them, including terms, polynomials,
equations and algebraic structures.
Together with geometry, analysis, topology, combinatorics, and number theory, algebra is one of
the main branches of pure mathematics.Elementary algebra, often part of the curriculum in
secondary education, introduces the concept of variables representing numbers.
Statements based on these variables are manipulated using the rules of operations that apply to
numbers, such as addition.
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This can be done for a variety of reasons, including equation solving. Algebra is much broader
than elementary algebra and studies what happens when different rules of operations are used
and when operations are devised for things other than numbers.
Addition and multiplication can be generalized and their precise definitions lead to structures such
as groups, rings and fields, studied in the area of mathematics called abstract algebra.Classification
Algebra may be divided roughly into the following categories:
Elementary algebra, in which the properties of operations on the real number system are
recorded using symbols as "place holders" to denote constants and variables, and the rules
governing mathematical expressions and equations involving these symbols are studied. This is
usually taught at school under the title algebra (or intermediate algebra and college algebra in
subsequent years). University-level courses in group theory may also be called elementary
Abstract algebra, sometimes also called modern algebra, in which algebraic structures such as
groups, rings and fields are axiomatically defined and investigated.
Linear algebra, in which the specific properties of vector spaces are studied (including matrices);
Universal algebra, in which properties common to all algebraic structures are studied.
Algebraic number theory, in which the properties of numbers are studied through algebraic
systems. Number theory inspired much of the original abstraction in algebra.
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Algebraic geometry applies abstract algebra to the problems of geometry.
Algebraic combinatorics, in which abstract algebraic methods are used to study combinatorial
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Free demo sessionTopics Covered in Algebra
Given below are some of the main topics covered in our Algebra Tutorial:
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed
to have no knowledge of mathematics beyond the basic principles of arithmetic. In arithmetic, only
numbers and their arithmetical operations (such as +, -, x, /) occur. In algebra, numbers are
often denoted by symbols (such as a, x, or y). This is useful because:
It allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and
thus is the first step to a systematic exploration of the properties of the real number system.
It allows the reference to "unknown" numbers, the formulation of equations and the study of how
to solve these. (For instance, "Find a number x such that 3x + 1 = 10" or going a bit further "Find a
number x such that ax+b=c". This step leads to the conclusion that it is not the nature of the
specific numbers that allows us to solve it, but that of the operations involved.)
It allows the formulation of functional relationships. (For instance, "If you sell x tickets, then your
profit will be 3x - 10 dollars, or f(x) = 3x - 10, where f is the function, and x is the number to which
the function is applied.")