algebra, branch of mathematics mathematics, deductive study of numbers, geometry, and various abstract constructs, or structures; the latter often "abstract" the features common to several models derived from the empirical, or applied, sciences, although many emerge from purely mathematical or
..... Click the link for more information. concerned with operations on sets of numbers number, entity describing the magnitude or position of a mathematical object or extensions of these concepts.

The Natural Numbers

Cardinal numbers describe the size of a collection of objects; two such collections have the same (cardinal) number of
..... Click the link for more information. or other elements that are often represented by symbols. Algebra is a generalization of arithmetic and gains much of its power from dealing symbolically with elements and operations (such as addition and multiplication) and relationships (such as equality) connecting the elements. Thus, a+a=2a and a+b=b+a no matter what numbers a and b represent.

Principles of Classical Algebra

In elementary algebra letters are used to stand for numbers. For example, in the equation equation, in mathematics, a statement, usually written in symbols, that states the equality of two quantities or algebraic expressions, e.g., x+3=5. The quantity x
..... Click the link for more information. ax²+bx+c=0, the letters a, b, and c stand for various known constant numbers called coefficients and the letter x is an unknown variable number whose value depends on the values of a, b, and c and may be determined by solving the equation. Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots root, in mathematics, number or quantity r for which an equation f(r)=0 holds true, where f is some function . If f is a polynomial , r is called a root of f; for example, r=3 and r
..... Click the link for more information. , or values of the unknowns, that upon substitution into the original equation will make it a numerical identity. For example, x=−2 is a root of x²−2x−8=0 because (−2)²−2(−2)−8=4+4−8=0; substitution will verify that x=4 is also a root of this equation.

The equations of elementary algebra usually involve polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a₀xⁿ+a
..... Click the link for more information. functions of one or more variables (see function function, in mathematics, a relation f that assigns to each member x of some set X a corresponding member y of some set Y; y is said to be a function of x, usually denoted f(x) (read "f of x
..... Click the link for more information. ). The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic quadratic, mathematical expression of the second degree in one or more unknowns (see polynomial ). The general quadratic in one unknown has the form ax²+bx+c, where a, b, and c are constants and x is the variable.
..... Click the link for more information. ). One method of finding the zeros of the polynomial function f(x), i.e., the roots of the equation f(x)=0, is to factor the polynomial, if possible. The polynomial x²−2x−8 has factors (x+2) and (x−4), since (x+2)(x−4)=x²−2x−8, so that setting either of these factors equal to zero will make the polynomial zero. In general, if (x−r) is a factor of a polynomial f(x), then r is a zero of the polynomial and a root of the equation f(x)=0. To determine if (x−r) is a factor, divide it into f(x); according to the Factor Theorem, if the remainder f(r)—found by substituting r for x in the original polynomial—is zero, then (x−r) is a factor of f(x). Although a polynomial has real coefficients, its roots may not be real numbers; e.g., x²−9 separates into (x+3)(x−3), which yields two zeros, x=−3 and x=+3, but the zeros of x²+9 are imaginary numbers.

The Fundamental Theorem of Algebra states that every polynomial f(x)=a_nxⁿ+a_n−1xⁿ⁻¹+ … +a₁x+a₀, with a_n≠0 and n≥1, has at least one complex root, from which it follows that the equation f(x)=0 has exactly n roots, which may be real or complex and may not all be distinct. For example, the equation x⁴+4x³+5x²+4x+4=0 has four roots, but two are identical and the other two are complex; the factors of the polynomial are (x+2)(x+2)(x+i)(x−i), as can be verified by multiplication.

Principles of Modern Algebra

Modern algebra is yet a further generalization of arithmetic than is classical algebra. It deals with operations that are not necessarily those of arithmetic and that apply to elements that are not necessarily numbers. The elements are members of a set set, in mathematics, collection of entities, called elements of the set, that may be real objects or conceptual entities. Set theory not only is involved in many areas of mathematics but has important applications in other fields as well, e.g.
..... Click the link for more information. and are classed as a group group, in mathematics, system consisting of a set of elements and a binary operation a+b defined for combining two elements such that the following requirements are satisfied: (1) The set is closed under the operation; i.e.
..... Click the link for more information. , a ring ring, in mathematics, system consisting of a set R of elements and two binary operations, such that addition makes R a commutative group and multiplication is associative and distributes over addition (see commutative law ; associative law ;
..... Click the link for more information. , or a field field, in algebra, set of elements (usually numbers) that may be combined under the operations of addition and multiplication so that it constitutes an additive group , the nonzero elements form a multiplicative group, and multiplication distributes over addition.
..... Click the link for more information. according to the axioms that are satisfied under the particular operations defined for the elements. Among the important concepts of modern algebra are those of a matrix matrix, in mathematics, a rectangular array of elements (e.g., numbers) considered as a single entity. A matrix is distinguished by the number of rows and columns it contains. The matrix is a 2×3 (read "2 by 3") matrix, because it contains 2 rows and 3 columns.
..... Click the link for more information. and of a vector U [−3,1] and V [5,2], one can add their corresponding components to find the resultant vector R [2,3], or one can graph U and V on a set of coordinate axes and complete the parallelogram formed with U and V
..... Click the link for more information. space.

Bibliography

See M. Artin, Algebra (1991).

algebra

Generalized version of arithmetic that uses variables to stand for unspecified numbers. Its purpose is to solve algebraic equations or systems of equations. Examples of such solutions are the quadratic formula (for solving a quadratic equation) and Gauss-Jordan elimination (for solving a system of equations in matrix form). In higher mathematics, an “algebra” is a structure consisting of a class of objects and a set of rules (analogous to addition and multiplication) for combining them. Basic and higher algebraic structures share two essential characteristics: (1) calculations involve a finite number of steps and (2) calculations involve abstract symbols (usually letters) representing more general objects (usually numbers). Higher algebra (also known as modern or abstract algebra) includes all of elementary algebra, as well as group theory, theory of rings, field theory, manifolds, and vector spaces.

(mathematics, logic)

algebra - 1. A loose term for an algebraic structure.

2. A vector space that is also a ring, where the vector space and the ring share the same addition operation and are related in certain other ways.

An example algebra is the set of 2x2 matrices with real numbers as entries, with the usual operations of addition and matrix multiplication, and the usual scalar multiplication. Another example is the set of all polynomials with real coefficients, with the usual operations.

In more detail, we have:

(1) an underlying set,

(2) a field of scalars,

(3) an operation of scalar multiplication, whose input is a scalar and a member of the underlying set and whose output is a member of the underlying set, just as in a vector space,

(4) an operation of addition of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a vector space or a ring,

(5) an operation of multiplication of members of the underlying set, whose input is an ordered pair of such members and whose output is one such member, just as in a ring.

This whole thing constitutes an `algebra' iff:

(1) it is a vector space if you discard item (5) and

(2) it is a ring if you discard (2) and (3) and

(3) for any scalar r and any two members A, B of the underlying set we have r(AB) = (rA)B = A(rB). In other words it doesn't matter whether you multiply members of the algebra first and then multiply by the scalar, or multiply one of them by the scalar first and then multiply the two members of the algebra. Note that the A comes before the B because the multiplication is in some cases not commutative, e.g. the matrix example.

Another example (an example of a Banach algebra) is the set of all bounded linear operators on a Hilbert space, with the usual norm. The multiplication is the operation of composition of operators, and the addition and scalar multiplication are just what you would expect.

Two other examples are tensor algebras and Clifford algebras.

[I. N. Herstein, "Topics in Algebra"].