algebra (redirected from Algebra 2)

algebra, branch of mathematics concerned with operations on sets of numbers 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 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, 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 functions of one or more variables (see function). The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic). 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 and are classed as a group, a ring, or a field 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 and of a vector space.

Bibliography

See M. Artin, Algebra (1991).

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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.

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algebra

a branch of mathematics in which arithmetical operations and relationships are generalized by using alphabetic symbols to represent unknown numbers or members of specified sets of numbers

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algebra [′al·jə·brə]

(mathematics)

A method of solving practical problems by using symbols, usually letters, for unknown quantities.

The study of the formal manipulations of equations involving symbols and numbers.

An abstract mathematical system consisting of a vector space together with a multiplication by which two vectors may be combined to yield a third, and some axioms relating this multiplication to vector addition and scalar multiplication. Also known as hypercomplex system.

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(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"].