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An algebraic equation is an equation involving polynomials, for which algebraic expressions may be solutions. If you restrict your set of constants to be numbers, any algebraic expression can be called an arithmetic expression. However, algebraic expressions can be used on more abstract objects such as in Abstract algebra.
In calculus and mathematical analysis, algebraic operation is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational exponent is an algebraic operation, but not the general exponentiation with a real or complex exponent. Also, the derivative is an operation that is ...
For example, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. [2] [3] Thus, in the expression 1 + 2 × 3, the multiplication is performed before addition, and the expression has the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9.
This object of algebra was called modern algebra or abstract algebra, as established by the influence and works of Emmy Noether. [36] 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: [14] group theory; field theory
The algebraic equations are the basis of a number of areas of modern mathematics: Algebraic number theory is the study of (univariate) algebraic equations over the rationals (that is, with rational coefficients). Galois theory was introduced by Évariste Galois to specify criteria for deciding if an algebraic equation may be solved in terms of ...
Algebra is the branch of mathematics that studies certain abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic operations other than the standard arithmetic operations, such as addition and multiplication.
An algebraic expression is an expression built up from algebraic constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by a rational number). [43] For example, 3x 2 − 2xy + c is an algebraic expression.
This is a method for removing surds from expressions (or at least moving them), applying to division by some combinations involving square roots. For example: The denominator of 5 3 + 4 {\displaystyle {\dfrac {5}{{\sqrt {3}}+4}}} can be rationalised as follows: