HOME

TheInfoList



OR:

In mathematics, an algebraic expression is an
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, ...
built up from integer
constants Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...
, variables, and the
algebraic operation Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
s (
addition Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
, subtraction, multiplication, division and exponentiation by an exponent that is a rational number). For example, is an algebraic expression. Since taking the square root is the same as raising to the power , the following is also an algebraic expression: :\sqrt An ''
algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
'' is an equation involving only algebraic expressions. By contrast,
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classe ...
s like and are not algebraic, since they are not derived from integer constants and algebraic operations. Usually, is constructed as a geometric relationship, and the definition of requires an ''infinite number'' of algebraic operations. A rational expression is an
expression Expression may refer to: Linguistics * Expression (linguistics), a word, phrase, or sentence * Fixed expression, a form of words with a specific meaning * Idiom, a type of fixed expression * Metaphorical expression, a particular word, phrase, ...
that may be rewritten to a rational fraction by using the properties of the arithmetic operations ( commutative properties and associative properties of addition and multiplication,
distributive property In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmeti ...
and rules for the operations on the fractions). In other words, a rational expression is an expression which may be constructed from the variables and the constants by using only the four operations of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers—addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th c ...
. Thus, :\frac is a rational expression, whereas :\sqrt is not. A rational equation is an equation in which two rational fractions (or rational expressions) of the form : \frac are set equal to each other. These expressions obey the same rules as fractions. The equations can be solved by cross-multiplying. Division by zero is undefined, so that a solution causing formal division by zero is rejected.


Terminology

Algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...
has its own terminology to describe parts of an expression:

1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, x, y - variables


In roots of polynomials

The
roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusing ...
of a polynomial expression of degree ''n'', or equivalently the solutions of a
polynomial equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
, can always be written as algebraic expressions if ''n'' < 5 (see
quadratic formula In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, gr ...
,
cubic function In mathematics, a cubic function is a function of the form f(x)=ax^3+bx^2+cx+d where the coefficients , , , and are complex numbers, and the variable takes real values, and a\neq 0. In other words, it is both a polynomial function of degree ...
, and quartic equation). Such a solution of an equation is called an
algebraic solution A solution in radicals or algebraic solution is a closed-form expression, and more specifically a closed-form algebraic expression, that is the solution of a polynomial equation, and relies only on addition, subtraction, multiplication, divisio ...
. But the
Abel–Ruffini theorem In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, ''general'' means t ...
states that algebraic solutions do not exist for all such equations (just for some of them) if ''n'' \ge 5.


Conventions


Variables

By convention, letters at the beginning of the alphabet (e.g. a, b, c) are typically used to represent
constants Constant or The Constant may refer to: Mathematics * Constant (mathematics), a non-varying value * Mathematical constant, a special number that arises naturally in mathematics, such as or Other concepts * Control variable or scientific const ...
, and those toward the end of the alphabet (e.g. x, y and z) are used to represent variables. They are usually written in italics.


Exponents

By convention, terms with the highest power ( exponent), are written on the left, for example, x^2 is written to the left of x. When a coefficient is one, it is usually omitted (e.g. 1x^2 is written x^2). Likewise when the exponent (power) is one, (e.g. 3x^1 is written 3x), and, when the exponent is zero, the result is always 1 (e.g. 3x^0 is written 3, since x^0 is always 1).Jerome E. Kaufmann, Karen L. Schwitters, ''Algebra for College Students'', Publisher Cengage Learning, 2010, , 9780538733540, 803 pages
page 222
/ref>


Algebraic and other mathematical expressions

The table below summarizes how algebraic expressions compare with several other types of mathematical expressions by the type of elements they may contain, according to common but not universal conventions. A ''rational algebraic expression'' (or ''rational expression'') is an algebraic expression that can be written as a
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of polynomials, such as . An ''irrational algebraic expression'' is one that is not rational, such as .


See also

*
Algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form :P = 0 where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term ''algebraic equation'' ...
*
Algebraic function In mathematics, an algebraic function is a function that can be defined as the root of a polynomial equation. Quite often algebraic functions are algebraic expressions using a finite number of terms, involving only the algebraic operations addit ...
*
Analytical expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
*
Arithmetic expression In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, fu ...
*
Closed-form expression In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th r ...
* Expression (mathematics) *
Precalculus In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus. Schools often distinguish between algebra and trigonomet ...
* Polynomial * Term (logic)


Notes


References

*


External links

* {{MathWorld, title=Algebraic Expression, id=AlgebraicExpression Elementary algebra