In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an algebraic expression is an
expression built up from
constants (usually,
algebraic number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s),
variables, and the basic
algebraic operation
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These o ...
s:
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
(+),
subtraction
Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...
(-),
multiplication
Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
(×),
division (÷), whole number
powers, and
root
In vascular plants, the roots are the plant organ, organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often bel ...
s (
fraction
A fraction (from , "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, thre ...
al powers).. For example, is an algebraic expression. Since taking the
square root
In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
is the same as raising to the power , the following is also an algebraic expression:
:
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 example, x^5-3x+1=0 is an algebraic equati ...
'' is an
equation
In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
involving
polynomials
In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative int ...
, for which algebraic expressions may be
solutions
Solution may refer to:
* Solution (chemistry), a mixture where one substance is dissolved in another
* Solution (equation), in mathematics
** Numerical solution, in numerical analysis, approximate solutions within specified error bounds
* Solutio ...
.
If you restrict your set of constants to be
numbers
A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
, 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 mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
. If you restrict your constants to
integers
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
, the set of numbers that can be described with an algebraic expression are called
Algebraic numbers.
By contrast,
transcendental number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . ...
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. More generally, expressions which are
algebraically independent
In abstract algebra, a subset S of a field L is algebraically independent over a subfield K if the elements of S do not satisfy any non- trivial polynomial equation with coefficients in K.
In particular, a one element set \ is algebraically i ...
from their constants and/or variables are called
transcendental.
Terminology
Algebra
Algebra is a branch of mathematics that deals with 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 ope ...
has its own terminology to describe parts of an expression:
1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant, - variables
Conventions
Variables
By convention, letters at the beginning of the alphabet (e.g.
) 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.
and
) are used to represent
variables. They are usually written in italics.
Exponents
By convention, terms with the highest power (
exponent
In mathematics, exponentiation, denoted , is an operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, i ...
), are written on the left, for example,
is written to the left of
. When a coefficient is one, it is usually omitted (e.g.
is written
). Likewise when the exponent (power) is one, (e.g.
is written
), and, when the exponent is zero, the result is always 1 (e.g.
is written
, since
is always
).
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 focusin ...
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 (mathematics), field, often the field of the rational numbers.
For example, x^5-3x+1=0 is a ...
, can always be written as algebraic expressions if ''n'' < 5 (see
quadratic formula
In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation. Other ways of solving quadratic equations, such as completing the square, yield the same solutions.
Given a general quadr ...
,
cubic function, and
quartic equation
In mathematics, a quartic equation is one which can be expressed as a ''quartic function'' equaling zero. The general form of a quartic equation is
:ax^4+bx^3+cx^2+dx+e=0 \,
where ''a'' ≠ 0.
The quartic is the highest order polynom ...
). Such a solution of an equation is called an
algebraic solution. 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''
5.
Rational expressions
Given two polynomials and
, their
quotient
In arithmetic, a quotient (from 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
is called a ''rational expression'' or simply ''
rational fraction''. A rational expression
is called ''proper'' if
, and ''improper'' otherwise. For example, the fraction
is proper, and the fractions
and
are improper. Any improper rational fraction can be expressed as the sum of a polynomial (possibly constant) and a proper rational fraction. In the first example of an improper fraction one has
:
where the second term is a proper rational fraction. The sum of two proper rational fractions is a proper rational fraction as well. The reverse process of expressing a proper rational fraction as the sum of two or more fractions is called resolving it into
partial fractions. For example,
:
Here, the two terms on the right are called partial fractions.
Irrational fraction
An ''irrational fraction'' is one that contains the variable under a fractional exponent.
An example of an irrational fraction is
:
The process of transforming an irrational fraction to a rational fraction is known as
rationalization. Every irrational fraction in which the radicals are
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called a power product or primitive monomial, is a product of powers of variables with n ...
s may be rationalized by finding the
least common multiple
In arithmetic and number theory, the least common multiple (LCM), lowest common multiple, or smallest common multiple (SCM) of two integers ''a'' and ''b'', usually denoted by , is the smallest positive integer that is divisible by both ''a'' and ...
of the indices of the roots, and substituting the variable for another variable with the least common multiple as exponent. In the example given, the least common multiple is 6, hence we can substitute
to obtain
:
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 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division (in th ...
of
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s, such as . An ''irrational algebraic expression'' is one that is not rational, such as .
See also
*
Algebraic function
*
Analytical expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
*
Closed-form expression
In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
*
Expression (mathematics)
In mathematics, an expression is a written arrangement of symbol (mathematics), symbols following the context-dependent, syntax (logic), syntactic conventions of mathematical notation. Symbols can denote numbers, variable (mathematics), variabl ...
*
Precalculus
In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level that is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish betwe ...
*
Term (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a wh ...
Notes
References
*
External links
*
{{Authority control
Elementary algebra