In
mathematics, an algebraic expression is an
expression built up from
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
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 operations (
addition,
subtraction,
multiplication,
division and
exponentiation by an exponent that is a
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
). 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:
:
An ''
algebraic equation'' is an
equation involving only algebraic expressions.
By contrast,
transcendental numbers 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 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 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,
:
is a rational expression, whereas
:
is not.
A rational equation is an equation in which two rational fractions (or rational expressions) of the form
:
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 areas of mathematics, 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 mathem ...
has its own terminology to describe parts of an expression:
1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 – constant,
- variables
In roots of polynomials
The
roots of a polynomial expression of
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathemati ...
''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 equati ...
, can always be written as algebraic expressions if ''n'' < 5 (see
quadratic formula,
cubic function, and
quartic equation). Such a solution of an equation is called an
algebraic solution. But the
Abel–Ruffini theorem states that algebraic solutions do not exist for all such equations (just for some of them) if ''n''
5.
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), 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
).
[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 of polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
s, such as . An ''irrational algebraic expression'' is one that is not rational, such as .
See also
* Algebraic equation
* Algebraic function
* 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, f ...
* 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 ro ...
* Expression (mathematics)
* Precalculus
* Polynomial
In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
* Term (logic)
Notes
References
*
External links
* {{MathWorld, title=Algebraic Expression, id=AlgebraicExpression
Elementary algebra