mathematical expressions
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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, an expression or mathematical expression is a finite combination of
symbols A symbol is a mark, sign, or word In linguistics Linguistics is the scientific study of language A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed languag ...
that is
well-formed Well-formed or wellformed indicate syntactic correctness and may refer to: * Well-formedness, quality of linguistic elements that conform to grammar rules * Well-formed formula, a string that is generated by a formal grammar in logic * Well-formed ...
according to rules that depend on the context. Mathematical symbols can designate numbers (
constant Constant or The Constant may refer to: Mathematics * Constant (mathematics) In mathematics, the word constant can have multiple meanings. As an adjective, it refers to non-variance (i.e. unchanging with respect to some other Value (mathematics ...
s), variables,
operation Operation or Operations may refer to: Arts, entertainment and media * Operation (game), ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * Operations (magazine ...
s,
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s,
bracket A bracket is either of two tall fore- or back-facing punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding ...
s, punctuation, and grouping to help determine
order of operations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

order of operations
and other aspects of logical syntax. Many authors distinguish an expression from a
formula In science Science () is a systematic enterprise that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe."... modern science is a discovery as well a ...

formula
, the former denoting a mathematical object, and the latter denoting a statement about mathematical objects. For example, 8x-5 is an expression, while 8x-5 \geq 5x-8 is a formula. However, in modern mathematics, and in particular in
computer algebra In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
, formulas are viewed as expressions that can be evaluated to ''true'' or ''false'', depending on the values that are given to the variables occurring in the expressions. For example 8x-5 \geq 5x-8 takes the value ''false'' if is given a value less than –1, and the value ''true'' otherwise.


Examples

The use of expressions ranges from the simple: ::3+8 ::8x-5   (
linear polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
) ::7+4x-10   (
quadratic polynomial In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
) ::\frac   (
rational fraction In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In ...
) to the complex: ::f(a)+\sum_^n\left.\frac\frac\_f(u(t)) + \int_0^1 \frac \frac f(u(t))\, dt.


Syntax versus semantics


Syntax

An expression is a syntactic construct. It must be
well-formed Well-formed or wellformed indicate syntactic correctness and may refer to: * Well-formedness, quality of linguistic elements that conform to grammar rules * Well-formed formula, a string that is generated by a formal grammar in logic * Well-formed ...
: the allowed
operator Operator may refer to: Mathematics * A symbol indicating a mathematical operation * Logical operator or logical connective in mathematical logic * Operator (mathematics), mapping that acts on elements of a space to produce elements of another sp ...
s must have the correct number of inputs in the correct places, the characters that make up these inputs must be valid, have a clear
order of operations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

order of operations
, etc. Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example, in the usual notation of
arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cr ...
, the expression ''1 + 2 × 3'' is well-formed, but the following expression is not: :\times4)x+,/y.


Semantics

Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions. In
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
, an expression may be used to designate a value, which might depend on values assigned to variables occurring in the expression. The determination of this value depends on the
semantics Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference Reference is a relationship between objects in which one object designates, or acts as a means by which to connect to or link to, another ...
attached to the symbols of the expression. The choice of semantics depends on the context of the expression. The same syntactic expression ''1 + 2 × 3'' can have different values (mathematically 7, but also 9), depending on the
order of operations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

order of operations
implied by the context (See also
Operations § Calculators
Operations § Calculators
). The semantic rules may declare that certain expressions do not designate any value (for instance when they involve division by 0); such expressions are said to have an undefined value, but they are well-formed expressions nonetheless. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an
equation In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

equation
that is to be solved, or it can be viewed as an object in its own right that can be manipulated according to certain rules. Certain expressions that designate a value simultaneously express a condition that is assumed to hold, for instance those involving the operator \oplus to designate an internal
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
.


Formal languages and lambda calculus

Formal languages allow formalizing the concept of well-formed expressions. In the 1930s, a new type of expressions, called
lambda expressions Lambda calculus (also written as ''λ''-calculus) is a formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theor ...
, were introduced by
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American American(s) may refer to: * American, something of, from, or related to the United States of America, commonly known as the United States The United States of America (US ...
and
Stephen Kleene Stephen Cole Kleene ( ; January 5, 1909 – January 25, 1994) was an American mathematician. One of the students of Alonzo Church, Kleene, along with Rózsa Péter, Alan Turing, Emil Post, and others, is best known as a founder of the branch of m ...
for formalizing
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
s and their evaluation. They form the basis for
lambda calculus Lambda calculus (also written as ''λ''-calculus) is a formal system A formal system is an used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, ar ...
, a
formal system A formal system is an abstract structure used for inferring theorems from axioms according to a set of rules. These rules, which are used for carrying out the inference of theorems from axioms, are the logical calculus of the formal system. A for ...
used in
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal sys ...
and the theory of programming languages. The equivalence of two lambda expressions is undecidable. This is also the case for the expressions representing real numbers, which are built from the integers by using the arithmetical operations, the logarithm and the exponential (
Richardson's theorem In mathematics, Richardson's theorem establishes a limit on the extent to which an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a compu ...
).


Variables

Many mathematical expressions include variables. Any variable can be classified as being either a
free variable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
or a
bound variable In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
. For a given combination of values for the free variables, an expression may be evaluated, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
whose inputs are the values assigned to the free variables and whose output is the resulting value of the expression. For example, the expression : x/y evaluated for ''x'' = 10, ''y'' = 5, will give 2; but it is
undefined Undefined may refer to: Mathematics * Undefined (mathematics), with several related meanings ** Indeterminate form, in calculus Computing * Undefined behavior, computer code whose behavior is not specified under certain conditions * Undefined v ...
for ''y'' = 0. The evaluation of an expression is dependent on the definition of the mathematical operators and on the system of values that is its context. Two expressions are said to be equivalent if, for each combination of values for the free variables, they have the same output, i.e., they represent the same function. Example: The expression :\sum_^ (2nx) has free variable ''x'', bound variable ''n'', constants 1, 2, and 3, two occurrences of an implicit multiplication operator, and a summation operator. The expression is equivalent to the simpler expression 12''x''. The value for ''x'' = 3 is 36.


See also

*
Algebraic closure In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
*
Algebraic expressionIn mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...
*
Analytic expression Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles". Analytic can also have the following meanings: Natural sciences Chemistry * ...
*
Closed-form expression In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...
* Combinator * Computer algebra#Expressions, Computer algebra expression * Defined and undefined * Equation * Expression (programming) * Formal grammar * Formula * Functional programming * Logical expression * Term (logic)


Notes


References

* {{Mathematical logic Abstract algebra Logical expressions bg:Израз pl:Wyrażenie algebraiczne