Editing Expression (mathematics) (section)

==Types of expressions==
===Algebraic expression===
An ''[[algebraic expression]]'' is an expression built up from [[Algebraic number|algebraic constants]], [[Variable (mathematics)|variables]], and the [[algebraic operation]]s ([[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]] and [[exponentiation]] by a [[rational number]]).<ref>{{cite book |last1=Morris |first1=Christopher G. |title=Academic Press dictionary of science and technology |publisher=Gulf Professional Publishing |page=[https://archive.org/details/academicpressdic00morr/page/74 74] |year=1992 |url=https://archive.org/details/academicpressdic00morr|url-access=registration |quote=algebraic expression over a field. }}</ref> For example, {{math|1=3''x''<sup>2</sup> − 2''xy'' + ''c''}} is an algebraic expression. Since taking the [[square root]] is the same as raising to the power {{sfrac|1|2}}, the following is also an algebraic expression:
:<math>\sqrt{\frac{1-x^2}{1+x^2}}</math>
See also: [[Algebraic equation]] and [[Algebraic closure]]

===Polynomial expression===

A [[polynomial expression]] is an expression built with [[scalar (mathematics)|scalar]]s (numbers of elements of some field), [[indeterminate (variable)|indeterminate]]s, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers; for example <math>3(x+1)^2 - xy.</math>

Using [[associativity]], [[commutativity]] and [[distributivity]], every polynomial expression is equivalent to a [[polynomial]], that is an expression that is a [[linear combination]] of products of integer powers of the indeterminates. For example the above polynomial expression is equivalent (denote the same polynomial as <math>3x^2-xy+6x+3.</math>

Many author do not distinguish polynomials and polynomial expressions. In this case the expression of a polynomial expression as a linear combination is called the ''canonical form'', ''normal form'', or ''expanded form'' of the polynomial.

===Computational expression===
{{Main|Expression (computer science)}}
In [[computer science]], an ''expression'' is a [[Syntax (programming languages)|syntactic]] entity in a [[programming language]] that may be evaluated to determine its [[value (computer science)|value]]<ref>[[John C. Mitchell|Mitchell, J.]] (2002). Concepts in Programming Languages. Cambridge: Cambridge University Press, ''3.4.1 Statements and Expressions'', p. 26</ref> or fail to terminate, in which case the expression is undefined.<ref>Maurizio Gabbrielli, Simone Martini (2010). Programming Languages - Principles and Paradigms. Springer London, ''6.1 Expressions'', p. 120</ref> It is a combination of one or more [[Constant (programming)|constants]], [[variable (programming)|variable]]s, [[function (programming)|function]]s, and [[operator (programming)|operator]]s that the programming language interprets (according to its particular [[Order of operations|rules of precedence]] and of [[Associative property|association]]) and computes to produce ("to return", in a [[state (computer science)|stateful]] environment) another value. This process, for mathematical expressions, is called ''evaluation''.
In simple settings, the [[return type|resulting value]] is usually one of various [[primitive data type|primitive types]], such as [[string (computer science)|string]], [[Boolean expression|Boolean]], or numerical (such as [[integer (computer science)|integer]], [[floating-point number|floating-point]], or [[complex data type|complex]]).

In [[computer algebra]], formulas are viewed as expressions that can be evaluated as a Boolean, depending on the values that are given to the variables occurring in the expressions. For example <math>8x-5 \geq 3</math> takes the value ''false'' if {{mvar|x}} is given a value less than 1, and the value ''true'' otherwise.

Expressions are often contrasted with [[Statement (computer science)|statements]]—syntactic entities that have no value (an instruction).

[[File:Cassidy.1985.015.gif|thumb|400px|Representation of the expression {{math|(8 − 6) × (3 + 1)}} as a [[Lisp (programming language)|Lisp]] tree, from a 1985 Master's Thesis<ref>{{cite thesis | type=Master's thesis | url=https://commons.wikimedia.org/wiki/File:The_feasibility_of_automatic_storage_reclamation_with_concurrent_program_execution_in_a_LISP_environment._(IA_feasibilityofaut00cass).pdf | first=Kevin G. |last=Cassidy | title=The Feasibility of Automatic Storage Reclamation with Concurrent Program Execution in a LISP Environment | institution=Naval Postgraduate School, Monterey/CA | date=Dec 1985 |page=15 |id=ADA165184}}</ref>]]
Except for [[number]]s and [[variable (mathematics)|variables]], every mathematical expression may be viewed as the symbol of an operator followed by a [[sequence]] of operands. In computer algebra software, the expressions are usually represented in this way. This representation is very flexible, and many things that seem not to be mathematical expressions at first glance, may be represented and manipulated as such. For example, an equation is an expression with "=" as an operator, a [[Matrix (mathematics)|matrix]] may be represented as an expression with "matrix" as an operator and its rows as operands.

See: [[Computer algebra#Expressions|Computer algebra expression]]

===Logical expression===
In [[mathematical logic]], a ''"logical expression"'' can refer to either [[Term (logic)|terms]] or [[Well-formed formula#Predicate logic|formulas]]. A term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula.

A [[first-order logic|first-order]] term is [[recursive definition|recursively constructed]] from constant symbols, variables, and [[function symbol (logic)|function symbols]].
An expression formed by applying a [[predicate (logic)|predicate symbol]] to an appropriate number of terms is called an [[atomic formula]], which evaluates to [[Truth#Truth in mathematics|true]] or [[False (logic)|false]] in [[Principle of bivalence|bivalent logics]], given an [[interpretation (logic)|interpretation]].
For example, {{tmath|(x+1)*(x+1)}} is a term built from the constant 1, the variable {{mvar|x}}, and the binary function symbols {{tmath|+}} and {{tmath|*}}; it is part of the atomic formula {{tmath|(x+1)*(x+1) \ge 0}} which evaluates to true for each [[real number|real-numbered]] value of {{mvar|x}}.

===Formal expression===
{{See also|Regular expression}}
A '''formal expression''' is a kind of [[String (computer science)|string]] of [[Symbol (formal)|symbols]], created by the same [[Expression (mathematics)#Formal definition|production rules]] as standard expressions, however, they are used without regard to the meaning of the expression. In this way, two ''formal expressions'' are considered equal only if they are [[Syntax (logic)|syntactically]] equal, that is, if they are the exact same expression.<ref>{{Cite book |last=McCoy |first=Neal H. |url=https://archive.org/details/introductiontomo00mcco/page/126/mode/2up?q=%22purely+formal+expression%22 |title=Introduction To Modern Algebra |publisher=[[Allyn & Bacon]] |year=1960 |location=Boston |pages=127 |lccn=68015225}}</ref><ref>{{Cite book |last=Fraleigh |first=John B. |url=https://archive.org/details/firstcourseinabs07edfral/page/198/mode/2up?q=%22formal+sum%22 |title=A first course in abstract algebra |date=2003 |publisher=Boston : Addison-Wesley |isbn=978-0-201-76390-4}}</ref> For instance, the formal expressions "2" and "1+1" are not equal.