Editing Expression (mathematics) (section)

{{Short description|Symbolic description of a mathematical object}}
{{other uses| Expression (disambiguation)}}
{{use dmy dates|date=July 2020|cs1-dates=y}}
[[File:Equation_vs_Expression.png|thumb|239x239px|In the [[equation]] 7x − 5 = 2, the [[Sides of an equation|sides of the equation]] are expressions.]]
In [[mathematics]], an '''expression''' is a written arrangement of [[symbol (mathematics)|symbol]]s following the context-dependent, [[syntax (logic)|syntactic]] conventions of [[mathematical notation]].  Symbols can denote [[numbers]], [[variable (mathematics)|variable]]s, [[operation (mathematics)|operation]]s, and [[function (mathematics)|function]]s.<ref>[[Oxford English Dictionary]], s.v. “[[doi:10.1093/OED/4555505636|Expression (n.), sense II.7]],” "''A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function.''"</ref> Other symbols include [[punctuation]] marks and [[bracket (mathematics)|bracket]]s, used for [[Symbols of grouping|grouping]] where there is not a well-defined [[order of operations]].

Expressions are commonly distinguished from ''[[mathematical formula|formulas]]'': expressions are a kind of [[mathematical object]], whereas formulas are statements ''about'' mathematical objects.<ref>{{cite book|first=Robert R.|last=Stoll|year=1963|title=Set Theory and Logic|publisher=Dover Publications|location=San Francisco, CA|isbn=978-0-486-63829-4}}</ref> This is analogous to [[natural language]], where a [[noun phrase]] refers to an object, and a whole [[Sentence (linguistics)|sentence]] refers to a [[fact]]. For example, <math>8x-5</math> is an expression, while the [[Inequality (mathematics)|inequality]] <math>8x-5 \geq 3 </math> is a formula.

To ''evaluate'' an expression means to find a numerical [[Value (mathematics)|value]] equivalent to the expression.<ref>[[Oxford English Dictionary]], s.v. "[[doi:10.1093/OED/3423541985|Evaluate (v.), sense a]]", "''Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation).''"</ref><ref>[[Oxford English Dictionary]], s.v. “[[doi:10.1093/OED/1018661347|Simplify (v.), sense 4.a]]”, "''To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables.''"</ref> Expressions can be ''evaluated'' or ''simplified'' by replacing [[Operation (mathematics)|operations]] that appear in them with their result. For example, the expression <math>8\times 2-5</math> simplifies to <math>16-5</math>, and evaluates to <math>11.</math>

An expression is often used to define a [[Function (mathematics)|function]], by taking the variables to be [[Argument of a function|arguments]], or inputs, of the function, and assigning the output to be the evaluation of the resulting expression.<ref name="Codd1970">{{cite journal |last1=Codd |first1=Edgar Frank 
|authorlink=Edgar F. Codd|date=June 1970 |title=A Relational Model of Data for Large Shared Data Banks |url=https://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |archive-url=https://web.archive.org/web/20040908011134/http://www.seas.upenn.edu/~zives/03f/cis550/codd.pdf |archive-date=2004-09-08 |url-status=live |journal=Communications of the ACM |volume=13 |issue=6 |pages=377–387 |doi=10.1145/362384.362685 |s2cid=207549016 |access-date=2020-04-29}}</ref> For example, <math>x\mapsto x^2+1</math> and <math>f(x) = x^2 + 1</math>  define the function that associates to each number its [[square function|square]] plus one. An expression with no variables would define a [[constant function]]. Usually, two expressions are considered [[Equality (mathematics)|equal]] or ''equivalent'' if they define the same function. Such an equality is called a "[[Formal semantics (natural language)|semantic]] equality", that is, both expressions "mean the same thing."