Editing Expression (mathematics) (section)

==Well-defined expressions==
{{Main|Well-defined expression}}

The [[language of mathematics]] exhibits a kind of [[grammar]] (called [[formal grammar]]) about how expressions may be written. There are two considerations for well-definedness of mathematical expressions, [[Syntax (logic)|syntax]] and [[Formal semantics (natural language)|semantics]]. Syntax is concerned with the rules used for constructing, or transforming the symbols of an expression without regard to any [[Interpretation (logic)|interpretation]] or [[Meaning (linguistics)|meaning]] given to them. Expressions that are syntactically correct are called [[Well-formedness|well-formed]]. Semantics is concerned with the meaning of these well-formed expressions. Expressions that are semantically correct are called [[Well-defined expression|well-defined]].

=== Well-formed ===
The syntax of mathematical expressions can be described somewhat informally as follows: the allowed [[operator (mathematics)|operator]]s must have the correct number of inputs in the correct places (usually written with [[infix notation]]), the sub-expressions that make up these inputs must be well-formed themselves, have a clear [[order of operations]], etc. Strings of symbols that conform to the rules of syntax are called [[Well-formedness|''well-formed'']], and those that are not well-formed are called, ''ill-formed'', and do not constitute mathematical expressions.<ref>{{cite book |last=Stoll |first=Robert R. |title=Set Theory and Logic |publisher=Dover Publications |isbn=978-0-486-63829-4 |location=San Francisco, CA |year=1963}}</ref>

For example, in [[arithmetic]], the expression ''1 + 2 × 3'' is well-formed, but  
:<math>\times4)x+,/y</math>.
is not.

However, being well-formed is not enough to be considered well-defined. For example in arithmetic, the expression <math display="inline">\frac{1}{0}</math> is well-formed, but it is not well-defined. (See [[Division by zero]]). Such expressions are called [[Undefined (mathematics)|undefined]].

===Well-defined===
[[Semantics]] is the study of meaning. [[Formal semantics (natural language)|Formal semantics]] is about attaching meaning to expressions. An expression that defines a unique [[Value (mathematics)|value]] or meaning is said to be [[Well-defined expression|well-defined]]. Otherwise, the expression is said to be ill defined or ambiguous.'''<ref name="MathWorld2">{{cite web |last=Weisstein |first=Eric W. |title=Well-Defined |url=http://mathworld.wolfram.com/Well-Defined.html |access-date=2 January 2013 |publisher=From MathWorld – A Wolfram Web Resource}}</ref>''' In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an [[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 <math>\oplus</math> to designate an internal [[direct sum]].

In [[algebra]], an expression may be used to designate a value, which might depend on values assigned to [[variable (mathematics)|variable]]s occurring in the expression. The determination of this value depends on the [[semantics]] 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]] implied by the context (See also [[Order of operations#Calculators|Operations § Calculators]]).

For [[real number]]s, the product <math>a \times b \times c</math> is unambiguous because <math>(a \times b)\times c = a \times (b \times c)</math>; hence the notation is said to be ''well defined''.<ref name="MathWorld2"/> This property, also known as [[associativity]] of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore, a specification of the sequence can be omitted. The [[subtraction]] operation is non-associative; despite that, there is a convention that <math>a-b-c</math> is shorthand for <math>(a-b)-c</math>, thus it is considered "well-defined". On the other hand, [[Division (mathematics)|Division]] is non-associative, and in the case of <math>a/b/c</math>, parenthesization conventions are not well established; therefore, this expression is often considered ill-defined.

Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of [[Operator precedence|precedence]], associativity of the operator). For example, in the programming language [[C (programming language)|C]], the operator <code>-</code> for subtraction is ''left-to-right-associative'', which means that <code>a-b-c</code> is defined as <code>(a-b)-c</code>, and the operator <code>=</code> for assignment is ''right-to-left-associative'', which means that <code>a=b=c</code> is defined as <code>a=(b=c)</code>.<ref>{{Cite web |date=2014-02-07 |title=Operator Precedence and Associativity in C |url=https://www.geeksforgeeks.org/operator-precedence-and-associativity-in-c/ |access-date=2019-10-18 |website=GeeksforGeeks |language=en-US}}</ref> In the programming language [[APL (programming language)|APL]] there is only one rule: from [[APL (programming language)#Design|right to left]] – but parentheses first.