Expression (mathematics)

The

The syntax of mathematical expressions can be described somewhat informally as follows: the allowed operators 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-formed, and those that are not well-formed are called, ill-formed, and do not constitute mathematical expressions.^[36]

For example, in arithmetic, the expression 1 + 2 × 3 is well-formed, but

${\displaystyle \times 4)x+,/y}$.

is not.

However, being well-formed is not enough to be considered well-defined. For example in arithmetic, the expression ${\textstyle {\frac {1}{0}}}$ is well-formed, but it is not well-defined. (See Division by zero). Such expressions are called undefined.

Semantics is the study of meaning. Formal semantics is about attaching meaning to expressions. An expression that defines a unique value or meaning is said to be well-defined. Otherwise, the expression is said to be ill defined or ambiguous.^[37] 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 ${\displaystyle \oplus }$ to designate an internal direct sum.

In 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 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 Operations § Calculators).

For real numbers, the product ${\displaystyle a\times b\times c}$ is unambiguous because ${\displaystyle (a\times b)\times c=a\times (b\times c)}$; hence the notation is said to be well defined.

operation is non-associative; despite that, there is a convention that ${\displaystyle a-b-c}$ is shorthand for ${\displaystyle (a-b)-c}$, thus it is considered "well-defined". On the other hand,

${\displaystyle a/b/c}$

Unlike with functions, notational ambiguities can be overcome by means of additional definitions (e.g., rules of

The term 'expression' is part of the language of mathematics, that is to say, it is not defined within mathematics, but taken as a primitive part of the language. To attempt to define the term would not be doing mathematics, but rather, one would be engaging in a kind of metamathematics (the metalanguage of mathematics), usually mathematical logic. Within mathematical logic, mathematics is usually described as a kind of formal language, and a well-formed expression can be defined recursively as follows:^[22]

The alphabet consists of:

• A set of individual constants: Symbols representing fixed objects in the domain of discourse, such as numerals (1, 2.5, 1/7, ...), sets (${\displaystyle \varnothing ,\{1,2,3\}}$, ...),
  truth values
  (T or F), etc.
• A set of individual variables: A countably infinite amount of symbols representing variables used for representing an unspecified object in the domain. (Usually letters like x, or y)
• A set of operations:
  unary operations
  )
• Brackets ( )

With this alphabet, the recursive rules for forming a well-formed expression (WFE) are as follows:

• Any constant or variable as defined are the atomic expressions, the simplest well-formed expressions (WFE's). For instance, the constant ${\displaystyle 2}$ or the variable ${\displaystyle x}$ are syntactically correct expressions.
• Let ${\displaystyle F}$ be a
  n-ary operation
  over the domain, and let ${\displaystyle \phi _{1},\phi _{2},...\phi _{n}}$ be metavariables for any WFE's.

Then ${\displaystyle F(\phi _{1},\phi _{2},...\phi _{n})}$ is also well-formed. For the most often used operations, more convenient notations (like infix notation) have been developed over the centuries.

For instance, if the domain of discourse is the real numbers, ${\displaystyle F}$ can denote the binary operation +, then ${\displaystyle \phi _{1}+\phi _{2}}$ is well-formed. Or ${\displaystyle F}$ can be the unary operation ${\displaystyle \surd }$ so ${\displaystyle {\sqrt {\phi _{1}}}}$ is well-formed.

Brackets are initially around each non-atomic expression, but they can be deleted in cases where there is a defined order of operations, or where order doesn't matter (i.e. where operations are associative).

A well-formed expression can be thought as a syntax tree.^[39] The leaf nodes are always atomic expressions. Operations ${\displaystyle +}$ and ${\displaystyle \cup }$ have exactly two child nodes, while operations ${\textstyle {\sqrt {x}}}$, ${\textstyle {\text{ln}}(x)}$ and ${\textstyle {\frac {d}{dx}}}$ have exactly one. There are countably infinitely many WFE's, however, each WFE has a finite number of nodes.

Formal languages allow formalizing the concept of well-formed expressions.

In the 1930s, a new type of expression, the

The equivalence of two lambda expressions is undecidable (but see unification (computer science)). 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).

An algebraic expression is an expression built up from algebraic constants, variables, and the algebraic operations (addition, subtraction, multiplication, division and exponentiation by a rational number).^[41] For example, 3x² − 2xy + c is an algebraic expression. Since taking the square root is the same as raising to the power ⁠1/2⁠, the following is also an algebraic expression:

${\displaystyle {\sqrt {\frac {1-x^{2}}{1+x^{2}}}}}$

See also: Algebraic equation and Algebraic closure

A

${\displaystyle 3(x+1)^{2}-xy.}$

Using

of products of integer powers of the indeterminates. For example the above polynomial expression is equivalent (denote the same polynomial as ${\displaystyle 3x^{2}-xy+6x+3.}$

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

In

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 ${\displaystyle 8x-5\geq 3}$ takes the value false if x is given a value less than 1, and the value true otherwise.

Expressions are often contrasted with statements—syntactic entities that have no value (an instruction).

Except for numbers and 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 may be represented as an expression with "matrix" as an operator and its rows as operands.

See: Computer algebra expression

In mathematical logic, a "logical expression" can refer to either terms or formulas. A term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula.

A

⁠${\displaystyle (x+1)*(x+1)}$⁠

⁠${\displaystyle +}$⁠

⁠${\displaystyle *}$⁠

⁠${\displaystyle (x+1)*(x+1)\geq 0}$⁠

A formal expression is a kind of string of symbols, created by the same 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 syntactically equal, that is, if they are the exact same expression.^[45][46] For instance, the formal expressions "2" and "1+1" are not equal.