Expression (mathematics)
Encyclopedia
In mathematics, an expression is a finite combination of symbols that is well-formed
according to rules that depend on the context. Symbols can designate numbers (constants
), variables
, operations
, functions
, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic
symbols. The use of expressions can range from the simple:
to the complex:
.
Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example:
would not be considered a mathematical expression but only a meaningless jumble.
In algebra
an expression may be used to designate a value, which might depend on values assigned to variable
s occurring in the expression; the determination of this value depends on the semantics
attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; 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
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
to designate an internal direct sum
.
Being an expression is a syntactic concept; although different mathematical fields have different notions of valid expressions, the values associated to variables does not play a role. See formal language
for general considerations on how expressions are constructed, and formal semantics for questions concerning attaching meaning (values) to expressions.
. Any variable can be classified as being either a free variable or a bound variable.
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
whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.
For example, the expression
evaluated for x = 10, y = 5, will give 2; but is undefined
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
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 with the simpler expression 12x. The value for x = 3 is 36.
The '+' and '−' (addition and subtraction) symbols have their usual meanings. Division can be expressed either with the '/' or with a horizontal dash. Thus
are perfectly valid. Also, for multiplication one can use the symbols '×' or a '·' (mid dot), or else simply omit it (multiplication is implicit); so:
are all acceptable. However, notice in the first example above how the "times" symbol resembles the letter 'x' and also how the '·' symbol resembles a decimal point, so to avoid confusion it's best to use one of the later two forms.
An expression must be well-formed
. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic.
Expressions and their evaluation were formalised
by Alonzo Church
and Stephen Kleene in the 1930s in their lambda calculus
. The lambda calculus has been a major influence in the development of modern mathematics and computer programming language
s.
One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable
. This is also true of any expression in any system that has power equivalent to the lambda calculus.
Well-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
according to rules that depend on the context. Symbols can designate numbers (constants
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
), variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
, operations
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
, functions
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
, and other mathematical symbols, as well as punctuation, symbols of grouping, and other syntactic
Syntax (logic)
In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them...
symbols. The use of expressions can range from the simple:
to the complex:
.Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example:
would not be considered a mathematical expression but only a meaningless jumble.
In algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
an expression may be used to designate a value, which might depend on values assigned to variable
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
s occurring in the expression; the determination of this value depends on the semantics
Semantics
Semantics is the study of meaning. It focuses on the relation between signifiers, such as words, phrases, signs and symbols, and what they stand for, their denotata....
attached to the symbols of the expression. These semantic rules may declare that certain expressions do not designate any value; 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
Equation
An equation is a mathematical statement that asserts the equality of two expressions. In modern notation, this is written by placing the expressions on either side of an equals sign , for examplex + 3 = 5\,asserts that x+3 is equal to 5...
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
to designate an internal direct sumDirect sum
In mathematics, one can often define a direct sum of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets , together with a suitably defined structure. More abstractly, the direct sum is often, but not always, the coproduct in the category in question...
.
Being an expression is a syntactic concept; although different mathematical fields have different notions of valid expressions, the values associated to variables does not play a role. See formal language
Formal language
A formal language is a set of words—that is, finite strings of letters, symbols, or tokens that are defined in the language. The set from which these letters are taken is the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar...
for general considerations on how expressions are constructed, and formal semantics for questions concerning attaching meaning (values) to expressions.
Variables
Many mathematical expressions include letters called variablesVariable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. Any variable can be classified as being either a free variable or a bound variable.
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 (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.
For example, the expression
evaluated for x = 10, y = 5, will give 2; but is undefined
Division by zero
In mathematics, division by zero is division where the divisor is zero. Such a division can be formally expressed as a / 0 where a is the dividend . Whether this expression can be assigned a well-defined value depends upon the mathematical setting...
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
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 with the simpler expression 12x. The value for x = 3 is 36.
The '+' and '−' (addition and subtraction) symbols have their usual meanings. Division can be expressed either with the '/' or with a horizontal dash. Thus
are perfectly valid. Also, for multiplication one can use the symbols '×' or a '·' (mid dot), or else simply omit it (multiplication is implicit); so:
are all acceptable. However, notice in the first example above how the "times" symbol resembles the letter 'x' and also how the '·' symbol resembles a decimal point, so to avoid confusion it's best to use one of the later two forms.
An expression must be well-formed
Well-formed formula
In mathematical logic, a well-formed formula, shortly wff, often simply formula, is a word which is part of a formal language...
. That is, the operators must have the correct number of inputs, in the correct places. The expression 2 + 3 is well formed; the expression * 2 + is not, at least, not in the usual notation of arithmetic.
Expressions and their evaluation were formalised
Formal system
In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive an expression from one or more other premises that are antecedently supposed or derived . The axioms and rules may be called a deductive apparatus...
by Alonzo Church
Alonzo Church
Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Frege–Church ontology, and the Church–Rosser theorem.-Life:Alonzo Church...
and Stephen Kleene in the 1930s in their lambda calculus
Lambda calculus
In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. The portion of lambda calculus relevant to computation is now called the untyped lambda calculus...
. The lambda calculus has been a major influence in the development of modern mathematics and computer programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
s.
One of the more interesting results of the lambda calculus is that the equivalence of two expressions in the lambda calculus is in some cases undecidable
Decision problem
In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem...
. This is also true of any expression in any system that has power equivalent to the lambda calculus.