expression (mathematics)
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an expression is a written arrangement of
symbol A symbol is a mark, Sign (semiotics), sign, or word that indicates, signifies, or is understood as representing an idea, physical object, object, or wikt:relationship, relationship. Symbols allow people to go beyond what is known or seen by cr ...
s following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers, variables, operations, and functions. Other symbols include
punctuation Punctuation marks are marks indicating how a piece of writing, written text should be read (silently or aloud) and, consequently, understood. The oldest known examples of punctuation marks were found in the Mesha Stele from the 9th century BC, c ...
marks and brackets, used for grouping where there is not a well-defined order of operations. Expressions are commonly distinguished from '' formulas'': expressions are a kind of mathematical object, whereas formulas are statements ''about'' mathematical objects. This is analogous to natural language, where a
noun phrase A noun phrase – or NP or nominal (phrase) – is a phrase that usually has a noun or pronoun as its head, and has the same grammatical functions as a noun. Noun phrases are very common cross-linguistically, and they may be the most frequently ...
refers to an object, and a whole sentence refers to a
fact A fact is a truth, true data, datum about one or more aspects of a circumstance. Standard reference works are often used to Fact-checking, check facts. Science, Scientific facts are verified by repeatable careful observation or measurement by ...
. For example, 8x-5 is an expression, while the inequality 8x-5 \geq 3 is a formula. To ''evaluate'' an expression means to find a numerical value equivalent to the expression. Expressions can be ''evaluated'' or ''simplified'' by replacing operations that appear in them with their result. For example, the expression 8\times 2-5 simplifies to 16-5, and evaluates to 11. An expression is often used to define a function, by taking the variables to be
arguments An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...
, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. For example, x\mapsto x^2+1 and f(x) = x^2 + 1 define the function that associates to each number its
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
plus one. An expression with no variables would define a constant function. Usually, two expressions are considered equal or ''equivalent'' if they define the same function. Such an equality is called a "
semantic Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
equality", that is, both expressions "mean the same thing."


History


Early written mathematics

The earliest written mathematics likely began with tally marks, where each mark represented one unit, carved into wood or stone. An example of early counting is the Ishango bone, found near the
Nile The Nile (also known as the Nile River or River Nile) is a major north-flowing river in northeastern Africa. It flows into the Mediterranean Sea. The Nile is the longest river in Africa. It has historically been considered the List of river sy ...
and dating back over 20,000 years ago, which is thought to show a six-month lunar calendar.Marshack, Alexander (1991). ''The Roots of Civilization'', Colonial Hill, Mount Kisco, NY.
Ancient Egypt Ancient Egypt () was a cradle of civilization concentrated along the lower reaches of the Nile River in Northeast Africa. It emerged from prehistoric Egypt around 3150BC (according to conventional Egyptian chronology), when Upper and Lower E ...
developed a symbolic system using hieroglyphics, assigning symbols for powers of ten and using addition and subtraction symbols resembling legs in motion. This system, recorded in texts like the Rhind Mathematical Papyrus (c. 2000–1800 BC), influenced other Mediterranean cultures. In
Mesopotamia Mesopotamia is a historical region of West Asia situated within the Tigris–Euphrates river system, in the northern part of the Fertile Crescent. Today, Mesopotamia is known as present-day Iraq and forms the eastern geographic boundary of ...
, a similar system evolved, with numbers written in a base-60 (
sexagesimal Sexagesimal, also known as base 60, is a numeral system with 60 (number), sixty as its radix, base. It originated with the ancient Sumerians in the 3rd millennium BC, was passed down to the ancient Babylonians, and is still used—in a modified fo ...
) format on clay tablets written in
Cuneiform Cuneiform is a Logogram, logo-Syllabary, syllabic writing system that was used to write several languages of the Ancient Near East. The script was in active use from the early Bronze Age until the beginning of the Common Era. Cuneiform script ...
, a technique originating with the Sumerians around 3000 BC. This base-60 system persists today in measuring time and
angle In Euclidean geometry, an angle can refer to a number of concepts relating to the intersection of two straight Line (geometry), lines at a Point (geometry), point. Formally, an angle is a figure lying in a Euclidean plane, plane formed by two R ...
s.


Syncopated stage

The "syncopated" stage of mathematics introduced symbolic abbreviations for commonly used operations and quantities, marking a shift from purely geometric reasoning. Ancient Greek mathematics, largely geometric in nature, drew on Egyptian numerical systems (especially Attic numerals), with little interest in algebraic symbols, until the arrival of Diophantus of
Alexandria Alexandria ( ; ) is the List of cities and towns in Egypt#Largest cities, second largest city in Egypt and the List of coastal settlements of the Mediterranean Sea, largest city on the Mediterranean coast. It lies at the western edge of the Nile ...
, who pioneered a form of syncopated algebra in his '' Arithmetica,'' which introduced symbolic manipulation of expressions. His notation represented unknowns and powers symbolically, but without modern symbols for relations (such as equality or inequality) or exponents.Boyer (1991). "Revival and Decline of Greek Mathematics". p. 178. "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation." An unknown number was called \zeta. The square of \zeta was \Delta^v; the cube was K^v; the fourth power was \Delta^v\Delta; the fifth power was \Delta K^v; and \pitchfork meant to subtract everything on the right from the left. So for example, what would be written in modern notation as: x^3 - 2x^2 + 10x -1, Would be written in Diophantus's syncopated notation as: : \Kappa^ \overline \; \zeta \overline \;\, \pitchfork \;\, \Delta^ \overline \; \Mu \overline \,\; In the 7th century,
Brahmagupta Brahmagupta ( – ) was an Indian Indian mathematics, mathematician and Indian astronomy, astronomer. He is the author of two early works on mathematics and astronomy: the ''Brāhmasphuṭasiddhānta'' (BSS, "correctly established Siddhanta, do ...
used different colours to represent the unknowns in algebraic equations in the '' Brāhmasphuṭasiddhānta''. Greek and other ancient mathematical advances, were often trapped in cycles of bursts of creativity, followed by long periods of stagnation, but this began to change as knowledge spread in the
early modern period The early modern period is a Periodization, historical period that is defined either as part of or as immediately preceding the modern period, with divisions based primarily on the history of Europe and the broader concept of modernity. There i ...
.


Symbolic stage and early arithmetic

The transition to fully symbolic algebra began with Ibn al-Banna' al-Marrakushi (1256–1321) and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī, (1412–1482) who introduced symbols for operations using Arabic characters. The plus sign (+) appeared around 1351 with Nicole Oresme, likely derived from the Latin ''et'' (meaning "and"), while the minus sign (−) was first used in 1489 by Johannes Widmann. Luca Pacioli included these symbols in his works, though much was based on earlier contributions by Piero della Francesca. The radical symbol (√) for square root was introduced by Christoph Rudolff in the 1500s, and parentheses for precedence by Niccolò Tartaglia in 1556. François Viète’s ''New Algebra'' (1591) formalized modern symbolic manipulation. The multiplication sign (×) was first used by William Oughtred and the division sign (÷) by Johann Rahn.
René Descartes René Descartes ( , ; ; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and Modern science, science. Mathematics was paramou ...
further advanced algebraic symbolism in '' La Géométrie'' (1637), where he introduced the use of letters at the end of the alphabet (x, y, z) for variables, along with the
Cartesian coordinate system In geometry, a Cartesian coordinate system (, ) in a plane (geometry), plane is a coordinate system that specifies each point (geometry), point uniquely by a pair of real numbers called ''coordinates'', which are the positive and negative number ...
, which bridged algebra and geometry.
Isaac Newton Sir Isaac Newton () was an English polymath active as a mathematician, physicist, astronomer, alchemist, theologian, and author. Newton was a key figure in the Scientific Revolution and the Age of Enlightenment, Enlightenment that followed ...
and
Gottfried Wilhelm Leibniz Gottfried Wilhelm Leibniz (or Leibnitz; – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat who is credited, alongside Sir Isaac Newton, with the creation of calculus in addition to ...
independently developed
calculus Calculus is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. Originally called infinitesimal calculus or "the ...
in the late 17th century, with Leibniz's notation becoming the standard.


Variables and evaluation

In
elementary algebra Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variable (mathematics ...
, a ''variable'' in an expression is a letter that represents a number whose value may change. To ''evaluate an expression'' with a variable means to find the value of the expression when the variable is assigned a given number. Expressions can be ''evaluated'' or ''simplified'' by replacing operations that appear in them with their result, or by combining like-terms. For example, take the expression 4 x^2 + 8; it can be evaluated at in the following steps: 4(3)^2 + 8, (replace x with 3) 4 \cdot (3 \cdot 3 ) + 8 (use definition of exponent) 4 \cdot 9 + 8 (evaluate inner multiplication) 36 + 8 (evaluate remaining multiplication) 44 (evaluate addition) A ''term'' is a constant or the product of a constant and one or more variables. Some examples include 7, \; 5x, \; 13x^2y, \; 4b The constant of the product is called the
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
. Terms that are either constants or have the same variables raised to the same powers are called '' like terms''. If there are like terms in an expression, one can simplify the expression by combining the like terms. One adds the coefficients and keeps the same variable. 4x + 7x +2x = 13x 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 an operation over constants and free variables and whose output is the resulting value of the expression.; here: Sect.1.3 For a non-formalized language, that is, in most mathematical texts outside of
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, for an individual expression it is not always possible to identify which variables are free and bound. For example, in \sum_ a_, depending on the context, the variable i can be free and k bound, or vice-versa, but they cannot both be free. Determining which value is assumed to be free depends on context and
semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
.


Equivalence

An expression is often used to define a function, or denote compositions of functions, by taking the variables to be
arguments An argument is a series of sentences, statements, or propositions some of which are called premises and one is the conclusion. The purpose of an argument is to give reasons for one's conclusion via justification, explanation, and/or persua ...
, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. For example, x\mapsto x^2+1 and f(x) = x^2 + 1 define the function that associates to each number its
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
plus one. An expression with no variables would define a constant function. In this way, 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. The equivalence between two expressions is called an identity and is sometimes denoted with \equiv. For example, in the expression \sum_^ (2nx), the variable is bound, and the variable is free. This expression is equivalent to the simpler expression ; that is \sum_^ (2nx)\equiv 12x. The value for is 36, which can be denoted \sum_^ (2nx)\Big, _= 36.


Polynomial evaluation

A polynomial consists of variables and
coefficient In mathematics, a coefficient is a Factor (arithmetic), multiplicative factor involved in some Summand, term of a polynomial, a series (mathematics), series, or any other type of expression (mathematics), expression. It may be a Dimensionless qu ...
s, that involve only the operations of
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
,
subtraction Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...
,
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
and
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
to nonnegative integer powers, and has a finite number of terms. The problem of polynomial evaluation arises frequently in practice. In computational geometry, polynomials are used to compute function approximations using Taylor polynomials. In
cryptography Cryptography, or cryptology (from "hidden, secret"; and ''graphein'', "to write", or ''-logy, -logia'', "study", respectively), is the practice and study of techniques for secure communication in the presence of Adversary (cryptography), ...
and
hash table In computer science, a hash table is a data structure that implements an associative array, also called a dictionary or simply map; an associative array is an abstract data type that maps Unique key, keys to Value (computer science), values. ...
s, polynomials are used to compute ''k''-independent hashing. In the former case, polynomials are evaluated using
floating-point arithmetic In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
, which is not exact. Thus different schemes for the evaluation will, in general, give slightly different answers. In the latter case, the polynomials are usually evaluated in a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), s ...
, in which case the answers are always exact. For evaluating the univariate polynomial a_nx^n+a_x^+\cdots +a_0, the most naive method would use n multiplications to compute a_nx^n, use n-1 multiplications to compute a_ x^ and so on for a total of \frac multiplications and n additions. Using better methods, such as Horner's rule, this can be reduced to n multiplications and n additions. If some preprocessing is allowed, even more savings are possible.


Computation

A
computation A computation is any type of arithmetic or non-arithmetic calculation that is well-defined. Common examples of computation are mathematical equation solving and the execution of computer algorithms. Mechanical or electronic devices (or, hist ...
is any type of
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
or non-arithmetic
calculation A calculation is a deliberate mathematical process that transforms a plurality of inputs into a singular or plurality of outputs, known also as a result or results. The term is used in a variety of senses, from the very definite arithmetical ...
that is "well-defined". The notion that mathematical statements should be 'well-defined' had been argued by mathematicians since at least the 1600s, but agreement on a suitable definition proved elusive. A candidate definition was proposed independently by several mathematicians in the 1930s. The best-known variant was formalised by the mathematician
Alan Turing Alan Mathison Turing (; 23 June 1912 – 7 June 1954) was an English mathematician, computer scientist, logician, cryptanalyst, philosopher and theoretical biologist. He was highly influential in the development of theoretical computer ...
, who defined a well-defined statement or calculation as any statement that could be expressed in terms of the initialisation parameters of a
Turing machine A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer algori ...
. Turing's definition apportioned "well-definedness" to a very large class of mathematical statements, including all well-formed algebraic statements, and all statements written in modern computer programming languages. Despite the widespread uptake of this definition, there are some mathematical concepts that have no well-defined characterisation under this definition. This includes
the halting problem In computability theory, the halting problem is the problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. The halting problem is '' undecidab ...
and the busy beaver game. It remains an open question as to whether there exists a more powerful definition of 'well-defined' that is able to capture both computable and 'non-computable' statements. All statements characterised in modern programming languages are well-defined, including C++, Python, and
Java Java is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea (a part of Pacific Ocean) to the north. With a population of 156.9 million people (including Madura) in mid 2024, proje ...
. Common examples of computation are basic
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
and the
execution Capital punishment, also known as the death penalty and formerly called judicial homicide, is the state-sanctioned killing of a person as punishment for actual or supposed misconduct. The sentence ordering that an offender be punished in ...
of computer
algorithms In mathematics and computer science, an algorithm () is a finite sequence of mathematically rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for per ...
. A
calculation A calculation is a deliberate mathematical process that transforms a plurality of inputs into a singular or plurality of outputs, known also as a result or results. The term is used in a variety of senses, from the very definite arithmetical ...
is a deliberate mathematical process that transforms one or more inputs into one or more outputs or ''results''. For example, multiplying 7 by 6 is a simple algorithmic calculation. Extracting the square root or the cube root of a number using mathematical models is a more complex algorithmic calculation.


Rewriting

Expressions can be computed by means of an ''
evaluation strategy In a programming language, an evaluation strategy is a set of rules for evaluating expressions. The term is often used to refer to the more specific notion of a ''parameter-passing strategy'' that defines the kind of value that is passed to the ...
.'' To illustrate, executing a function call f(a,b) may first evaluate the arguments a and b, store the results in
references A reference is a relationship between Object (philosophy), objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to ''refer to'' the second object. ...
or memory locations ref_a and ref_b, then evaluate the function's body with those references passed in. This gives the function the ability to look up the original argument values passed in through dereferencing the parameters (some languages use specific operators to perform this), to modify them via assignment as if they were local variables, and to return values via the references. This is the call-by-reference evaluation strategy. Evaluation strategy is part of the semantics of the programming language definition. Some languages, such as PureScript, have variants with different evaluation strategies. Some
declarative language In computer science, declarative programming is a programming paradigm—a style of building the structure and elements of computer programs—that expresses the logic of a computation without describing its control flow. Many languages that app ...
s, such as Datalog, support multiple evaluation strategies. Some languages define a calling convention. In rewriting, a reduction strategy or rewriting strategy is a relation specifying a rewrite for each object or term, compatible with a given reduction relation. A rewriting strategy specifies, out of all the reducible subterms ( redexes), which one should be reduced (''contracted'') within a term. One of the most common systems involves
lambda calculus In mathematical logic, the lambda calculus (also written as ''λ''-calculus) is a formal system for expressing computability, computation based on function Abstraction (computer science), abstraction and function application, application using var ...
.


Well-defined expressions

The language of mathematics exhibits a kind of
grammar In linguistics, grammar is the set of rules for how a natural language is structured, as demonstrated by its speakers or writers. Grammar rules may concern the use of clauses, phrases, and words. The term may also refer to the study of such rul ...
(called
formal grammar A formal grammar is a set of Terminal and nonterminal symbols, symbols and the Production (computer science), production rules for rewriting some of them into every possible string of a formal language over an Alphabet (formal languages), alphabe ...
) about how expressions may be written. There are two considerations for well-definedness of mathematical expressions,
syntax In linguistics, syntax ( ) is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituenc ...
and
semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
. Syntax is concerned with the rules used for constructing, or transforming the symbols of an expression without regard to any interpretation or meaning given to them. Expressions that are syntactically correct are called well-formed. Semantics is concerned with the meaning of these well-formed expressions. Expressions that are semantically correct are called well-defined.


Well-formed

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 Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—"infixed operators"—such as the plus sign in . Usage Binary relations are ...
), 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. For example, in
arithmetic Arithmetic is an elementary branch of mathematics that deals with numerical operations like addition, subtraction, multiplication, and division. In a wider sense, it also includes exponentiation, extraction of roots, and taking logarithms. ...
, the expression ''1 + 2 × 3'' is well-formed, but :\times4)x+,/y. is not. However, being well-formed is not enough to be considered well-defined. For example in arithmetic, the expression \frac is well-formed, but it is not well-defined. (See
Division by zero In mathematics, division by zero, division (mathematics), division where the divisor (denominator) is 0, zero, is a unique and problematic special case. Using fraction notation, the general example can be written as \tfrac a0, where a is the di ...
). Such expressions are called undefined.


Well-defined

Semantics Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
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. In general the meaning of expressions is not limited to designating values; for instance, an expression might designate a condition, or an
equation In mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for ...
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 \oplus to designate an internal
direct sum The direct sum is an operation between structures in abstract algebra, a branch of mathematics. It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups A and B is anothe ...
. In
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
, 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 Semantics is the study of linguistic Meaning (philosophy), meaning. It examines what meaning is, how words get their meaning, and how the meaning of a complex expression depends on its parts. Part of this process involves the distinction betwee ...
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 number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, the product a \times b \times c is unambiguous because (a \times b)\times c = a \times (b \times c); hence the notation is said to be ''well defined''. This property, also known as
associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
of multiplication, guarantees the result does not depend on the sequence of multiplications; therefore, a specification of the sequence can be omitted. The
subtraction Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...
operation is non-associative; despite that, there is a convention that a-b-c is shorthand for (a-b)-c, thus it is considered "well-defined". On the other hand, Division is non-associative, and in the case of a/b/c, 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 precedence, associativity of the operator). For example, in the programming language C, the operator - for subtraction is ''left-to-right-associative'', which means that a-b-c is defined as (a-b)-c, and the operator = for assignment is ''right-to-left-associative'', which means that a=b=c is defined as a=(b=c). In the programming language APL there is only one rule: from right to left – but parentheses first.


Formal definition

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 Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
. Within mathematical logic, mathematics is usually described as a kind of
formal language In logic, mathematics, computer science, and linguistics, a formal language is a set of strings whose symbols are taken from a set called "alphabet". The alphabet of a formal language consists of symbols that concatenate into strings (also c ...
, and a well-formed expression can be defined recursively as follows: The
alphabet An alphabet is a standard set of letter (alphabet), letters written to represent particular sounds in a spoken language. Specifically, letters largely correspond to phonemes as the smallest sound segments that can distinguish one word from a ...
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 (\varnothing, \, ...), truth values (T or F), etc. * A set of individual variables: A
countably infinite In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbe ...
amount of symbols representing variables used for representing an unspecified object in the domain. (Usually letters like , or ) * A set of operations: Function symbols representing operations that can be performed on elements over the domain, like addition (+), multiplication (×), or set operations like union (∪), or intersection (∩). (Functions can be understood as 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 2 or the variable x are syntactically correct expressions. * Let F be a metavariable for any n-ary operation over the domain, and let \phi_1, \phi_2, ... \phi_n be metavariables for any WFE's. :Then F(\phi_1, \phi_2, ... \phi_n) is also well-formed. For the most often used operations, more convenient notations (like
infix notation Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands—"infixed operators"—such as the plus sign in . Usage Binary relations are ...
) have been developed over the centuries. :For instance, if the domain of discourse is the
real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...
s, F can denote the
binary operation In mathematics, a binary operation or dyadic operation is a rule for combining two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two. More specifically, a binary operation ...
+, then \phi_1 + \phi_2 is well-formed. Or F can be the unary operation \surd so \sqrt 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 In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement for express ...
). A well-formed expression can be thought as a syntax tree. The leaf nodes are always atomic expressions. Operations + and \cup have exactly two child nodes, while operations \sqrt , \text(x) and \frac have exactly one. There are countably infinitely many WFE's, however, each WFE has a finite number of nodes.


Lambda calculus

Formal languages allow formalizing the concept of well-formed expressions. In the 1930s, a new type of expression, the lambda expression, was introduced by
Alonzo Church Alonzo Church (June 14, 1903 – August 11, 1995) was an American computer scientist, mathematician, logician, and philosopher who made major contributions to mathematical logic and the foundations of theoretical computer science. He is bes ...
and Stephen Kleene for formalizing functions and their evaluation. The lambda operators (lambda abstraction and function application) form the basis for lambda calculus, a formal system used in
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
and
programming language theory Programming language theory (PLT) is a branch of computer science that deals with the design, implementation, analysis, characterization, and classification of formal languages known as programming languages. Programming language theory is clos ...
. The equivalence of two lambda expressions is undecidable (but see
unification (computer science) In logic and computer science, specifically automated reasoning, unification is an algorithmic process of solving equations between symbolic expression (mathematics), expressions, each of the form ''Left-hand side = Right-hand side''. For example, ...
). 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).


Types of expressions


Algebraic expression

An '' algebraic expression'' is an expression built up from algebraic constants, variables, and the algebraic operations (
addition Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol, +) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication, and Division (mathematics), divis ...
,
subtraction Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that repre ...
,
multiplication Multiplication is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division (mathematics), division. The result of a multiplication operation is called a ''Product (mathem ...
, division and
exponentiation In mathematics, exponentiation, denoted , is an operation (mathematics), operation involving two numbers: the ''base'', , and the ''exponent'' or ''power'', . When is a positive integer, exponentiation corresponds to repeated multiplication ...
by a
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...
). For example, is an algebraic expression. Since taking the square root is the same as raising to the power , the following is also an algebraic expression: :\sqrt See also:
Algebraic equation In mathematics, an algebraic equation or polynomial equation is an equation of the form P = 0, where ''P'' is a polynomial with coefficients in some field, often the field of the rational numbers. For example, x^5-3x+1=0 is an algebraic equati ...
and Algebraic closure


Polynomial expression

A
polynomial expression In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (mathematics), ring formed from the set (mathematics), set of polynomials in one or more indeterminate (variable), indeterminates (traditionally ...
is an expression built with scalars (numbers of elements of some field), indeterminates, and the operators of addition, multiplication, and exponentiation to nonnegative integer powers; for example 3(x+1)^2 - xy. Using
associativity In mathematics, the associative property is a property of some binary operations that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a Validity (logic), valid rule of replaceme ...
,
commutativity In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a p ...
and
distributivity In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary ...
, every polynomial expression is equivalent to a
polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
, that is an expression that is a
linear combination In mathematics, a linear combination or superposition is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' a ...
of products of integer powers of the indeterminates. For example the above polynomial expression is equivalent (denote the same polynomial as 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
computer science Computer science is the study of computation, information, and automation. Computer science spans Theoretical computer science, theoretical disciplines (such as algorithms, theory of computation, and information theory) to Applied science, ...
, an ''expression'' is a syntactic entity in a
programming language A programming language is a system of notation for writing computer programs. Programming languages are described in terms of their Syntax (programming languages), syntax (form) and semantics (computer science), semantics (meaning), usually def ...
that may be evaluated to determine its value or fail to terminate, in which case the expression is undefined. It is a combination of one or more constants, variables, functions, and operators that the programming language interprets (according to its particular rules of precedence and of association) and computes to produce ("to return", in a stateful environment) another value. This process, for mathematical expressions, is called ''evaluation''. In simple settings, the resulting value is usually one of various
primitive types In computer science, primitive data types are a set of basic data types from which all other data types are constructed. Specifically it often refers to the limited set of data representations in use by a particular processor, which all compiled ...
, such as string, Boolean, or numerical (such as
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
,
floating-point In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a ''significand'' (a Sign (mathematics), signed sequence of a fixed number of digits in some Radix, base) multiplied by an integer power of that ba ...
, or
complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...
). In
computer algebra In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating expression (mathematics), ...
, 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 8x-5 \geq 3 takes the value ''false'' if 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
number A number is a mathematical object used to count, measure, and label. The most basic examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, individual numbers can ...
s and variables, every mathematical expression may be viewed as the symbol of an operator followed by a
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is cal ...
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


Logical expression

In
mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, 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 first-order term is recursively constructed from constant symbols, variables, and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of .


Formal expression

A formal expression is a kind of string of
symbols A symbol is a mark, sign, or word that indicates, signifies, or is understood as representing an idea, object, or relationship. Symbols allow people to go beyond what is known or seen by creating linkages between otherwise different concep ...
, 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. For instance, the formal expressions "2" and "1+1" are not equal.


See also

* Analytic expression *
Closed-form expression In mathematics, an expression or equation is in closed form if it is formed with constants, variables, and a set of functions considered as ''basic'' and connected by arithmetic operations (, and integer powers) and function composition. ...
* Formal calculation *
Functional programming In computer science, functional programming is a programming paradigm where programs are constructed by Function application, applying and Function composition (computer science), composing Function (computer science), functions. It is a declarat ...
* Infinite expression *
Number sentence In mathematics education, a number sentence is an equation or inequality expressed using numbers and mathematical symbols. The term is used in primary level mathematics teaching in the US, Canada, UK, Australia, New Zealand and South Africa. Usa ...
* Rewriting * Signature (logic)


Notes


References


Works Cited

{{Mathematical logic Abstract algebra Logical expressions Elementary algebra