Expression (mathematics)

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In the equation 7x − 5 = 2, the sides of the equation are expressions.

In mathematics, an expression is a written arrangement of symbols following the context-dependent, syntactic conventions of mathematical notation. Symbols can denote numbers (constants), variables, operations, and functions. Other symbols include punctuation 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 refers to an object, and a whole sentence refers to a fact. For example, ${\displaystyle 8x-5}$ is an expression, while the inequality ${\displaystyle 8x-5\ge 3}$ is a formula.

To evaluate or simplify 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 ${\displaystyle 8\times 2-5}$ simplifies to ${\displaystyle 16-5}$, and evaluates to ${\displaystyle 11.}$

An expression is often used to define a function, by taking the variables to be arguments, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. For example, ${\displaystyle x\mapsto x^2+1}$ and ${\displaystyle f(x)=x^2+1}$ define the function that associates to each number its square 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 equality", that is, both expressions "mean the same thing."

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. For instance, the formal expressions "2" and "1+1" are not equal.

Variables and evaluation

In elementary algebra, 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 ${\displaystyle 4x^2+8}$; it can be evaluated at x = 3 in the following steps:

$4(3{)}^2+3$, (replace x with 3)

${\displaystyle 4\cdot (3\cdot 3)+8}$ (use definition of exponent)

${\displaystyle 4\cdot 9+8}$ (simplify)

${\displaystyle 36+8}$

${\displaystyle 44}$

A term is a constant or the product of a constant and one or more variables. Some examples include ${\displaystyle 7,5x,13x^{2}y,4b}$ The constant of the product is called the coefficient. 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, you can simplify the expression by combining the like terms. We add the coefficients and keep the same variable.

${\displaystyle 4x+7x+2x=15x}$

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.

For a non-formalized language, that is, in most mathematical texts outside of mathematical logic, for an individual expression it is not always possible to identify which variables are free and bound. For example, in $\sum _{i<k}{a}_{ik}$, 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.

Equivalence

Main article: Identity (mathematics)

An expression is often used to define a function, or denote compositions of funtions, by taking the variables to be arguments, or inputs, of the function, and assigning the output to be the evaluation of the resulting expression. For example, ${\displaystyle x\mapsto x^2+1}$ and ${\displaystyle f(x)=x^2+1}$ define the function that associates to each number its square 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 ${\displaystyle \equiv .}$

For example, in the expression $\sum _{n=1}^3(2nx),$ the variable n is bound, and the variable x is free. This expression is equivalent to the simpler expression 12 x; that is $${\displaystyle \sum _{n=1}^3(2nx)\equiv 12x.}$$ The value for x = 3 is 36, which can be denoted $${\displaystyle \sum _{n=1}^3(2nx){|}_{x=3}=36.}$$

Polynomial evaluation

Main article: Polynomial evaluation

A polynomial consists of variables and coefficients, that involve only the operations of addition, subtraction, multiplication and exponentiation 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 and hash tables, polynomials are used to compute k-independent hashing.

In the former case, polynomials are evaluated using floating-point arithmetic, 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 which case the answers are always exact.

For evaluating the univariate polynomial $a_n{x}^n+a_{n-1}{x}^{n-1}+\cdots +a_0,$ the most naive method would use ${\displaystyle n}$ multiplications to compute ${\displaystyle a_n{x}^n}$, use $n-1$ multiplications to compute ${\displaystyle a_{n-1}{x}^{n-1}}$ and so on for a total of $\frac{n(n+1)}{2}$ multiplications and ${\displaystyle n}$ additions. Using better methods, such as Horner's rule, this can be reduced to ${\displaystyle n}$ multiplications and ${\displaystyle n}$ additions. If some preprocessing is allowed, even more savings are possible.

Computation

Main article: Computation

A computation is any type of arithmetic or non-arithmetic calculation 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, 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. 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 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.

Common examples of computation are basic arithmetic and the execution of computer algorithms. A calculation 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. To illustrate, executing a function call f(a,b) may first evaluate the arguments a and b, store the results in references 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 languages, 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.

Well-defined expressions

Main article: 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 and semantics. 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), 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 are do not constitute mathematical expressions.

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 $\frac{1}{0}$ is well-formed, but it is not well-defined. (See Division by zero). Such expressions are called undefined.

Well-defined

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. 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. 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 ${\displaystyle a-b-c}$ is shorthand for ${\displaystyle (a-b)-c}$, thus it is considered "well-defined". On the other hand, Division is non-associative, and in the case of ${\displaystyle a/b/c}$, 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. Within mathematical logic, mathematics is usually described as a kind of formal language, and a well-formed expression can be defined recursively as follows:

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: 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 ${\displaystyle 2}$ or the variable ${\displaystyle x}$ are syntactically correct expressions.

• Let ${\displaystyle F}$ be a metavariable for any n-ary operation over the domain, and let ${\displaystyle {\varphi }_1,{\varphi }_2,...{\varphi }_n}$ be metavariables for any WFE's.

Then ${\displaystyle F({\varphi }_1,{\varphi }_2,...{\varphi }_n)}$ 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 ${\displaystyle {\varphi }_1+{\varphi }_2}$ is well-formed. Or ${\displaystyle F}$ can be the unary operation ${\displaystyle \surd }$ so ${\displaystyle \sqrt{{\varphi }_1}}$ 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. The leaf nodes are always atomic expressions. Operations ${\displaystyle +}$ and ${\displaystyle \cup }$ have exactly two child nodes, while operations $\sqrt{x}$, $\text{ln}(x)$ and $\frac{d}{dx}$ have exactly one. There are countably infinitely many WFE's, however, each WFE has a finite number of nodes.

Lambda calculus

Main article: Lambda calculus

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

In the 1930s, a new type of expressions, called lambda expressions, were introduced by Alonzo Church and Stephen Kleene for formalizing functions and their evaluation.  They form the basis for lambda calculus, a formal system used in mathematical logic and the theory of programming languages.

The equivalence of two lambda expressions is undecidable. 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).

History

For broader coverage of this topic, see History of mathematics and History of mathematical notation.

Early written mathematics

The Ishango bone on exhibition at the Royal Belgian Institute of Natural Sciences

Written mathematics began with numbers expressed as tally marks, with each tally representing a single unit. The numerical symbols consisted probably of strokes or notches cut in wood or stone, and intelligible alike to all nations. For example, one notch in a bone represented one animal, or person, or anything else. The Ishango bone, found near the headwaters of the Nile river (northeastern Congo), may be more than 20,000 years old and generally considered an early evidence of counting. The ordered engravings have led many to speculate the meaning behind these marks, including interpretations like mathematical significance or astrological relevance. Common interpretations are that the Ishango bone shows a six-month lunar calendar.

Image of Problem 14 from the Moscow Mathematical Papyrus. The problem includes a diagram indicating the dimensions of the truncated pyramid.

The Ancient Egyptians had a symbolic mathematics which was the numeration by Hieroglyphics. The Egyptian mathematics had a symbol for one, ten, one hundred, one thousand, ten thousand, one hundred thousand, and one million. Later, the Egyptians used hieratic instead of hieroglyphic script to show numbers. For example, the four vertical lines used to represent four were replaced by a single horizontal line. This is found in the Rhind Mathematical Papyrus (c. 2000–1800 BC) and the Moscow Mathematical Papyrus (c. 1890 BC). The system the Egyptians used was discovered and modified by many other civilizations in the Mediterranean. The Egyptians also had symbols for basic operations: legs going forward represented addition, and legs walking backward to represent subtraction.

Babylonian tablet (c. 1800–1600 BCE), showing an expression for an approximation to the square root of 2 (1 24 51 10 w: sexagesimal) in the context of Pythagoras' Theorem for an isosceles triangle.

The Mesopotamians had symbols for each power of ten. Later, they wrote their numbers in almost exactly the same way done in modern times. Instead of having symbols for each power of ten, they would just put the coefficient of that number. Each digit was separated by only a space, but by the time of Alexander the Great, they had created a symbol that represented zero and was a placeholder. The Mesopotamians also used a sexagesimal system, that is base sixty. It is this system that is used in modern times when measuring time and angles. Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and the system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.

Syncopated stage

The "syncopated" stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. During antiquity and the medieval periods, bursts of mathematical creativity were often followed by centuries of stagnation. As the early modern age opened and the worldwide spread of knowledge began, written examples of mathematical developments came to light.

Greek mathematics, which originated with the study of geometry, employed Attic numeration, which was based on the system of the Egyptians and was later adapted and used by the Romans. Greek mathematical reasoning was almost entirely geometric (albeit often used to reason about non-geometric subjects such as number theory), and hence the Greeks had no interest in algebraic symbols. The great exception was Diophantus of Alexandria, the great algebraist. His Arithmetica was one of the texts to manipulate mathematical expressions and equations symbolically; he used what is now known as syncopated algebra. It was not completely symbolic, but was much more so than previous books. An unknown number was called ${\displaystyle \zeta }$. The square of ${\displaystyle \zeta }$ was ${\displaystyle {\mathrm{\Delta }}^v}$; the cube was ${\displaystyle K^v}$; the fourth power was ${\displaystyle {\mathrm{\Delta }}^v\mathrm{\Delta }}$; and the fifth power was ${\displaystyle \mathrm{\Delta }{K}^v}$. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponents. So for example, what would be written in modern notation as $${\displaystyle x^3-2x^2+10x-1,}$$Would be written in Diophantus's syncopated notation as:

${\displaystyle {\mathrm{K}}^{\upsilon }\overline{\alpha }\zeta \overline{\iota }⋔{\mathrm{\Delta }}^{\upsilon }\overline{\beta }\mathrm{M}\overline{\alpha }}$

Symbolic stage and early arithmetic

The transition to symbolic algebra, where only symbols are used, can first be seen in the work of Ibn al-Banna' al-Marrakushi (1256–1321) and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī (1412–1482). In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations, al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.

The 1489 use of the plus and minus signs in print.

The 14th century saw the development of new mathematical concepts to investigate a wide range of problems. The two widely used arithmetic symbols are addition and subtraction, + and −. The plus sign was used starting around 1351 by Nicole Oresme and publicized in his 1360 in his work Algorismus proportionum. It is thought an abbreviation for "et", meaning "and" in Latin, in much the same way the ampersand sign also began as "et". Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of the distance covered by a body undergoing uniformly accelerated motion, asserting that the area under the line depicting the constant acceleration and represented the total distance traveled. The minus sign was used in 1489 by Johannes Widmann in Mercantile Arithmetic or Behende und hüpsche Rechenung auff allen Kauffmanschafft,. Widmann used the minus symbol with the plus symbol, to indicate deficit and surplus, respectively. In Summa de arithmetica, geometria, proportioni e proportionalità, Luca Pacioli used symbols for plus and minus symbols and contained algebra, though much of the work originated from Piero della Francesca whom he appropriated and purloined.

The radical symbol (√), for square root was introduced by Christoph Rudolff in the early 1500s.Michael Stifel's important work Arithmetica integra contained important innovations in written mathematics. In 1556, Niccolò Tartaglia used parentheses for precedence grouping. In 1557 Robert Recorde published Simon Stevin's book De Thiende ('the art of tenths'), published in Dutch in 1585, contained a systematic treatment of decimal notation, which influenced all later work on the real number system. The new algebra (1591) of François Viète introduced the modern notational manipulation of algebraic expressions.

William Oughtred, introduced the multiplication sign (×). Johann Rahn introduced the division sign (÷, an obelus variant repurposed) .

Types of expressions

Algebraic expression

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). 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

Polynomial expression

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

Using associativity, commutativity and distributivity, every polynomial expression is equivalent to a polynomial, that is an expression that is a linear combination 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

Main article: Expression (computer science)

In computer science, an expression is a syntactic entity in a programming language 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, such as string, Boolean, or numerical (such as integer, floating-point, or complex).

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 ${\displaystyle 8x-5\ge 3}$ 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).

Representation of the expression (8 − 6) × (3 + 1) as a Lisp tree, from a 1985 Master's Thesis

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

Logical 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 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, ⁠${\displaystyle (x+1)\ast (x+1)}$⁠ is a term built from the constant 1, the variable x, and the binary function symbols ⁠${\displaystyle +}$⁠ and ⁠${\displaystyle \ast }$⁠; it is part of the atomic formula ⁠${\displaystyle (x+1)\ast (x+1)\ge 0}$⁠ which evaluates to true for each real-numbered value of x.