Expression (mathematics)

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:

3 + 5

to the complex:

$f(a)+\sum_{k=1}^n\left.\frac{1}{k!}\frac{d^k}{dt^k}\right|_{t=0}f(u(t)) + \int_0^1 \frac{(1-t)^n }{n!} \frac{d^{n+1}}{dt^{n+1}} f(u(t))\, dt.$.

Strings of symbols that violate the rules of syntax are not well-formed and are not valid mathematical expressions. For example:

$\times4)x+,/y$

would not be considered a mathematical expression but only a meaningless jumble.^[1]

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. 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 $\oplus$ 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.

Variables

Many mathematical expressions include letters called variables. 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.^[2]

For example, the expression

x / y

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

$\sum_{n=1}^{3} (2nx)$

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

$x/2 \text{ or } {x \over 2}$

are perfectly valid. Also, for multiplication one can use the symbols '×' or a '·' (mid dot), or else simply omit it (multiplication is implicit); so:

$x \times 2 \text{ or } x\cdot2 \text{ or } x2 \text{ or } 2x$

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^[3] in the 1930s in their lambda calculus. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages.^[4]

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.

See also

• Algebraic closure
• Algebraic expression
• Combinator
• Defined and undefined
• Expression (programming)
• Formula
• Formal grammar
• Functional programming

References

1. ^ Introduction to Algebra
2. ^ TalkTalk Reference Encyclopedia
3. ^ Biographical Memoir of Stephen Kleene
4. ^ Programming Languages and Lambda Calculi