An expression is a combination of numbers, operators, grouping symbols (such as brackets and parentheses) and/or free variables and bound variables arranged in a meaningful way which can be evaluated. Bound variables are assigned values within the expression (they are for internal use) while free variables can take on values from outside the expression. 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 expression may be undefined. Thus an expression represents a function whose inputs are the values assigned the free variables and whose output is the resulting value of the expression. For example, the expression
- <math> x/y </math>
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
- <math>\sum_{n=1}^{3} 2nx</math>
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. 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 languages. 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
External links
- Axiomatic Theory of Formulas - theory of expressions on high abstraction level.
- Plot mathematical expressions this system plots math equations, graphs, diagrams, and even animated cartoons of transformation of math expressions and arithmetic operations. Knowledge of TeX not required.
