Expression (mathematics)

For other uses, see Expression (disambiguation).

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, variables, operations, and functions.[1] 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.[2] This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, is an expression, while the inequality is a formula.

To evaluate an expression means to find a numerical value equivalent to the expression.[3][4] Expressions can be evaluated or simplified by replacing operations that appear in them with their result. For example, the expression simplifies to , and evaluates to

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.[5] For example, and 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.[6][7] For instance, the formal expressions "2" and "1+1" are not equal.

  1. ^ Oxford English Dictionary, s.v. “Expression (n.), sense II.7,” "A group of symbols which together represent a numeric, algebraic, or other mathematical quantity or function."
  2. ^ Stoll, Robert R. (1963). Set Theory and Logic. San Francisco, CA: Dover Publications. ISBN 978-0-486-63829-4.
  3. ^ Oxford English Dictionary, s.v. "Evaluate (v.), sense a", "Mathematics. To work out the ‘value’ of (a quantitative expression); to find a numerical expression for (any quantitative fact or relation)."
  4. ^ Oxford English Dictionary, s.v. “Simplify (v.), sense 4.a”, "To express (an equation or other mathematical expression) in a form that is easier to understand, analyse, or work with, e.g. by collecting like terms or substituting variables."
  5. ^ Codd, Edgar Frank (June 1970). "A Relational Model of Data for Large Shared Data Banks" (PDF). Communications of the ACM. 13 (6): 377–387. doi:10.1145/362384.362685. S2CID 207549016. Archived (PDF) from the original on 2004-09-08. Retrieved 2020-04-29.
  6. ^ McCoy, Neal H. (1960). Introduction To Modern Algebra. Boston: Allyn & Bacon. p. 127. LCCN 68015225.
  7. ^ Fraleigh, John B. (2003). A first course in abstract algebra. Boston : Addison-Wesley. ISBN 978-0-201-76390-4.