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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.
A simple fraction (also known as a common fraction or vulgar fraction) [n 1] is a rational number written as a/b or , where a and b are both integers. [9] As with other fractions, the denominator (b) cannot be zero. Examples include 1 / 2 , − 8 / 5 , −8 / 5 , and 8 / −5 .
For example, 1 / 4 , 5 / 6 , and −101 / 100 are all irreducible fractions. On the other hand, 2 / 4 is reducible since it is equal in value to 1 / 2 , and the numerator of 1 / 2 is less than the numerator of 2 / 4 . A fraction that is reducible can be reduced by dividing both the numerator ...
They may also be performed, in a similar way, on variables, algebraic expressions, [7] and more generally, on elements of algebraic structures, such as groups and fields. [8] An algebraic operation may also be defined more generally as a function from a Cartesian power of a given set to the same set.
In computer science, Backus–Naur form (BNF; / ˌ b æ k ə s ˈ n aʊər /; Backus normal form) is a notation used to describe the syntax of programming languages or other formal languages. It was developed by John Backus and Peter Naur. BNF can be described as a metasyntax notation for context-free grammars. Backus–Naur form is applied ...
The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
Below is a simple grammar, defined using the notation of regular expressions and Extended Backus–Naur form. It describes the syntax of S-expressions, a data syntax of the programming language Lisp, which defines productions for the syntactic categories expression, atom, number, symbol, and list:
Abstract syntax tree of the arithmetic expression "a^2+4*b" wrt. the example grammar (top) and its Chomsky normal form (bottom) The following grammar, with start symbol Expr, describes a simplified version of the set of all syntactical valid arithmetic expressions in programming languages like C or Algol60.