In mathematics, the word expression is a term for any well-formed In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal combination of mathematical symbols[citation needed]. An algebraic expression is only a phrase, not a whole sentence, so it cannot contain an equality sign (=).[1] For example,
- x2 + 3x − 4
is an expression, while
- y = x2 + 3x − 4
is an equation An equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in but not an expression. Neither is
- )x) / y
an expression because the parentheses are not balanced.
Being an expression is a syntactic In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning concept – the meaning of the variables is irrelevant, but different fields have different notions of validity. See formal language A formal language is a set of words, i.e. finite strings of letters, symbols, or tokens. The set from which these letters are taken is called the alphabet over which the language is defined. A formal language is often defined by means of a formal grammar ; accordingly, words that belong to a formal language are sometimes called well-formed words ( for how expressions are constructed, and formal semantics Formal semantics is the study of the semantics, or interpretations, of formal and also natural languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols and a set of formation rules (also called a formal grammar) which determine which strings of symbols are well-formed formulas for meaning.
Variables
Many mathematical expressions include letters called variables A variable is a symbol that stands for a value that may vary; the term usually occurs in opposition to constant, which is a symbol for a non-varying value, i.e. completely fixed or fixed in the context of use. The concepts of constants and variables are fundamental to all modern mathematics, science, engineering, and computer programming. Variables are classified as either free or bound In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation that specifies places in an expression where substitution may take place. The idea is related to a placeholder , or a wildcard character that stands for an unspecified symbol.
For a given combination of values for the free variables, an expression may be evaluated Evaluation is the process of characterizing and appraising something of interest or of determining the value of an expression, although for some combinations of values of the free variables, the value of the expression may be undefined. Thus an expression represents a function In mathematics, a function is a relation between a given set of elements and another set of elements (the codomain), which associates each element in the domain with exactly one element in the codomain. The elements so related can be any kind of thing (words, objects, qualities) but are typically mathematical quantities, such as real numbers whose inputs are the value assigned the free variables and whose output is the resulting value of the expression.
For example, the expression
- x / y
evaluated for x = 10, y = 5, will give 2; but is undefined In mathematics, a division is called a division by zero if the divisor is zero. Such a division can be formally expressed as a/0 where a is the dividend. Whether this expression can be assigned a well-defined value depends upon the mathematical setting. In ordinary arithmetic, the expression has no meaning 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. See Formal semantics Formal semantics is the study of the semantics, or interpretations, of formal and also natural languages. A formal language can be defined apart from any interpretation of it. This is done by designating a set of symbols and a set of formation rules (also called a formal grammar) which determine which strings of symbols are well-formed formulas and Interpretation (logic) An interpretation is an assignment of "meaning" to the symbols of a language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any "meaning" until they are given some interpretation. The general study of interpretations of formal for the study of this question in logic.
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
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, i.e.:
x / 2 or
are perfectly valid. Also, for multiplication one can use the symbols or a "." (dot), or else simply omit it (multiplication is implicit); so:
or or or
are all acceptable (please notice in the first example above how the "times" symbol resembles an "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 In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal. 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 In formal logic, a formal system consists of a formal language and a set of inference rules, used to derive (to conclude) one expression from one or more other expressions (premises) antecedently supposed (axioms) or derived (theorems). The axioms and rules may be called a deductive apparatus. A formal system may be formulated and studied for its by Alonzo Church Alonzo Church was an American mathematician and logician who made major contributions to mathematical logic and the foundations of theoretical computer science. He is best known for the lambda calculus, Church–Turing thesis, Church Frege ontology, and the Church-Rosser theorem and Stephen Kleene Stephen Cole Kleene was an American mathematician who helped lay the foundations for theoretical computer science. One of many distinguished students of Alonzo Church, Kleene, along with Alan Turing, Emil Post, and others, is best known as a founder of the branch of mathematical logic known as recursion theory. Kleene's work grounds the study of in the 1930s in their lambda calculus In mathematical logic and computer science, lambda calculus, also written as λ-calculus, is a formal system for function definition, function application and recursion. It was introduced by Alonzo Church in the 1930s as part of an investigation into the foundations of mathematics. After the original system was shown to be logically inconsistent ,. The lambda calculus has been a major influence in the development of modern mathematics and computer programming languages A programming language is an artificial language designed to express computations that can be performed by a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine, to express algorithms precisely, or as a mode of human communication.
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 In computability theory and computational complexity theory, a decision problem is a question in some formal system with a yes-or-no answer, depending on the values of some input parameters. For example, the problem "given two numbers x and y, does x evenly divide y?" is a decision problem. The answer can be either 'yes' or 'no', and. This is also true of any expression in any system that has power equivalent to the lambda calculus.
See also
- Algebraic closure In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics
- Combinator Combinatory logic was originally intended as a 'pre-logic' that would clarify the role of quantified variables in logic, essentially by eliminating them. Another way of eliminating quantified variables is Quine's predicate functor logic. While the expressive power of combinatory logic typically exceeds that of first-order logic, the expressive
- Defined and undefined In mathematics, defined and undefined are used to explain whether or not expressions have meaningful, sensible, and unambiguous values. Whether an expression has a meaningful value depends on the context of the expression. For example the value of 4-5 is undefined if an positive integer result is required
- Expression (programming) An expression in a programming language is a combination of values, variables, operators, and functions that are interpreted according to the particular rules of precedence and of association for a particular programming language, which computes and then produces (returns, in a stateful environment) another value. The expression is said to
- Equation An equation is a mathematical statement, in symbols, that two things are exactly the same . Equations are written with an equal sign, as in
- Formula In mathematics and other sciences, a formula is a concise way of expressing information symbolically (as in a mathematical or chemical formula), or a general relationship between quantities. One of many famous formulae is Albert Einstein's E = mc2 (see special relativity)
- Formal grammar A formal grammar is a set of rules for forming strings in a formal language. These rules that make up the grammar describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings—only their location and the ways that they can be manipulated
- Functional programming In computer science, functional programming is a programming paradigm that treats computation as the evaluation of mathematical functions and avoids state and mutable data. It emphasizes the application of functions, in contrast to the imperative programming style, which emphasizes changes in state. Functional programming has its roots in the
- Inequation In mathematics, an inequation is a statement that two objects or expressions are not the same, or do not represent the same value. This relation is written with a crossed-out equal sign, like
- Syntax (logic) In logic, syntax is anything having to do with formal languages or formal systems without regard to any interpretation or meaning given to them. Syntax is concerned with the rules used for constructing, or transforming the symbols and words of a language, as contrasted with the semantics of a language which is concerned with its meaning
- Well-defined In mathematics, the term well-defined is used to specify that a certain concept or object is defined in a mathematical or logical way using a set of base axioms in an entirely unambiguous way and satisfies the properties it is required to satisfy. Usually definitions are stated unambiguously, and it is clear they satisfy the required properties
References
- ^ The language of Algebra, definitions accessed July 6, 2009; What is Algebra accessed July 6, 2009.
Categories: Abstract algebra | Algebra Algebra is a branch of mathematics which may be defined as a generalization and extension of arithmetic, and is focused on finding patterns between groups of numbers, operators, and other mathematical objects. The following category includes articles about algebra. Algebra is also a branch of mathematics that substitutes letters for numbers, | Evaluation
FLFF
Fri, 16 Oct 2009 10:08:01 GM
But I never have that problem with Science, thanks to the fact that I screw up 40% of the questions -inserts smoulderingly unhappy . expression. -. But this will change -cough-. Anyway yes today~ . Mathematics. was actually good. ...
Q. Add. 1/xy + 3/y Divide: a^2 +2a/ a^2 +3a +2 (divide by) a^2 - 3a/ 2a +2
Asked by Eloo B - Wed Apr 16 18:27:03 2008 - - 2 Answers - 0 Comments
A. Add. 1/xy + 3/y 1/xy + 3x/yx (1+3x)/xy Divide: a^2 +2a/ a^2 +3a +2 (divide by) a^2 - 3a/ 2a +2 Divide: a(a+2)/ (a+2)(a+3)(divide by) a(a-3)/ 2(a+1) Divide: a/ (a+3)times 2(a+1)/a(a-3) 1/ (a+3)times 2(a+1)/(a-3) 2(a+1)/(a+3)(a-3)
Answered by mom - Wed Apr 16 18:32:54 2008
