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In computing, a regular expression, also referred to as regex or regexp, provides a concise and flexible means for matching strings of text, such as particular characters, words, or patterns of characters. A regular expression is written in a formal language that can be interpreted by a regular expression processor, a program that either serves as a parser generator or examines text and identifies parts that match the provided specification. The following examples illustrate a few specifications that could be expressed in a regular expression: The sequence of characters "car" appearing consecutively in any context, such as in "car", "cartoon", or "bicarbonate" The sequence of characters "car" occurring in that order with other characters between them, such as in "Icelander" or "chandler" The word "car" when it appears as an isolated word The word "car" when preceded by the word "blue" or "red" The word "car" when not preceded by the word "motor" A dollar sign immediately followed by one or more digits, and then optionally a period and exactly two more digits (for example, "$100" or "$245.99"). Regular expressions can be much more complex than these examples. Regular expressions are used by many text editors, utilities, and programming languages to search and manipulate text based on patterns. Some of these languages, including Perl, Ruby, Awk, and Tcl, have fully integrated regular expressions into the syntax of the core language itself. Others like C, C++,.NET, Java, and Python instead provide access to regular expressions only through libraries. Utilities provided by Unix distributionsâ€”including the editor ed and the filter grepâ€”were the first to popularize the concept of regular expressions. As an example of the syntax, the regular expression \bex can be used to search for all instances of the string "ex" that occur after "word boundaries" (signified by the \b). In layman's terms, \bex will find the matching string "ex" in two possible locations, (1) at the beginning of words, and (2) between two characters in a string, where one is a word character and the other is not a word character. Thus, in the string "Texts for experts", \bex matches the "ex" in "experts" but not in "Texts" (because the "ex" occurs inside a word and not immediately after a word boundary). Many modern computing systems provide wildcard characters in matching filenames from a file system. This is a core capability of many commandline shells and is also known as globbing. Wildcards differ from regular expressions in generally expressing only limited forms of patterns. Basic concepts A regular expression, often called a pattern, is an expression that describes a set of strings. They are usually used to give a concise description of a set, without having to list all elements. For example, the set containing the three strings "Handel", "HÃ¤ndel", and "Haendel" can be described by the pattern H(Ã¤ae?)ndel (or alternatively, it is said that the pattern matches each of the three strings). In most formalisms, if there is any regex that matches a particular set then there is an infinite number of such expressions. Most formalisms provide the following operations to construct regular expressions. Boolean "or" A vertical bar separates alternatives. For example, graygrey can match "gray" or "grey". Grouping Parentheses are used to define the scope and precedence of the operators (among other uses). For example, graygrey and gr(ae)y are equivalent patterns which both describe the set of "gray" and "grey". Quantification A quantifier after a token (such as a character) or group specifies how often that preceding element is allowed to occur. The most common quantifiers are the question mark ?, the asterisk * (derived from the Kleene star), and the plus sign + (Kleene cross). These constructions can be combined to form arbitrarily complex expressions, much like one can construct arithmetical expressions from numbers and the operations +, âˆ’, Ã—, and Ã·. For example, H(ae?Ã¤)ndel and H(aaeÃ¤)ndel are both valid patterns which match the same strings as the earlier example, H(Ã¤ae?)ndel. The precise syntax for regular expressions varies among tools and with context; more detail is given in the Syntax section. History The origins of regular expressions lie in automata theory and formal language theory, both of which are part of theoretical computer science. These fields study models of computation (automata) and ways to describe and classify formal languages. In the 1950s, mathematician Stephen Cole Kleene described these models using his mathematical notation called regular sets. The SNOBOL language was an early implementation of pattern matching, but not identical to regular expressions. Ken Thompson built Kleene's notation into the editor QED as a means to match patterns in text files. He later added this capability to the Unix editor ed, which eventually led to the popular search tool grep's use of regular expressions ("grep" is a word derived from the command for regular expression searching in the ed editor: g/re/p where re stands for regular expression). Since that time, many variations of Thompson's original adaptation of regular expressions have been widely used in Unix and Unixlike utilities including expr, AWK, Emacs, vi, and lex. Perl and Tcl regular expressions were derived from a regex library written by Henry Spencer, though Perl later expanded on Spencer's library to add many new features. Philip Hazel developed PCRE (Perl Compatible Regular Expressions), which attempts to closely mimic Perl's regular expression functionality and is used by many modern tools including PHP and Apache HTTP Server. Part of the effort in the design of Perl 6 is to improve Perl's regular expression integration, and to increase their scope and capabilities to allow the definition of parsing expression grammars. The result is a minilanguage called Perl 6 rules, which are used to define Perl 6 grammar as well as provide a tool to programmers in the language. These rules maintain existing features of Perl 5.x regular expressions, but also allow BNFstyle definition of a recursive descent parser via subrules. The use of regular expressions in structured information standards for document and database modeling started in the 1960s and expanded in the 1980s when industry standards like ISO SGML (precursored by ANSI "GCA 1011983") consolidated. The kernel of the structure specification language standards are regular expressions. Simple use is evident in the DTD element group syntax. See also Pattern matching: History. Formal language theory Definition Regular expressions describe regular languages in formal language theory. They have thus the same expressive power as regular grammars. Regular expressions consist of constants and operators that denote sets of strings and operations over these sets, respectively. The following definition is standard, and found as such in most textbooks on formal language theory. Given a finite alphabet Î£, the following constants are defined: (empty set) denoting the set . (empty string) Îµ denoting the set containing only the "empty" string, which has no characters at all. (literal character) a in Î£ denoting the set containing only the character a. The following operations are defined: (concatenation) RS denoting the set { Î±Î²  Î± in R and Î² in S }. For example {"ab", "c"}{"d", "ef"} = {"abd", "abef", "cd", "cef"}. (alternation) R  S denoting the set union of R and S. For example {"ab", "c"}{"ab", "d", "ef"} = {"ab", "c", "d", "ef"}. (Kleene star) R* denoting the smallest superset of R that contains Îµ and is closed under string concatenation. This is the set of all strings that can be made by concatenating any finite number (including zero) of strings from R. For example, {"0","1"}* is the set of all finite binary strings (including the empty string), and {"ab", "c"}* = {Îµ, "ab", "c", "abab", "abc", "cab", "cc", "ababab", "abcab", ... }. To avoid parentheses it is assumed that the Kleene star has the highest priority, then concatenation and then set union. If there is no ambiguity then parentheses may be omitted. For example, (ab
Mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics, the physical sciences, engineering, and economics. Mathematical notations include relatively simple symbolic representations, such as numbers 1 and 2, function symbols sin and +; conceptual symbols, such as lim, dy/dx, equations and variables; and complex diagrammatic notations such as Penrose graphical notation and CoxeterDynkin diagrams.
Definition
A mathematical notation is a writing system used for recording concepts in mathematics.
 The notation uses symbols or symbolic expressions which are intended to have a precise semantic meaning.
 In the history of mathematics, these symbols have denoted numbers, shapes, patterns, and change. The notation can also include symbols for parts of the conventional discourse between mathematicians, when viewing mathematics as a language.
The media used for writing are recounted below, but common materials currently include paper and pencil, board and chalk (or dryerase marker), and electronic media. Systematic adherence to mathematical concepts is a fundamental concept of mathematical notation. (See also some related concepts: Logical argument, Mathematical logic, and Model theory.)
Expressions
A mathematical expression is a sequence of symbols which can be evaluated. For example, if the symbols represent numbers, the expressions are evaluated according to a conventional order of operations which provides for calculation, if possible, of any expressions within parentheses, followed by any exponents and roots, then multiplications and divisions and finally any additions or subtractions, all done from left to right. In a computer language, these rules are implemented by the compilers. For more on expression evaluation, see the computer science topics: eager evaluation, lazy evaluation, and evaluation operator.
Precise semantic meaning
Modern mathematics needs to be precise, because ambiguous notations do not allow formal proofs. Suppose that we have statements, denoted by some formal sequence of symbols, about some objects (for example, numbers, shapes, patterns). Until the statements can be shown to be valid, their meaning is not yet resolved. While reasoning, we might let the symbols refer to those denoted objects, perhaps in a model. The semantics of that object has a heuristic side and a deductive side. In either case, we might want to know the properties of that object, which we might then list in an intensional definition.
Those properties might then be expressed by some wellknown and agreedupon symbols from a table of mathematical symbols. This mathematical notation might include annotation such as
 "All x", "No x", "There is an x" (or its equivalent, "Some x"), "A set", "A function"
 "A mapping from the real numbers to the complex numbers"
In different contexts, the same symbol or notation can be used to represent different concepts. Therefore, to fully understand a piece of mathematical writing, it is important to first check the definitions that an author gives for the notations that are being used. This may be problematic if the author assumes the reader is already familiar with the notation in use.
History
Counting
It is believed that a mathematical notation to represent counting was first developed at least 50,000 years ago â€” early mathematical ideas such as finger counting have also been represented by collections of rocks, sticks, bone, clay, stone, wood carvings, and knotted ropes. The tally stick is a timeless way of counting. Perhaps the oldest known mathematical texts are those of ancient Sumer. The Census Quipu of the Andes and the Ishango Bone from Africa both used the tally mark method of accounting for numerical concepts.
The development of zero as a number is one of the most important developments in early mathematics. It was used as a placeholder by the Babylonians and Greek Egyptians, and then as an integer by the Mayans, Indians and Arabs. (See From Yahoo Answers
Answers:the left hand side 6x2(1+x) =6x 2 2x =2 +4x = left hand side so they are equvalent
Answers:they're eqivilant to eachother!!!
Answers:Rather than chance contaminating you with my definitions, I've listed 2 sources for the ones you're after. Should be fine. Good luck
Answers:we know that cot = 1 / tan => tan * cot = 1 => tan 2 * cot 2 = 1 we know that tan 2 = 2 tan / ( 1  tan^2 ) ( a  b ) ( a + b ) = a^2  b^2 let here a = 1 and b = tan then 1  tan^2 = ( 1  tan )( 1 + tan ) therefore ..........................2 tan tan 2 = ______________________ ...............( 1  tan ) ( 1 + tan ) tan 2 * cot 2 = 1 .....................................2 tan cot 2 tan 2 * cot 2 = ______________________ = 1 ............................( 1  tan ) ( 1 + tan ) .........2 tan cot 2 ______________________ = 1 ..( 1  tan ) ( 1 + tan ) .........2 tan cot 2 ______________________ = ( 1 + tan ) ..........( 1 + tan )
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