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Bracket (mathematics)
In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets nbsp; braces and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Sometimes, for the clarity of reading, different kinds of brackets are used to express the same meaning of precedence in a single expression with deep nesting of sub-expressions. Historically, other notations, such as the vinculum, were similarly used for grouping. In present-day use, these notations all have specific meanings. The earliest use of brackets to indicate aggregation (i.e. grouping) was suggested in 1608 by Christopher Clavius, and in 1629 by Albert Girard. Symbols for representing angle brackets A variety of different symbols are used to represent angle brackets. In e-ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Expression
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. 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. This is analogous to natural language, where a noun phrase refers to an object, and a whole sentence refers to a fact. For example, 8x-5 is an expression, while the inequality 8x-5 \geq 3 is a formula. To ''evaluate'' an expression means to find a numerical value equivalent to the expression. Expressions can be ''evaluated'' or ''simplified'' by replacing operations that appear in them with their result. For example, the expression 8\times 2-5 simplifies to 16-5, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Set (mathematics)
In mathematics, a set is a collection of different things; the things are '' elements'' or ''members'' of the set and are typically mathematical objects: numbers, symbols, points in space, lines, other geometric shapes, variables, or other sets. A set may be finite or infinite. There is a unique set with no elements, called the empty set; a set with a single element is a singleton. Sets are ubiquitous in modern mathematics. Indeed, set theory, more specifically Zermelo–Fraenkel set theory, has been the standard way to provide rigorous foundations for all branches of mathematics since the first half of the 20th century. Context Before the end of the 19th century, sets were not studied specifically, and were not clearly distinguished from sequences. Most mathematicians considered infinity as potentialmeaning that it is the result of an endless processand were reluctant to consider infinite sets, that is sets whose number of members is not a natural number. Specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discrete Mathematics
Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continuous functions). Objects studied in discrete mathematics include integers, Graph (discrete mathematics), graphs, and Statement (logic), statements in Mathematical logic, logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumeration, enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets (finite sets or sets with the same cardinality as the natural numbers). However, there is no exact definition of the term "discrete mathematics". The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extended Real Number Line
In mathematics, the extended real number system is obtained from the real number system \R by adding two elements denoted +\infty and -\infty that are respectively greater and lower than every real number. This allows for treating the potential infinities of infinitely increasing sequences and infinitely decreasing series as actual infinities. For example, the infinite sequence (1,2,\ldots) of the natural numbers increases ''infinitively'' and has no upper bound in the real number system (a potential infinity); in the extended real number line, the sequence has +\infty as its least upper bound and as its limit (an actual infinity). In calculus and mathematical analysis, the use of +\infty and -\infty as actual limits extends significantly the possible computations. It is the Dedekind–MacNeille completion of the real numbers. The extended real number system is denoted \overline, \infty,+\infty/math>, or \R\cup\left\. When the meaning is clear from context, the symbol +\inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number Line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The association between numbers and points on the line links arithmetical operations on numbers to geometric relations between points, and provides a conceptual framework for learning mathematics. In elementary mathematics, the number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the circle constant : Every point of the number line corresponds to a unique real number, and every real number to a unique point. Using a number line, numerical concepts can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is denoted by \infty, called the infinity symbol. From the time of the Ancient Greek mathematics, ancient Greeks, the Infinity (philosophy), philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including Guillaume de l'Hôpital, l'Hôpital and Johann Bernoulli, Bernoulli) regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or Magnitude (mathematics), magnitude and, if so, how this could be done. At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semicolon
The semicolon (or semi-colon) is a symbol commonly used as orthographic punctuation. In the English language, a semicolon is most commonly used to link (in a single sentence) two independent clauses that are closely related in thought, such as when restating the preceding idea with a different expression. When a semicolon joins two or more ideas in one sentence, those ideas are then given equal rank. Semicolons can also be used in place of commas to separate items in a list, particularly when the elements of the list themselves have embedded commas. The semicolon is one of the least understood of the standard marks, and is not frequently used by many English speakers. In the QWERTY keyboard layout, the semicolon resides in the unshifted homerow beneath the little finger of the right hand and has become widely used in programming languages as a statement separator or terminator. History In 1496, the semicolon is attested in Pietro Bembo's book ' printed by Aldo Manuz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decimal Separator
FIle:Decimal separators.svg, alt=Four types of separating decimals: a) 1,234.56. b) 1.234,56. c) 1'234,56. d) ١٬٢٣٤٫٥٦., Both a comma and a full stop (or period) are generally accepted decimal separators for international use. The apostrophe and Arabic decimal separator are also used in certain contexts. A decimal separator is a symbol that separates the integer part from the fractional part of a number written in decimal form. Different countries officially designate different symbols for use as the separator. The choice of symbol can also affect the choice of symbol for the #Digit grouping, thousands separator used in digit grouping. Any such symbol can be called a decimal mark, decimal marker, or decimal sign. Symbol-specific names are also used; decimal point and decimal comma refer to a dot (either Baseline dot, baseline or Middle dot, middle) and comma respectively, when it is used as a decimal separator; these are the usual terms used in English, with the aforem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inner Product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in \langle a, b \rangle. Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or ''scalar product'' of Cartesian coordinates. Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898. An inner product naturally induces an associated norm, (denoted , x, and , y, in the pictu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
