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Nonnegative
In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero. In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero. The word "sign" is also often used to indicate binary aspects of mathematical or scientific objects, such as odd and even ( sign of a permutation), sense of orientation or rotation ( cw/ccw), one sided limits, and other concepts described in below. Sign of a number Numbers from various number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and in many other branches of mathematics), in particular by their role in the classical definitions of limits, continuity and derivatives. The set of real numbers, sometimes called "the reals", is traditionally denoted by a bold , often using blackboard bold, . The adjective ''real'', used in the 17th century by René Descartes, distinguishes real numbers from imaginary numbers such as the square roots of . The real numbers include the rational numbers, such as the integer and the fraction . The rest of the real numbers are called irrational numbers. Some irrational numbers (as well as all the rationals) a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Ring
In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'': * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''ab''. Examples Ordered rings are familiar from arithmetic. Examples include the integers, the rationals and the real numbers. (The rationals and reals in fact form ordered fields.) The complex numbers, in contrast, do not form an ordered ring or field, because there is no inherent order relationship between the elements 1 and ''i''. Positive elements In analogy with the real numbers, we call an element ''c'' of an ordered ring ''R'' positive if 0 < ''c'', and negative if ''c'' < 0. 0 is considered to be neither positive nor negative. The set of positive elements of an ordered ring ''R'' is often denoted by ''R''+. An alternative notation, favored in some disciplines, is to use ''R''+ for the set of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signed Number Representations
In computing, signed number representations are required to encode negative numbers in binary number systems. In mathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in RAM or CPU registers, numbers are represented only as sequences of bits, without extra symbols. The four best-known methods of extending the binary numeral system to represent signed numbers are: sign–magnitude, ones' complement, two's complement, and offset binary. Some of the alternative methods use implicit instead of explicit signs, such as negative binary, using the base −2. Corresponding methods can be devised for other bases, whether positive, negative, fractional, or other elaborations on such themes. There is no definitive criterion by which any of the representations is universally superior. For integers, the representation used in most current computing devices is two's complement, although the Unisys ClearPath Dorado series mainf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octonion
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of Hypercomplex number, hypercomplex Number#Classification, number system. The octonions are usually represented by the capital letter O, using boldface or blackboard bold \mathbb O. Octonions have eight dimension (vector space), dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are commutative property, noncommutative and associative property, nonassociative, but satisfy a weaker form of associativity; namely, they are alternative algebra, alternative. They are also Power associativity, power associative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Octonions are related to exceptional structures in mathematics, among them the Simple Lie group#Exceptional cases, exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and qu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0 (number)
0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and complex numbers, as well as other algebraic structures. Multiplying any number by 0 results in 0, and consequently division by zero has no meaning in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it indicates that the power of ten corresponding to the place containing a 0 does not contribute to the total. For example, "205" in decimal means two hundreds, no tens, and five ones. The same principle applies in place-value notations that uses a base other than ten, such as binary and hexadecimal. The modern use of 0 in this manner derives from Indian mathematics that was transmitted to Europe via medieval Islamic mathematicians and popularized by Fibonacci. It was independently used by the Maya. Common names for th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quaternions is often denoted by (for ''Hamilton''), or in blackboard bold by \mathbb H. Quaternions are not a field, because multiplication of quaternions is not, in general, commutative. Quaternions provide a definition of the quotient of two vectors in a three-dimensional space. Quaternions are generally represented in the form a + b\,\mathbf i + c\,\mathbf j +d\,\mathbf k, where the coefficients , , , are real numbers, and , are the ''basis vectors'' or ''basis elements''. Quaternions are used in pure mathematics, but also have practical uses in applied mathematics, particularly for calculations involving three-dimensional rotations, such as in three-dimensional computer graphics, computer vision, robotics, magnetic resonance i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negative Number
In mathematics, a negative number is the opposite (mathematics), opposite of a positive real number. Equivalently, a negative number is a real number that is inequality (mathematics), less than 0, zero. Negative numbers are often used to represent the Magnitude (mathematics), magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as ''positive'' and ''negative''. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −(−3) = 3 because the opposite of an opposite is the original value. Negative numbers are usually written with a Plus and minus signs, minus sig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Inverse
In mathematics, the additive inverse of an element , denoted , is the element that when added to , yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element. In elementary mathematics, the additive inverse is often referred to as the opposite number, or its negative. The unary operation of arithmetic negation is closely related to '' subtraction'' and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers. Common examples When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1. The concept can also be extended to algebraic expressions, which is often used when balancing equations. Relation to subtraction The additive inverse is closely related to subtraction, which can be viewed as an add ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Order
In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( reflexive). # If a \leq b and b \leq c then a \leq c ( transitive). # If a \leq b and b \leq a then a = b ( antisymmetric). # a \leq b or b \leq a ( strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term ''chain'' is sometimes defined as a synonym of ''totally ordered set'', but generally refers to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floating-point Representation
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a '' significand'' (a signed sequence of a fixed number of digits in some base) multiplied by an integer power of that base. Numbers of this form are called floating-point numbers. For example, the number 2469/200 is a floating-point number in base ten with five digits: 2469/200 = 12.345 = \! \underbrace_\text \! \times \! \underbrace_\text\!\!\!\!\!\!\!\overbrace^ However, 7716/625 = 12.3456 is not a floating-point number in base ten with five digits—it needs six digits. The nearest floating-point number with only five digits is 12.346. And 1/3 = 0.3333… is not a floating-point number in base ten with any finite number of digits. In practice, most floating-point systems use base two, though base ten (decimal floating point) is also common. Floating-point arithmetic operations, such as addition and division, approximate the corresponding real number arithmetic operations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Éléments De Mathématique
''Éléments de mathématique'' (English: ''Elements of Mathematics'') is a series of mathematics books written by the pseudonymous French collective Nicolas Bourbaki. Begun in 1939, the series has been published in several volumes, and remains in progress. The series is noted as a large-scale, self-contained, formal treatment of mathematics. The members of the Bourbaki group originally intended the work as a textbook on analysis, with the working title ''Traité d'analyse'' (''Treatise on Analysis''). While planning the structure of the work they became more ambitious, expanding its scope to cover several branches of modern mathematics. Once the plan of the work was expanded to treat other fields in depth, the title ''Éléments de mathématique'' was adopted. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups and Lie algebras. The unusual singular "mathématique" (mathematic) of the title is deliberate, meant to convey the a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicolas Bourbaki
Nicolas Bourbaki () is the collective pseudonym of a group of mathematicians, predominantly French alumni of the École normale supérieure (Paris), École normale supérieure (ENS). Founded in 1934–1935, the Bourbaki group originally intended to prepare a new textbook in mathematical analysis, analysis. Over time the project became much more ambitious, growing into a large series of textbooks published under the Bourbaki name, meant to treat modern pure mathematics. The series is known collectively as the ''Éléments de mathématique'' (''Elements of Mathematics''), the group's central work. Topics treated in the series include set theory, abstract algebra, topology, analysis, Lie groups, and Lie algebras. Bourbaki was founded in response to the effects of the First World War which caused the death of a generation of French mathematicians; as a result, young university instructors were forced to use dated texts. While teaching at the University of Strasbourg, Henri Cartan co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
