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Arbitrarily Small
In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers (and its subsets thereof), though its meaning can differ from that of "sufficiently" and "infinitely". Examples The statement : "f(x) is non-negative for arbitrarily large ''x''." is a shorthand for: : "For every real number ''n'', f(x) is non-negative for some value of ''x'' greater than ''n''." In the common parlance, the term "arbitrarily long" is often used in the context of sequence of numbers. For example, to say that there are "arbitrarily long arithmetic progressions of prime numbers" does not mean that there exists any infinitely long arithmetic progression of prime numbers (there is not), nor that there exists any particular arithmetic progression of prime numbers that is in s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Number
In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real number can be almost uniquely represented by an infinite decimal expansion. The real numbers are fundamental in calculus (and more generally in all mathematics), in particular by their role in the classical definitions of limit (mathematics), limits, continuous function, continuity and derivatives. The set of real numbers is mathematical notation, denoted or \mathbb and is sometimes called "the reals". The adjective ''real'' in this context was introduced in the 17th century by René Descartes to distinguish real numbers, associated with physical reality, from imaginary numbers (such as the square roots of ), which seemed like a theoretical contrivance unrelated to physical reality. The real numbers subset, include t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ''B''. The relationship of one set being a subset of another is called inclusion (or sometimes containment). ''A'' is a subset of ''B'' may also be expressed as ''B'' includes (or contains) ''A'' or ''A'' is included (or contained) in ''B''. A ''k''-subset is a subset with ''k'' elements. The subset relation defines a partial order on sets. In fact, the subsets of a given set form a Boolean algebra under the subset relation, in which the join and meet are given by intersection and union, and the subset relation itself is the Boolean inclusion relation. Definition If ''A'' and ''B'' are sets and every element of ''A'' is also an element of ''B'', then: :*''A'' is a subset of ''B'', denoted by A \subseteq B, or equivalently, :* ''B'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primes In Arithmetic Progressions
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a_n = 3 + 4n for 0 \le n \le 2. According to the Green–Tao theorem, there exist arbitrarily long sequences of primes in arithmetic progression. Sometimes the phrase may also be used about primes which belong to an arithmetic progression which also contains composite numbers. For example, it can be used about primes in an arithmetic progression of the form an + b, where ''a'' and ''b'' are coprime which according to Dirichlet's theorem on arithmetic progressions contains infinitely many primes, along with infinitely many composites. For integer ''k'' ≥ 3, an AP-''k'' (also called PAP-''k'') is any sequence of ''k'' primes in arithmetic progression. An AP-''k'' can be written as ''k'' primes of the form ''a''·''n'' + ''b'', for fixed integers ''a'' (called t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sufficiently Large
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set (such as sequences and subsets of \mathbb). Notation The general form where the phrase eventually (or sufficiently large) is found appears as follows: :P is ''eventually'' true for x (P is true for ''sufficiently large'' x), where \forall and \exists are the universal and existential quantifiers, which is actually a shorthand for: :\exists a \in \mathbb such that P is true \forall x \ge a or somewhat more formally: :\exists a \in \mathbb: \forall x \in \mathbb:x \ge a \Rightarrow P(x) This does not necessarily mean that any particular val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclid's Theorem
Euclid's theorem is a fundamental statement in number theory that asserts that there are infinitely many prime numbers. It was first proved by Euclid in his work '' Elements''. There are several proofs of the theorem. Euclid's proof Euclid offered a proof published in his work ''Elements'' (Book IX, Proposition 20), which is paraphrased here. Consider any finite list of prime numbers ''p''1, ''p''2, ..., ''p''''n''. It will be shown that at least one additional prime number not in this list exists. Let ''P'' be the product of all the prime numbers in the list: ''P'' = ''p''1''p''2...''p''''n''. Let ''q'' = ''P'' + 1. Then ''q'' is either prime or not: *If ''q'' is prime, then there is at least one more prime that is not in the list, namely, ''q'' itself. *If ''q'' is not prime, then some prime factor ''p'' divides ''q''. If this factor ''p'' were in our list, then it would divide ''P'' (since ''P'' is the product of every number ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitely Large
Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was 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 l'Hôpital and 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 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 infinite sets and infinite numbers, showing that they can be of various sizes. For example, if a line is viewed as the set of all of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sufficiently Large
In the mathematical areas of number theory and analysis, an infinite sequence or a function is said to eventually have a certain property, if it doesn't have the said property across all its ordered instances, but will after some instances have passed. The use of the term "eventually" can be often rephrased as "for sufficiently large numbers", and can be also extended to the class of properties that apply to elements of any ordered set (such as sequences and subsets of \mathbb). Notation The general form where the phrase eventually (or sufficiently large) is found appears as follows: :P is ''eventually'' true for x (P is true for ''sufficiently large'' x), where \forall and \exists are the universal and existential quantifiers, which is actually a shorthand for: :\exists a \in \mathbb such that P is true \forall x \ge a or somewhat more formally: :\exists a \in \mathbb: \forall x \in \mathbb:x \ge a \Rightarrow P(x) This does not necessarily mean that any particular val ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Jargon
The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in lectures, and sometimes in print, as informal shorthand for rigorous arguments or precise ideas. Much of this is common English, but with a specific non-obvious meaning when used in a mathematical sense. Some phrases, like "in general", appear below in more than one section. Philosophy of mathematics ; abstract nonsense:A tongue-in-cheek reference to category theory, using which one can employ arguments that establish a (possibly concrete) result without reference to any specifics of the present problem. For that reason, it's also known as ''general abstract nonsense'' or ''generalized abstract nonsense''. ; canonical:A reference to a standard or choice-free presentation of some mathematical object (e.g., canonical map, canonica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
