Surreal number  

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- +In [[mathematics]], the '''surreal number''' system is an [[Linear continuum|arithmetic continuum]] containing the [[real number]]s as well as [[Infinity|infinite]] and [[infinitesimal|infinitesimal numbers]], respectively larger or smaller in [[absolute value]] than any positive real number. The surreals share many properties with the reals, including a [[total order]] ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an [[ordered field]].
-'''Infinity''' (symbol: '''<big>∞</big>''') refers to something ''without any limit'', and is a concept relevant in a number of fields, predominantly [[mathematics]] and [[physics]]. The English word ''infinity'' derives from [[Latin]] ''infinitas'', which can be translated as "unboundedness", itself [[calque]]d from the Greek word ''apeiros'', meaning "endless".+
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-==History==+
-Ancient cultures had various ideas about the nature of infinity. The [[Maurya Empire|ancient Indians]] and [[ancient Greece|Greeks]], unable to codify infinity in terms of a formalized mathematical system, approached infinity as a philosophical concept.+
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-===Early Greek===+
-The earliest attestable accounts of mathematical infinity come from [[Zeno of Elea]] (c.&nbsp;490&nbsp;BCE? – c.&nbsp;430&nbsp;BCE?), a [[Pre-Socratic philosophy|pre-Socratic]] Greek philosopher of southern Italy and member of the [[Eleatics|Eleatic]] School founded by [[Parmenides]]. [[Aristotle]] called him the inventor of the [[dialectic]]. He is best known for his [[Zeno's paradoxes|paradoxes]], described by [[Bertrand Russell]] as "immeasurably subtle and profound".+
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-In accordance with the traditional view of Aristotle, the [[Hellenistic]] Greeks generally preferred to distinguish the [[potential infinity]] from the [[actual infinity]]; for example, instead of saying that there are an infinity of primes, [[Euclid]] prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers ([[Euclid's Elements|Elements]], Book&nbsp;IX, Proposition&nbsp;20).+
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-However, recent readings of the [[Archimedes Palimpsest]] have hinted that Archimedes at least had an intuition about actual infinite quantities.+
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-===Early Indian===+
-The [[Isha Upanishad]] of the [[Yajurveda]] (c.&nbsp;4th&nbsp;to 3rd&nbsp;century&nbsp;BCE?) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".+
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-The [[Indian mathematics|Indian mathematical]] text [[Sūryaprajñapti|Surya Prajnapti]] (c. 3rd–4th century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:+
-* Enumerable: lowest, intermediate, and highest+
-* Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable+
-* Infinite: nearly infinite, truly infinite, infinitely infinite+
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-In the Indian work on the theory of sets, two basic types of infinite numbers are distinguished. On both physical and [[Ontology|ontological]] grounds, a distinction was made between [[Asaṃkhyeya|{{IAST|''asaṃkhyāta''}}]] ("countless, innumerable") and ''ananta'' ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.+
-<!--=== Buddhism ===+
-In some [[Buddhist]] imagery, including [[Tibetan Buddhism|Tibetan]] Buddhist [[thangka]] and [[vajrayana]] meditation deities such as [[Chenrezig]], the deity is pictured holding a [[Buddhist prayer beads|mala]] twisted in the middle to form a figure&nbsp;8. This represents the endless (infinite) cycle of existence, of birth, death, and rebirth, i.e., the [infinity of] [[samsara]].{{Citation needed|date=May 2010}}-->+
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-===Cosmology===+
-In 1584, the Italian philosopher and astronomer [[Giordano Bruno]] proposed an unbounded universe in ''On the Infinite Universe and Worlds'': "Innumerable suns exist; innumerable earths revolve around these suns in a manner similar to the way the seven planets revolve around our sun. Living beings inhabit these worlds."+
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-[[Cosmology|Cosmologists]] have long sought to discover whether infinity exists in our physical [[universe]]: Are there an infinite number of stars? Does the universe have infinite volume? Does space [[Shape of the Universe|"go on forever"]]? This is an open question of [[physical cosmology|cosmology]]. Note that the question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar [[topology]]. If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough.+
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-If, on the other hand, the universe were not curved like a sphere but had a flat topology, it could be both unbounded and infinite. The curvature of the universe can be measured through [[multipole moments]] in the spectrum of the [[Cosmic microwave background radiation|cosmic background radiation]]. As to date, analysis of the radiation patterns recorded by the [[WMAP]] spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe. The [[Planck (spacecraft)|Planck spacecraft]] launched in 2009 is expected to record the cosmic background radiation with 10 times higher precision, and will give more insight into the question of whether the universe is infinite or not.+
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-==Logic==+
-In [[logic]] an [[infinite regress]] argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form&nbsp;A) no such series exists or (form&nbsp;B) were it to exist, the thesis would lack the role (e.g.,&nbsp;of justification) that it is supposed to play."+
==See also== ==See also==
-* [[0.999...]]+* [[Hyperreal number]]
-* [[Aleph number]]+* [[Non-standard analysis]]
-* [[Infinite monkey theorem]]+* [[Dehn planes]]
-* [[Infinite set]]+
-* [[Paradoxes of infinity]]+
-* [[Surreal number]]+
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In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field.

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