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| | It can be non-spherical as per the shape of gutters. Keep in mind, if this is a replacement roof due to storm damage, the insurance company will only be paying the claim on the original roof. Repairs keep your home in normal condition and do not add value to your home or prolong its life. Lack of maintenance can lead to deterioration of your Fort Worth roofing contractor, ([http://www.manta.com/c/mx4d4d0/fort-worth-roofing-contractors www.manta.com]) roofing. Making this decision depends on how much time, money and experience you may have. <br><br>Deciding on the type of roofing can be daunting and you should consider several things. To fix a small crack, use RT-600 in a caulk gun to place a layer of the tile roof sealer into the crack and allow it to flow just over the crack. A professional could easily do the repair job, but it would be an [http://meyeroimz.livejournal.com/20093.html arlington commercial roofing,] expensive undertaking for the homeowner because it would be a job that needs to be done regularly. You want to find a roofing company that offers professional dedication to your job, this way you know they are focused on getting your roof completed securely and safely with a positive attitude. Make sure you verify there professions, ask for a list of references and proof of insurance. <br><br>A proper cleaner p - H balancing chemical for the sealer and a ladder is required while doing pressure wash. With these excellent accreditations, you will know you have found a spectacular roofing Portland contractor for all of your roofing needs. Mild to severe tornados pass through Georgia every year. Connect the downspout to the gutter at the place where you have marked its hole. Optimus sent the Autobots after the humans, and he himself intercepted the vehicle they were in. <br><br>For more complex home improvement projects, an extension ladder may be the answer. This style of roofing is easy to maintain and once fitted you will not need to do anything else to it. Proceed by applying pressure to the cemented areas to help the cement to take hold. Once the fire is established, smaller pieces of wood can be added; controlling the combustion air with the dampers. See how long the roofing contractors have been carrying out a Flat Roof Repair service and make sure they are members of the Consumer Protection Association for your safety. <br><br>Moreover, it is easy to [https://Www.Google.com/search?hl=en&gl=us&tbm=nws&q=identify identify] the issue by following some basic troubleshooting steps whereas solving the issue is a task for which you can trust a professional only. The item is not only [http://kerrywfcv.jigsy.com/entries/general/quick-advice-in-best-roofers-in-fort-worth---an-update Arlington roofing contractor,] to wash any debris in the roofing but to allow you to walk the entire top of the roofing specialist that will help you feel or hear any details of the roofing that might have rotted. If you do have an asbestos roof you will need to seek an asbestos removals specialist. Repairing a flat roof is not difficult even for complete novices. Talking to the contractor about the roofing needs is vital as they will decide what is to be done for the tough, strong and simple yet attractive roofing installation. |
| [[File:Surreal number tree.svg|thumb|The surreal number tree visualization.]]
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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]]. (Strictly speaking, the surreals are not a [[Set (mathematics)|set]], but a proper [[class (set theory)|class]].<ref>In the original formulation using [[von Neumann–Bernays–Gödel set theory]], the surreals form a proper class, rather than a set, so the term [[field (mathematics)|field]] is not precisely correct; where this distinction is important, some authors use Field or FIELD to refer to a proper class that has the arithmetic properties of a field. One can obtain a true field by limiting the construction to a [[Grothendieck universe]], yielding a set with the cardinality of some [[strongly inaccessible cardinal]], or by using a form of set theory in which constructions by [[transfinite recursion]] stop at some countable ordinal such as [[Epsilon numbers (mathematics)|epsilon nought]].</ref>) If formulated in [[Von Neumann–Bernays–Gödel set theory]], the surreal numbers are the largest possible ordered field; all other ordered fields, such as the rationals, the reals, the [[rational function]]s, the [[Levi-Civita field]], the [[superreal number]]s, and the [[hyperreal number]]s, can be realized as subfields of the surreals.<ref name=bajnok>{{cite book|last=Bajnok|first=Béla|title=An Invitation to Abstract Mathematics|year=2013|quote=Theorem 24.29. The surreal number system is the largest ordered field}}</ref> It has also been shown (in [[Von Neumann–Bernays–Gödel set theory]]) that the maximal class hyperreal field is isomorphic to the maximal class surreal field; in theories without the [[axiom of global choice]], this need not be the case, and in such theories it is not necessarily true that the surreals are the largest ordered field. The surreals also contain all [[transfinite number|transfinite]] [[ordinal number]]s; the arithmetic on them is given by the [[Ordinal arithmetic#Natural operations|natural operations]].
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| The definition and construction of the surreals is due to [[John Horton Conway]]. They were introduced in [[Donald Knuth]]'s 1974 book ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness''. This book is a mathematical novelette, and is notable as one of the rare cases where a new mathematical idea was first presented in a work of fiction. In his book, which takes the form of a dialogue, Knuth coined the term ''surreal numbers'' for what Conway had simply called ''numbers'' originally. Conway liked the new name, and later adopted it himself. Conway then described the surreal numbers and used them for analyzing games in his 1976 book ''[[On Numbers and Games]]''.
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| ==Overview==
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| The surreal numbers are constructed in stages, along with an ordering ≤ such that for any two surreal numbers ''a'' and ''b'' either ''a'' ≤ ''b'' or ''b'' ≤ ''a''. (Both may hold, in which case ''a'' and ''b'' are equivalent and denote the same number.) Numbers are formed by pairing subsets of numbers already constructed: given subsets ''L'' and ''R'' of numbers such that all the members of ''L'' are strictly less than all the members of ''R'', then the pair {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}} represents a number intermediate in value between all the members of ''L'' and all the members of ''R''.
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| Different subsets may end up defining the same number: {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}} and {{nowrap begin}}{ ''L′'' | ''R′'' }{{nowrap end}} may define the same number even if ''L'' ≠ ''L′'' and ''R'' ≠ ''R′''. (A similar phenomenon occurs when [[rational numbers]] are defined as quotients of integers: 1/2 and 2/4 are different representations of the same rational number.) So strictly speaking, the surreal numbers are [[equivalence class]]es of representations of form {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}} that designate the same number.
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| In the first stage of construction, there are no previously existing numbers so the only representation must use the empty set: {{nowrap begin}}{ | }{{nowrap end}}. This representation, where ''L'' and ''R'' are both empty, is called 0. Subsequent stages yield forms like:
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| :{{nowrap begin}}{ 0 | }{{nowrap end}} = 1
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| :{{nowrap begin}}{ 1 | }{{nowrap end}} = 2
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| :{{nowrap begin}}{ 2 | }{{nowrap end}} = 3
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| and
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| :{{nowrap begin}}{ | 0 }{{nowrap end}} = −1
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| :{{nowrap begin}}{ | −1 }{{nowrap end}} = −2
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| :{{nowrap begin}}{ | −2 }{{nowrap end}} = −3
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| The integers are thus contained within the surreal numbers. Similarly, representations arise like:
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| :{{nowrap begin}}{ 0 | 1 }{{nowrap end}} = 1/2
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| :{{nowrap begin}}{ 0 | 1/2 }{{nowrap end}} = 1/4
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| :{{nowrap begin}}{ 1/2 | 1 }{{nowrap end}} = 3/4
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| so that the [[dyadic rational]]s (rational numbers whose denominators are powers of 2) are contained within the surreal numbers.
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| After an infinite number of stages, infinite subsets become available, so that any [[real numbers|real number]] ''a'' can be represented by {{nowrap begin}}{ ''L<sub>a</sub>'' | ''R<sub>a</sub>'' },{{nowrap end}}
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| where ''L<sub>a</sub>'' is the set of all dyadic rationals less than ''a'' and
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| ''R<sub>a</sub>'' is the set of all dyadic rationals greater than ''a'' (reminiscent of a [[Dedekind cut]]). Thus the real numbers are also embedded within the surreals.
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| But there are also representations like
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| :{{nowrap begin}}{ 0, 1, 2, 3, … | }{{nowrap end}} = ω
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| :{{nowrap begin}}{ 0 | 1, 1/2, 1/4, 1/8, … }{{nowrap end}} = ε
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| where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. Moreover, the standard arithmetic operations (addition, subtraction, multiplication, and division) can be extended to these non-real numbers in a manner that turns the collection of surreal numbers into an ordered field, so that one can talk about 2ω or ω − 1 and so forth.
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| ==Construction==
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| Surreal numbers are [[inductive definition|constructed inductively]] as [[equivalence class]]es of [[ordered pair|pair]]s of sets of surreal numbers, restricted by the condition that each element of the first set is smaller than each element of the second set. The construction consists of three interdependent parts: the construction rule, the comparison rule and the equivalence rule.
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| ===Forms===
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| A ''form'' is a pair of sets of surreal numbers, called its ''left set'' and its ''right set''. A form with left set ''L'' and right set ''R'' is written {{nowrap begin}}{ ''L'' | ''R'' }{{nowrap end}}. When ''L'' and ''R'' are given as lists of elements, the braces around them are omitted.
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| Either or both of the left and right set of a form may be the empty set. The form {{nowrap begin}}{ { } | { } }{{nowrap end}} with both left and right set empty is also written {{nowrap begin}}{ | }{{nowrap end}}.
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| ===Numeric forms===
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| '''Construction Rule'''
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| :A form { ''L'' | ''R'' } is ''numeric'' if the intersection of ''L'' and ''R'' is the empty set and each element of ''R'' is greater than every element of ''L'', according to the [[order theory|order relation]] ≤ given by the comparison rule below.
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| ===Equivalence classes of numeric forms===
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| The numeric forms are placed in equivalence classes; each such equivalence class is a ''surreal number''. The elements of the left and right set of a form are drawn from the universe of the surreal numbers (not of ''forms'', but of their ''equivalence classes'').
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| '''Equivalence Rule'''
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| : Two numeric forms ''x'' and ''y'' are forms of the same number (lie in the same equivalence class) if and only if both ''x'' ≤ ''y'' and ''y'' ≤ ''x''.
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| An [[Order theory|ordering relationship]] must be [[antisymmetric relation|antisymmetric]], i.e., it must have the property that ''x'' = ''y'' (i. e., ''x'' ≤ ''y'' and ''y'' ≤ ''x'' are both true) only when ''x'' and ''y'' are the same object. This is not the case for surreal number ''forms'', but is true by construction for surreal ''numbers'' (equivalence classes).
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| The equivalence class containing {{nowrap begin}}{ | }{{nowrap end}} is labeled 0; in other words, {{nowrap begin}}{ | }{{nowrap end}} is a form of the surreal number 0.
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| ===Order===
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| The recursive definition of surreal numbers is completed by defining comparison:
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| Given numeric forms ''x'' = { ''X<sub>L</sub>'' | ''X<sub>R</sub>'' } and ''y'' = { ''Y<sub>L</sub>'' | ''Y<sub>R</sub>'' }, ''x'' ≤ ''y'' if and only if:
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| *there is no <math>\scriptstyle x_L \in X_L</math> such that ''y'' ≤ <math>\scriptstyle x_L</math> (every element in the left part of ''x'' is smaller than ''y''), and
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| *there is no <math>\scriptstyle y_R \in Y_R</math> such that <math>\scriptstyle y_R</math> ≤ ''x'' (every element in the right part of ''y'' is bigger than ''x'').
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| A comparison ''y'' ≤ ''c'' between a form ''y'' and a surreal number ''c'' is performed by choosing a form ''z'' from the equivalence class ''c'' and evaluating ''y'' ≤ ''z''; and likewise for ''c'' ≤ ''x'' and for comparison ''b'' ≤ ''c'' between two surreal numbers.
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| ===Induction===
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| This group of definitions is [[recursion|recursive]], and requires some form of [[mathematical induction]] to define the universe of objects (forms and numbers) that occur in them. The only surreal numbers reachable via ''finite induction'' are the [[Dyadic rational|dyadic fractions]]; a wider universe is reachable given some form of [[transfinite induction]].
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| '''Induction Rule'''
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| * There is a generation ''S''<sub>0</sub> = { 0 }, in which 0 consists of the single form { | }.
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| * Given any [[ordinal number]] ''n'', the generation ''S''<sub>''n''</sub> is the set of all surreal numbers that are generated by the construction rule from subsets of <math>\cup_{i < n} \scriptstyle S_i</math>.
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| The base case is actually a special case of the induction rule, with 0 taken as a label for the "least ordinal". Since there exists no ''S<sub>i</sub>'' with ''i'' < 0, the expression <math>\cup_{i < 0} \scriptstyle S_i</math> is the empty set; the only subset of the empty set is the empty set, and therefore ''S''<sub>0</sub> consists of a single surreal form { | } lying in a single equivalence class 0.
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| For every finite ordinal number ''n'', ''S<sub>n</sub>'' is [[well-order]]ed by the ordering induced by the comparison rule on the surreal numbers.
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| The first iteration of the induction rule produces the three numeric forms { | 0 } < { | } < { 0 | } (the form { 0 | 0 } is non-numeric because 0≤0). The equivalence class containing { 0 | } is labeled 1 and the equivalence class containing { | 0 } is labeled −1. These three labels have a special significance in the axioms that define a [[ring (mathematics)|ring]]; they are the additive identity (0), the multiplicative identity (1), and the additive inverse of 1 (−1). The arithmetic operations defined below are consistent with these labels.
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| For every ''i'' < ''n'', since every valid form in ''S''<sub>''i''</sub> is also a valid form in ''S''<sub>''n''</sub>, all of the numbers in ''S<sub>i</sub>'' also appear in ''S<sub>n</sub>'' (as supersets of their representation in ''S<sub>i</sub>''). (The set union expression appears in our construction rule, rather than the simpler form ''S<sub>n-1</sub>'', so that the definition also makes sense when ''n'' is a [[limit ordinal]].) Numbers in ''S<sub>n</sub>'' that are a superset of some number in ''S<sub>i</sub>'' are said to have been ''inherited'' from generation ''i''. The smallest value of α for which a given surreal number appears in ''S''<sub>α</sub> is called its ''birthday''. For example, the birthday of 0 is 0, and the birthday of −1 is 1.
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| A second iteration of the construction rule yields the following ordering of equivalence classes:
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| : { | −1 } = { | −1, 0 } = { | −1, 1 } = { | −1, 0, 1 }
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| : < { | 0 } = { | 0, 1 }
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| : < { −1 | 0 } = { −1 | 0, 1 }
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| : < { | } = { −1 | } = { | 1 } = { −1 | 1 }
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| : < { 0 | 1 } = { −1, 0 | 1 }
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| : < { 0 | } = { −1, 0 | }
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| : < { 1 | } = { 0, 1 | } = { −1, 1 | } = { −1, 0, 1 | }
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| Comparison of these equivalence classes is consistent irrespective of the choice of form. Three observations follow:
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| # ''S''<sub>2</sub> contains four new surreal numbers. Two contain extremal forms: { | −1, 0, 1 } contains all numbers from previous generations in its right set, and { −1, 0, 1 | } contains all numbers from previous generations in its left set. The others have a form that partitions all numbers from previous generations into two non-empty sets.
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| # Every surreal number ''x'' that existed in the previous "generation" exists also in this generation, and includes at least one new form: a partition of all numbers ''other than'' ''x'' from previous generations into a left set (all numbers less than ''x'') and a right set (all numbers greater than ''x'').
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| # The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set.
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| The informal interpretations of { 1 | } and { | −1 } are "the number just after 1" and "the number just before −1" respectively; their equivalence classes are labeled 2 and −2. The informal interpretations of { 0 | 1 } and { −1 | 0 } are "the number halfway between 0 and 1" and "the number halfway between −1 and 0" respectively; their equivalence classes are labeled <sup>1</sup>/<sub>2</sub> and −<sup>1</sup>/<sub>2</sub>. These labels will also be justified by the rules for surreal addition and multiplication below.
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| The equivalence classes at each stage ''n'' of induction may be characterized by their ''n''-''complete forms'' (each containing as many elements as possible of previous generations in its left and right sets). Either this complete form contains ''every'' number from previous generations in its left or right set, in which case this is the first generation in which this number occurs; or it contains all numbers from previous generations but one, in which case it is a new form of this one number. We retain the labels from the previous generation for these "old" numbers, and write the ordering above using the old and new labels:
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| : −2 < −1 < −<sup>1</sup>/<sub>2</sub> < 0 < <sup>1</sup>/<sub>2</sub> < 1 < 2.
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| The third observation extends to all surreal numbers with finite left and right sets. (For infinite left or right sets, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element.) The number { 1, 2 | 5, 8 } is therefore equivalent to { 2 | 5 }; one can establish that these are forms of 3 by using the ''birthday property'', which is a consequence of the rules above.
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| '''Birthday property'''
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| : A form ''x'' = { ''L'' | ''R'' } occurring in generation ''n'' represents a number inherited from an earlier generation ''i'' < ''n'' if and only if there is some number in ''S<sub>i</sub>'' that is greater than all elements of ''L'' and less than all elements of the ''R''. (In other words, if ''L'' and ''R'' are already separated by a number created at an earlier stage, then ''x'' does not represent a new number but one already constructed.) If ''x'' represents a number from any generation earlier than ''n'', there is a least such generation ''i'', and exactly one number ''c'' with this least ''i'' as its birthday lies between ''L'' and ''R''. ''x'' is a form of this ''c'', i. e., it lies in the equivalence class in ''S<sub>n</sub>'' that is a superset of the representation of ''c'' in generation ''i''.
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| ==Arithmetic==
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| The addition, negation (additive inverse), and multiplication of surreal number ''forms'' ''x'' = { ''X<sub>L</sub>'' | ''X<sub>R</sub>'' } and ''y'' = { ''Y<sub>L</sub>'' | ''Y<sub>R</sub>'' } are defined by three recursive formulas.
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| ===Negation===
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| Negation of a given number {{nowrap begin}}''x'' = { ''X<sub>L</sub>'' | ''X<sub>R</sub>'' }{{nowrap end}} is defined by
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| :<math>-x = - \{ X_L | X_R \} = \{ -X_R | -X_L \}</math>,
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| where the negation of a set ''S'' of numbers is given by the set of the negated elements of ''S'':
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| :<math>-S = \{ -s: s \in S \}</math>.
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| This formula involves the negation of the surreal ''numbers'' appearing in the left and right sets of ''x'', which is to be understood as the result of choosing a form of the number, evaluating the negation of this form, and taking the equivalence class of the resulting form. This only makes sense if the result is the same irrespective of the choice of form of the operand. This can be proven inductively using the fact that the numbers occurring in ''X<sub>L</sub>'' and ''X<sub>R</sub>'' are drawn from generations earlier than that in which the form ''x'' first occurs, and observing the special case:
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| : -0 = - { | } = { | } = 0.
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| ===Addition===
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| The definition of addition is also a recursive formula:
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| : <math>x + y = \{ X_L | X_R \} + \{ Y_L | Y_R \} = \{ X_L + y, x + Y_L | X_R + y, x + Y_R \}</math>,
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| where
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| :<math>X + y = \{ x + y: x \in X \} , x + Y = \{ x + y: y \in Y \}</math>.
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| This formula involves sums of one of the original operands and a surreal ''number'' drawn from the left or right set of the other. These are to be understood as the result of choosing a form of the numeric operand, performing the sum of the two forms, and taking the equivalence class of the resulting form. This only makes sense if the result is the same irrespective of the choice of form of the numeric operand. This can also be proven inductively with the special cases:
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| : 0 + 0 = { | } + { | } = { | } = 0
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| : ''x'' + 0 = x + { | } = { ''X<sub>L</sub>'' + 0 | ''X<sub>R</sub>'' + 0 } = { ''X<sub>L</sub>'' | ''X<sub>R</sub>'' } = ''x''
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| : 0 + ''y'' = { | } + ''y'' = { 0 + ''Y<sub>L</sub>'' | 0 + ''Y<sub>R</sub>'' } = { ''Y<sub>L</sub>'' | ''Y<sub>R</sub>'' } = ''y''
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| (The latter two cases are of course themselves proven inductively.)
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| ===Multiplication===
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| The recursive formula for multiplication contains arithmetic expressions involving the operands and their left and right sets, such as the expression <math>X_R y + x Y_R - X_R Y_R</math> that appears in the left set of the product of ''x'' and ''y''. This is to be understood as the set of surreal numbers resulting from choosing one number from each set that appears in the expression and evaluating the expression on these numbers. (In each individual evaluation of the expression, only one number is chosen from each set, and is substituted in each place where that set appears in the expression.)
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| This depends, in turn, on the ability to (a) multiply pairs of surreal ''numbers'' drawn from the left and right sets of ''x'' and ''y'' to get a surreal number, and negate the result; (b) multiply the surreal number ''form'' ''x'' or ''y'' and a surreal ''number'' drawn from the left or right set of the other operand to get a surreal number; and (c) add the resulting surreal numbers. This again involves special cases, this time containing 0 = { | }, the multiplicative identity 1 = { 0 | }, and its additive inverse -1 = { | 0 }.
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| : <math>\begin{align}
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| xy & = \{ X_L | X_R \} \{ Y_L | Y_R \} \\
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| & = \left\{ X_L y + x Y_L - X_L Y_L, X_R y + x Y_R - X_R Y_R | X_L y + x Y_R - X_L Y_R, x Y_L + X_R y - X_R Y_L \right\} \\
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| \end{align}</math>
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| ===Consistency===
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| It can be shown that the definitions of negation, addition and multiplication are consistent, in the sense that:
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| * addition and negation are defined recursively in terms of "simpler" addition and negation steps, so that operations on numbers with birthday ''n'' will eventually be expressed entirely in terms of operations on numbers with birthdays less than ''n'';
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| * multiplication is defined recursively in terms of additions, negations, and "simpler" multiplication steps, so that the product of numbers with birthday ''n'' will eventually be expressed entirely in terms of sums and differences of products of numbers with birthdays less than ''n'';
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| * as long as the operands are well-defined surreal number forms (each element of the left set is less than each element of the right set), the results are again well-defined surreal number forms;
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| * when forms are gathered into equivalence classes using the "birthday rule", the result of negating ''x'' or adding or multiplying ''x'' and ''y'' does not depend on the choice of form of ''x'' and ''y''; and
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| * these operations obey the associativity, commutativity, additive inverse, and distributivity axioms in the definition of a [[field (mathematics)|field]], with additive identity 0 = { | } and multiplicative identity 1 = { 0 | }.
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| With these rules one can now verify that the numbers found in the first few generations were properly labeled. The construction rule is repeated to obtain more generations of surreals:
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| : ''S<sub>0</sub>'' = { 0 }
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| : ''S<sub>1</sub>'' = { −1 < 0 < 1 }
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| : ''S<sub>2</sub>'' = { −2 < −1 < −<sup>1</sup>/<sub>2</sub> < 0 < <sup>1</sup>/<sub>2</sub> < 1 < 2}
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| : ''S<sub>3</sub>'' = { −3 < −2 < −<sup>3</sup>/<sub>2</sub> < −1 < −<sup>3</sup>/<sub>4</sub> < −<sup>1</sup>/<sub>2</sub> < −<sup>1</sup>/<sub>4</sub> < 0 < <sup>1</sup>/<sub>4</sub> < <sup>1</sup>/<sub>2</sub> < <sup>3</sup>/<sub>4</sub> < 1 < <sup>3</sup>/<sub>2</sub> < 2 < 3 }
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| : ''S<sub>4</sub>'' = { -4 < −3 < ... < -<sup>1</sup>/<sub>8</sub> < 0 < <sup>1</sup>/<sub>8</sub> < <sup>1</sup>/<sub>4</sub> < <sup>3</sup>/<sub>8</sub> < <sup>1</sup>/<sub>2</sub> < <sup>5</sup>/<sub>8</sub> < <sup>3</sup>/<sub>4</sub> < <sup>7</sup>/<sub>8</sub> < 1 < <sup>5</sup>/<sub>4</sub> < <sup>3</sup>/<sub>2</sub> < <sup>7</sup>/<sub>4</sub> < 2 < <sup>5</sup>/<sub>2</sub> < 3 < 4 }
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| ===Arithmetic closure===
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| For each [[natural number]] (finite ordinal) ''n'', all numbers generated in ''S<sub>n</sub>'' are [[dyadic fraction]]s, i.e., can be written as an [[irreducible fraction]] <math>\scriptstyle \,\frac{a}{2^b}</math>
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| where ''a'' and ''b'' are [[integer]]s and 0 ≤ ''b'' < ''n''.
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| The class of all surreal numbers that are generated in some ''S<sub>n</sub>'' for finite ''n'' may be denoted as ''S<sub>*</sub>'' = <math>\cup_{n \in N} \scriptstyle S_n</math>. One may form the three classes ''S<sub>0</sub>'' = { 0 }, ''S<sub>+</sub>'' = <math>\scriptstyle { x \in S_*: x > 0 }</math>, and ''S<sub>-</sub>'' = <math>\scriptstyle { x \in S_*: x < 0 }</math>, and state that ''S<sub>*</sub>'' is the union of these three classes. No individual ''S<sub>n</sub>'' is closed under addition and multiplication (except ''S<sub>0</sub>''), but ''S<sub>*</sub>'' is; it is the subring of the rationals consisting of all dyadic fractions.
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| At an appropriate stage of transfinite induction, the surreal numbers may be expected to form a category on which the addition and multiplication operations (as well as the surreal construction step) are closed, and in which the multiplicative inverse of every nonzero number can be found. Assuming that one can find such a class, the surreal numbers, with their ordering and these algebraic operations, constitute an [[ordered field]], with the caveat that they do not form a [[Set (mathematics)|set]] but a proper [[mathematical class|class]]. In fact, it is a very special ordered field: the biggest one. Every other ordered field can be embedded in the surreals. (See also the definition of [[rational number]]s and [[real number]]s.)
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| == "To Infinity ..." ==
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| Let there be an ordinal ω greater than the natural numbers, and define ''S''<sub>ω</sub> as the set of all surreal numbers generated by the construction rule from subsets of ''S<sub>*</sub>''. (This is the same inductive step as before, since the ordinal number ω is the smallest ordinal that is larger than all natural numbers; however, the set union appearing in the inductive step is now an infinite union of finite sets, and so this step can only be performed in a set theory that allows such a union.) A unique infinitely large positive number occurs in ''S''<sub>ω</sub>:
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| : <math>\scriptstyle \omega = \{ S_* | \} = \{ 1, 2, 3, 4, ... | \}. </math>
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| ''S''<sub>ω</sub> also contains objects that can be identified as the [[rational number]]s. For example, the ω-complete form of the fraction <sup>1</sup>/<sub>3</sub> is given by:
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| : <math>\scriptstyle \tfrac{1} {3} = \{ y \in S_*: 3 y < 1 | y \in S_*: 3 y > 1 \}</math>.
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| The product of this form of <sup>1</sup>/<sub>3</sub> with any form of 3 is a form whose left set contains only numbers less than 1 and whose right set contains only numbers greater than 1; the birthday property implies that this product is a form of 1.
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| Not only do all the rest of the [[rational number]]s appear in ''S''<sub>ω</sub>; the remaining finite [[real number]]s do too. For example
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| : <math>\scriptstyle \pi = \{ 3, \frac{25}{8},\frac{201}{64}, ... | 4, \frac{7}{2}, \frac{13}{4}, \frac{51}{16},... \}</math>.
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| The only infinities in ''S''<sub>ω</sub> are ω and -ω; but there are other non-real numbers in ''S''<sub>ω</sub> among the reals. Consider the smallest positive number in ''S''<sub>ω</sub>:
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| : <math>\scriptstyle \epsilon = \{ S_- \cup S_0 | S_+ \} = \{ 0 | 1, \tfrac{1}{2}, \tfrac{1}{4}, \tfrac{1}{8}, ... \} = \{ 0 | y \in S_* : y > 0 \}</math>.
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| This number is larger than zero but less than all positive dyadic fractions. It is therefore an [[infinitesimal]] number, often labeled ε. The ω-complete form of ε (resp. -ε) is the same as the ω-complete form of 0, except that 0 is included in the left (resp. right) set. The only "pure" infinitesimals in ''S''<sub>ω</sub> are ε and its additive inverse -ε; adding them to any dyadic fraction ''y'' produces the numbers ''y''±ε, which also lie in ''S''<sub>ω</sub>.
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| One can determine the relationship between ω and ε by multiplying particular forms of them to obtain:
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| : ω · ε = { ε · ''S''<sub>+</sub> | ω · ''S''<sub>+</sub> + ''S''<sub>*</sub> + ε · ''S''<sub>*</sub> }.
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| This expression is only well-defined in a set theory which permits transfinite induction up to <math>S_{\omega^2}</math>. In such a system, one can demonstrate that all the elements of the left set of ω · ε are positive infinitesimals and all the elements of the right set are positive infinities, and therefore ω · ε is the oldest positive finite number, i. e., 1. Consequently,
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| : <sup>1</sup>/<sub>ε</sub> = ω.
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| Some authors systematically use ω<sup>−1</sup> in place of the symbol ε.
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| ===Contents of ''S''<sub>ω</sub>===
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| Given any ''x'' = { ''L'' | ''R'' } in ''S''<sub>ω</sub>, exactly one of the following is true:
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| * ''L'' and ''R'' are both empty, in which case ''x'' = 0;
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| * ''R'' is empty and some integer ''n''≥0 is greater than every element of ''L'', in which case ''x'' equals the smallest such integer ''n'';
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| * ''R'' is empty and no integer ''n'' is greater than every element of ''L'', in which case ''x'' equals +ω;
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| * ''L'' is empty and some integer ''n''≤0 is less than every element of ''R'', in which case ''x'' equals the largest such integer ''n'';
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| * ''L'' is empty and no integer ''n'' is less than every element of ''R'', in which case ''x'' equals -ω;
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| * ''L'' and ''R'' are both non-empty, and:
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| ** some dyadic fraction ''y'' is "strictly between" ''L'' and ''R'' (greater than all elements of ''L'' and less than all elements of ''R''), in which case ''x'' equals the oldest such dyadic fraction ''y'';
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| ** no dyadic fraction ''y'' lies strictly between ''L'' and ''R'', but some dyadic fraction <math>\scriptstyle y \in L</math> is greater than or equal to all elements of ''L'' and less than all elements of ''R'', in which case ''x'' equals ''y''+ε;
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| ** no dyadic fraction ''y'' lies strictly between ''L'' and ''R'', but some dyadic fraction <math>\scriptstyle y \in R</math> is greater than all elements of ''L'' and less than or equal to all elements of ''R'', in which case ''x'' equals ''y''-ε;
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| ** every dyadic fraction is either greater than some element of ''R'' or less than some element of ''L'', in which case ''x'' is some real number that has no representation as a dyadic fraction.
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| ''S''<sub>ω</sub> is not an algebraic field, because it is not closed under arithmetic operations; consider ω+1, whose form <math>\scriptstyle \{ 1, 2, 3, 4, ... | \} + \{ 0 | \} = \{ 1, 2, 3, 4, ..., \omega | \}</math> does not lie in any number in ''S''<sub>ω</sub>. The maximal subset of ''S''<sub>ω</sub> that is closed under (finite series of) arithmetic operations is the field of real numbers, obtained by leaving out the infinities ±ω, the infinitesimals ±ε, and the infinitesimal neighbors ''y''±ε of each nonzero dyadic fraction ''y''.
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| This construction of the real numbers differs from the [[Dedekind cut]]s of [[standard analysis]] in that it starts from dyadic fractions rather than general rationals and naturally identifies each dyadic fraction in ''S''<sub>ω</sub> with its forms in previous generations. (The ω-complete forms of real elements of ''S''<sub>ω</sub> are in one-to-one correspondence with the reals obtained by Dedekind cuts, under the proviso that Dedekind reals corresponding to rational numbers are represented by the form in which the cut point is omitted from both left and right sets.) The rationals are not an identifiable stage in the surreal construction; they are merely the subset ''Q'' of ''S''<sub>ω</sub> containing all elements ''x'' such that ''x'' ''b'' = ''a'' for some ''a'' and some nonzero ''b'', both drawn from ''S''<sub>*</sub>. By demonstrating that ''Q'' is closed under individual repetitions of the surreal arithmetic operations, one can show that it is a field; and by showing that every element of ''Q'' is reachable from ''S''<sub>*</sub> by a finite series (no longer than two, actually) of arithmetic operations ''including multiplicative inversion'', one can show that ''Q'' is strictly smaller than the subset of ''S''<sub>ω</sub> identified with the reals.
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| == "... And Beyond." ==
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| Continuing to perform transfinite induction beyond ''S''<sub>ω</sub> produces more ordinal numbers α, each represented as the largest surreal number having birthday α. (This is essentially a definition of the ordinal numbers resulting from transfinite induction.) The first such ordinal is ω+1 = { ω | }. There is another positive infinite number in generation ω+1:
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| : ω−1 = { 1, 2, 3, 4, ... | ω }.
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| It is important to observe that the surreal number ω−1 is not an ordinal; the ordinal ω is not the successor of any ordinal. This is a surreal number with birthday ω+1, which is labeled ω−1 on the basis that it coincides with the sum of ω = { 1, 2, 3, 4, ... | } and −1 = { | 0 }. Similarly, there are two new infinitesimal numbers in generation ω+1:
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| : 2ε = ε + ε = { ε | 1+ε, <sup>1</sup>/<sub>2</sub>+ε, <sup>1</sup>/<sub>4</sub>+ε, <sup>1</sup>/<sub>8</sub>+ε, ... } and
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| : ε/2 = ε · <sup>1</sup>/<sub>2</sub> = { 0 | ε }.
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| At a later stage of transfinite induction, there is a number larger than ω+''k'' for all natural numbers ''k'':
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| : 2ω = ω + ω = { ω+1, ω+2, ω+3, ω+4, ... | }
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| This number may be labeled ω + ω both because its birthday is ω + ω (the first ordinal number not reachable from ω by the successor operation) and because it coincides with the surreal sum of ω and ω; it may also be labeled 2ω because it coincides with the product of ω = { 1, 2, 3, 4, ... | } and 2 = { 1 | }. It is the second limit ordinal; reaching it from ω via the construction step requires a transfinite induction on <math>\bigcup_{k < \omega} S_{\omega + k}</math>. This involves an infinite union of infinite sets, which is a "stronger" set theoretic operation than the previous transfinite induction required.
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| Note that the ''conventional'' addition and multiplication of ordinals does not always coincide with these operations on their surreal representations. The sum of ordinals 1 + ω equals ω, but the surreal sum is commutative and produces 1 + ω = ω + 1 > ω. The addition and multiplication of the surreal numbers associated with ordinals coincides with the [[ordinal arithmetic#Natural operations|natural sum and natural product]] of ordinals.
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| Just as 2ω is bigger than ω+''n'' for any natural number ''n'', there is a surreal number <sup>ω</sup>/<sub>2</sub> that is infinite but smaller than ω−''n'' for any natural number ''n''.
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| : <sup>ω</sup>/<sub>2</sub> = { ''S''<sub>*</sub> | ω − ''S''<sub>*</sub> }
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| where ''x'' − ''Y'' = { ''x'' − ''y'' | ''y'' in ''Y'' }. It can be identified as the product of ω and the form { 0 | 1 } of <sup>1</sup>/<sub>2</sub>. The birthday of <sup>ω</sup>/<sub>2</sub> is the limit ordinal 2ω.
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| == Powers of ω ==
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| To classify the "orders" of infinite and infinitesimal surreal numbers, also known as [[archimedean property|archimedean]] classes, Conway associated to each surreal number ''x'' the surreal number
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| * ω<sup>''x''</sup> = { 0, ''r'' ω<sup>''x''<sub>L</sub></sup> | ''s'' ω<sup>''x''<sub>R</sub></sup> },
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| where ''r'' and ''s'' range over the positive real numbers. If 0 ≤ ''x'' < ''y'' then ω<sup>''y''</sup> is "infinitely greater" than ω<sup>''x''</sup>, in that it is greater than ''r'' ω<sup>''x''</sup> for all real numbers ''r''. Powers of ω also satisfy the conditions
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| * ω<sup>''x''</sup> ω<sup>''y''</sup> = ω<sup>''x+y''</sup>,
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| * ω<sup>−''x''</sup> = 1/ω<sup>''x''</sup>,
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| so they behave the way one would expect powers to behave.
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| Each power of ω also has the redeeming feature of being the ''simplest'' surreal number in its archimedean class; conversely, every archimedean class within the surreal numbers contains a unique simplest member. Thus, for every positive surreal number ''x'' there will always exist some positive real number ''r'' and some surreal number ''y'' so that ''x'' − ''r'' ω<sup>''y''</sup> is "infinitely smaller" than ''x''. The exponent ''y'' is the "base ω logarithm" of ''x'', defined on the positive surreals; it can be demonstrated that log<sub>ω</sub> maps the positive surreals onto the surreals and that log<sub>ω</sub>(''xy'') = log<sub>ω</sub>(''x'') + log<sub>ω</sub>(''y'').
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| This gets extended by transfinite induction so that every surreal number ''x'' has a "normal form" analogous to the [[Ordinal arithmetic#Cantor normal form|Cantor normal form]] for ordinal numbers. Every surreal number may be uniquely written as
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| * ''x'' = ''r''<sub>0</sub> ω<sup>''y''<sub>0</sub></sup> + ''r''<sub>1</sub> ω<sup>''y''<sub>1</sub></sup> + …,
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| where every ''r''<sub>α</sub> is a nonzero real number and the ''y''<sub>α</sub>s form a strictly decreasing sequence of surreal numbers. This "sum", however, may have infinitely many terms, and in general has the length of an arbitrary ordinal number. (Zero corresponds of course to the case of an empty sequence, and is the only surreal number with no leading exponent.)
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| Looked at in this manner, the surreal numbers resemble a [[Formal power series|power series field]], except that the decreasing sequences of exponents must be bounded in length by an ordinal and are not allowed to be as long as the class of ordinals.
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| == Surcomplex numbers ==
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| A '''surcomplex number''' is a number of the form <math>a+bi </math>, where ''a'' and ''b'' are surreal numbers.<ref>[http://jamespropp.org/surreal/text.ps.gz Surreal vectors and the game of Cutblock], James Propp, August 22, 1994.</ref><ref>N. L. Alling, ''Foundations of analysis over surreal number fields'', N. L. Alling, Amsterdam: North-Holland, 1987. ISBN 0-444-70226-1.</ref> The surcomplex numbers form an [[algebraically closed]] [[field (mathematics)|field]] (except for being a proper class), [[isomorphic (mathematics)|isomorphic]] to the [[algebraic closure]] of the field generated by extending the [[rational numbers]] by a [[proper class]] of [[algebraically independent]] [[transcendental (mathematics)|transcendental]] elements. Up to field [[isomorphism]], this fact characterizes the field of surcomplex numbers within any fixed set theory.<ref>Theorem 27, ''On Numbers and Games'', John H. Conway, 2nd ed., Natick, Massachusetts: [[A K Peters, Ltd.]], 2000. ISBN 1-56881-127-6.</ref>
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| ==Games==
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| {{main|combinatorial game theory}}
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| The definition of surreal numbers contained one restriction: each element of L must be strictly less than each element of R. If this restriction is dropped we can generate a more general class known as ''games''. All games are constructed according to this rule:
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| :'''Construction Rule'''<br/> If ''L'' and ''R'' are two sets of games then { ''L'' | ''R'' } is a game.
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| Addition, negation, and comparison are all defined the same way for both surreal numbers and games.
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| Every surreal number is a game, but not all games are surreal numbers, e.g. the game [[star (game theory)|{ '''0''' | '''0''' }]] is not a surreal number. The class of games is more general than the surreals, and has a simpler definition, but lacks some of the nicer properties of surreal numbers. The class of surreal numbers forms a [[field (mathematics)|field]], but the class of games does not. The surreals have a [[total order]]: given any two surreals, they are either equal, or one is greater than the other. The games have only a [[partial order]]: there exist pairs of games that are neither equal, greater than, nor less than each other. Each surreal number is either positive, negative, or zero. Each game is either positive, negative, ''[[zero game|zero]]'', or ''[[fuzzy game|fuzzy]]'' (incomparable with zero, such as {1|−1}).
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| A move in a game involves the player whose move it is choosing a game from those available in L (for the left player) or R (for the right player) and then passing this chosen game to the other player. A player who cannot move because the choice is from the empty set has lost. A positive game represents a win for the left player, a negative game for the right player, a zero game for the second player to move, and a [[fuzzy game]] for the first player to move.
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| If ''x'', ''y'', and ''z'' are surreals, and ''x''=''y'', then ''x'' ''z''=''y'' ''z''. However, if ''x'', ''y'', and ''z'' are games, and ''x''=''y'', then it is not always true that ''x'' ''z''=''y'' ''z''. Note that "=" here means equality, not identity.
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| ==Application to combinatorial game theory==
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| The surreal numbers were originally motivated by studies of the game [[Go (board game)|Go]],<ref>{{citation | url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Conway.html | title = Conway Biography | last = O'Connor | first = J.J. | last2 = Robertson | first2 = E.F. | accessdate = 2008-01-24 }}</ref> and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized ''Game'' for the mathematical object {L|R}, and the lowercase ''game'' for recreational games like [[Chess]] or [[Go (board game)|Go]].
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| We consider games with these properties:
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| * Two players (named ''Left'' and ''Right'')
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| * [[Deterministic]] (the game at each step will completely depend on the choices the players make, rather than a random factor)
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| * No hidden information (such as cards or tiles that a player hides)
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| * Players alternate taking turns (the game may or may not allow multiple moves in a turn)
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| * Every game must end in a finite number of moves
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| * As soon as there are no legal moves left for a player, the game ends, and that player loses
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| For most games, the initial board position gives no great advantage to either player. As the game progresses and one player starts to win, board positions will occur where that player has a clear advantage. For analyzing games, it is useful to associate a Game with every board position. The value of a given position will be the Game {L|R}, where L is the set of values of all the positions that can be reached in a single move by Left. Similarly, R is the set of values of all the positions that can be reached in a single move by Right.
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| The zero Game (called 0) is the Game where L and R are both empty, so the player to move next (L or R) immediately loses. The sum of two Games G = { L1 | R1 } and H = { L2 | R2 } is defined as the Game G + H = { L1 + H, G + L2 | R1 + H, G + R2 } where the player to move chooses which of the Games to play in at each stage, and the loser is still the player who ends up with no legal move. One can imagine two chess boards between two players, with players making moves alternatively, but with complete freedom as to which board to play on. If G is the Game {L | R}, -G is the game {-R | -L}, i.e. with the role of the two players reversed. It is easy to show G - G = 0 for all Games G (where G - H is defined as G + (-H)).
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| This simple way to associate Games with games yields a very interesting result. Suppose two perfect players play a game starting with a given position whose associated Game is ''x''. We can classify all Games into four classes as follows:
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| * If x > 0 then Left will win, regardless of who plays first.
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| * If x < 0 then Right will win, regardless of who plays first.
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| * If x = 0 then the player who goes second will win.
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| * If x || 0 then the player who goes first will win.
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| More generally, we can define G > H as G - H > 0, and similarly for <, = and ||.
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| The notation G || H means that G and H are incomparable. G || H is equivalent to G−H || 0, i.e. that G > H, G < H and G = H are all false. Incomparable games are sometimes said to be ''confused'' with each other, because one or the other may be preferred by a player depending on what is added to it. A game confused with zero is said to be [[fuzzy game|fuzzy]], as opposed to [[sign (mathematics)|positive, negative, or zero]]. An example of a fuzzy game is [[star (game theory)|star (*)]].
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| Sometimes when a game nears the end, it will decompose into several smaller games that do not interact, except in that each player's turn allows moving in only one of them. For example, in Go, the board will slowly fill up with pieces until there are just a few small islands of empty space where a player can move. Each island is like a separate game of Go, played on a very small board. It would be useful if each subgame could be analyzed separately, and then the results combined to give an analysis of the entire game. This doesn't appear to be easy to do. For example, you might have two subgames where whoever moves first wins, but when they are combined into one big game, it's no longer the first player who wins. Fortunately, there is a way to do this analysis. Just use the following remarkable theorem:
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| :If a big game decomposes into two smaller games, and the small games have associated Games of ''x'' and ''y'', then the big game will have an associated Game of ''x''+''y''.
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| A game composed of smaller games is called the [[disjunctive sum]] of those smaller games, and the theorem states that the method of addition we defined is equivalent to taking the disjunctive sum of the addends.
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| Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing [[Go terms#Yose|Go endgames]], and realized that it would be useful to have some way to combine the analyses of non-interacting subgames into an analysis of their [[disjunctive sum]]. From this he invented the concept of a Game and the addition operator for it. From there he moved on to developing a definition of negation and comparison. Then he noticed that a certain class of Games had interesting properties; this class became the surreal numbers. Finally, he developed the multiplication operator, and proved that the surreals are actually a field, and that it includes both the reals and ordinals.
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| ==Alternative realizations==
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| Since Conway first introduced surreal numbers, several alternative constructions
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| have been developed.
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| ===Sign expansion===
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| ====Definitions====
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| In one alternative realization, called the ''sign-expansion'' or ''sign-sequence'' of a surreal number, a surreal number is a [[function (mathematics)|function]] whose [[domain of a function|domain]] is an [[ordinal number|ordinal]] and whose [[range (mathematics)|range]] is { − 1, + 1 }.
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| Define the binary predicate "simpler than" on numbers by ''x'' is simpler than ''y'' if ''x'' is a [[subset|proper subset]] of ''y'', ''i.e.'' if dom(''x'') < dom(''y'') and ''x''(α) = ''y''(α) for all α < dom(''x'').
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| For surreal numbers define the binary relation < to be lexicographic order (with the convention that "undefined values" are greater than −1 and less than 1). So ''x'' < ''y'' if one of the following holds:
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| * ''x'' is simpler than ''y'' and ''y''(dom(''x'')) = + 1;
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| * ''y'' is simpler than ''x'' and ''x''(dom(''y'')) = − 1;
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| * there exists a number ''z'' such that ''z'' is simpler than ''x'', ''z'' is simpler than ''y'', ''x''(dom(''z'')) = − 1 and ''y''(dom(''z'')) = + 1.
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| Equivalently, let δ(''x'',''y'') = min({ dom(''x''), dom(''y'')} ∪ { α :
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| α < dom(''x'') ∧ α < dom(''y'') ∧ ''x''(α) ≠ ''y''(α) }),
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| so that ''x'' = ''y'' if and only if δ(''x'',''y'') = dom(''x'') = dom(''y''). Then, for numbers ''x'' and ''y'', ''x'' < ''y'' if and only if one of the following holds:
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| * δ(''x'',''y'') = dom(''x'') ∧ δ(''x'',''y'') < dom(''y'') ∧ ''y''(δ(''x'',''y'')) = + 1;
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| * δ(''x'',''y'') < dom(''x'') ∧ δ(''x'',''y'') = dom(''y'') ∧ ''x''(δ(''x'',''y'')) = − 1;
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| * δ(''x'',''y'') < dom(''x'') ∧ δ(''x'',''y'') < dom(''y'') ∧ ''x''(δ(''x'',''y'')) = − 1 ∧ ''y''(δ(''x'',''y'')) = + 1.
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| For numbers ''x'' and ''y'', ''x'' ≤ ''y'' if and only if ''x'' < ''y'' ∨ ''x'' = ''y'', and ''x'' > ''y'' if and only if ''y'' < ''x''. Also ''x'' ≥ ''y'' if and only if ''y'' ≤ ''x''.
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| The relation < is [[transitive relation|transitive]], and for all numbers ''x'' and ''y'', exactly one of ''x'' < ''y'', ''x'' = ''y'', ''x'' > ''y'', holds (law of [[trichotomy (mathematics)|trichotomy]]). This means that < is a [[linear order]] (except that < is a proper class).
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| For sets of numbers, ''L'' and ''R'' such that ∀''x'' ∈ ''L'' ∀''y'' ∈ ''R'' (''x'' < ''y''), there exists a unique number ''z'' such that
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| * ∀''x'' ∈ ''L'' (''x'' < ''z'') ∧ ∀''y'' ∈ ''R'' (''z'' < ''y''),
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| * For any number ''w'' such that ∀''x'' ∈ ''L'' (''x'' < ''w'') ∧ ∀''y'' ∈ ''R'' (''w'' < ''y''), ''w'' = ''z'' or ''z'' is simpler than ''w''.
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| Furthermore, ''z'' is constructible from ''L'' and ''R'' by transfinite induction. ''z'' is the simplest number between ''L'' and ''R''. Let the unique number ''z'' be denoted by σ(''L'',''R'').
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| For a number ''x'', define its left set ''L''(''x'') and right set ''R''(''x'') by
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| * ''L''(''x'') = { ''x''|<sub>α</sub> : α < dom(''x'') ∧ ''x''(α) = + 1 };
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| * ''R''(''x'') = { ''x''|<sub>α</sub> : α < dom(''x'') ∧ ''x''(α) = − 1 },
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| then σ(''L''(''x''),''R''(''x'')) = ''x''.
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| One advantage of this alternative realization is that equality is identity, not an inductively defined relation. Unlike Conway's realization of the surreal numbers, however, the sign-expansion requires a prior construction of the ordinals, while in Conway's realization, the ordinals are constructed as particular cases of surreals.
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| However, similar definitions can be made that obviate the need for prior construction of the ordinals. For instance, we could let the surreals be the (recursively-defined) class of functions whose domain is a subset of the surreals satisfying the transitivity rule ∀''g'' ∈ dom ''f'' (∀''h'' ∈ dom ''g'' (''h'' ∈ dom ''f'' )) and whose range is { −, + }. "Simpler than" is very simply defined now—''x'' is simpler than ''y'' if ''x'' ∈ dom ''y''. The total ordering is defined by considering ''x'' and ''y'' as sets of ordered pairs (as a function is normally defined): Either ''x'' = ''y'', or else the surreal number ''z'' = ''x'' ∩ ''y'' is in the domain of ''x'' or the domain of ''y'' (or both, but in this case the signs must disagree). We then have ''x'' < ''y'' if ''x''(''z'') = − or ''y''(''z'') = + (or both). Converting these functions into sign sequences is a straightforward task; arrange the elements of dom ''f'' in order of simplicity (i.e., inclusion), and then write down the signs that ''f'' assigns to each of these elements in order. The ordinals then occur naturally as those surreal numbers whose range is { + }.
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| ====Addition and multiplication====
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| The sum ''x'' + ''y'' of two numbers, ''x'' and ''y'', is defined by induction on dom(''x'') and dom(''y'') by ''x'' + ''y'' = σ(''L'',''R''), where
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| * ''L'' = { ''u'' + ''y'' : ''u'' ∈ ''L''(''x'') } ∪{ ''x'' + ''v'' : ''v'' ∈ ''L''(''y'') },
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| * ''R'' = { ''u'' + ''y'' : ''u'' ∈ ''R''(''x'') } ∪{ ''x'' + ''v'' : ''v'' ∈ ''R''(''y'') }.
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| The additive identity is given by the number 0 = { }, ''i.e.'' the number 0 is the unique function whose domain is the ordinal 0, and the additive inverse of the number ''x'' is the number − ''x'', given by dom(− ''x'') = dom(''x''), and, for α < dom(''x''), (− ''x'')(α) = − 1 if ''x''(α) = + 1, and (− ''x'')(α) = + 1 if ''x''(α) = − 1.
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| It follows that a number ''x'' is [[Positive number|positive]] if and only if 0 < dom(''x'') and ''x''(0) = + 1, and ''x'' is [[negative number|negative]] if and only if 0 < dom(''x'') and ''x''(0) = − 1.
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| The product ''xy'' of two numbers, ''x'' and ''y'', is defined by induction on dom(''x'') and dom(''y'') by ''xy'' = σ(''L'',''R''), where
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| * ''L'' = { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''L''(''y'') } ∪ { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''R''(''y'') },
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| * ''R'' = { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''L''(''x''), ''v'' ∈ ''R''(''y'') } ∪ { ''uy'' + ''xv'' − ''uv'' : ''u'' ∈ ''R''(''x''), ''v'' ∈ ''L''(''y'') }.
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| The multiplicative identity is given by the number 1 = { (0,+ 1) }, ''i.e.'' the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.
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| ====Correspondence with Conway====
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| The map from Conway's realization to sign expansions is given by ''f''({ ''L'' | ''R'' }) = σ(''M'',''S''), where ''M'' = { ''f''(''x'') : ''x'' ∈ ''L'' } and ''S'' = { ''f''(''x'') : ''x'' ∈ ''R'' }.
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| The inverse map from the alternative realization to Conway's realization is given by ''g''(''x'') = { ''L'' | ''R'' }, where ''L'' = { ''g''(''y'') : ''y'' ∈ ''L''(''x'') } and ''R'' = { ''g''(''y'') : ''y'' ∈ ''R''(''x'') }.
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| ===Axiomatic approach===
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| In another approach to the surreals, given by Alling,<ref name="Alling">{{cite book | last = Alling | first = Norman L. | title = Foundations of Analysis over Surreal Number Fields | publisher = North-Holland | series = Mathematics Studies 141 | year = 1987 | isbn = 0-444-70226-1}}</ref> explicit construction is bypassed altogether. Instead, a set of axioms is given that any particular approach to the surreals must satisfy. Much like the [[Real numbers#Axiomatic approach|axiomatic approach]] to the reals, these axioms guarantee uniqueness up to isomorphism.
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| A triple <math>\langle \mathbf{No}, \mathrm{<}, b \rangle</math> is a surreal number system if and only if the following hold:
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| * < is a [[total order]] over '''No'''
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| * ''b'' is a function from '''No''' onto the class of all ordinals (''b'' is called the "birthday function" on '''No''').
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| * Let ''A'' and ''B'' be subclasses of '''No''' such that for all ''x'' ∈ ''A'' and ''y'' ∈ ''B'', ''x'' < ''y'' (using Alling's terminology, 〈 ''A'',''B'' 〉 is a "Conway cut" of '''No'''). Then there exists a unique ''z'' ∈ '''No''' such that ''b(z)'' is minimal and for all ''x'' ∈ ''A'' and all ''y'' ∈ ''B'', ''x'' < ''z'' < ''y''. (This axiom is often referred to as "Conway's Simplicity Theorem".)
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| * Furthermore, if an ordinal ''α'' is greater than ''b(x)'' for all ''x'' ∈ ''A'', ''B'', then ''b(z)'' ≤ ''α''. (Alling calls a system that satisfies this axiom a "full surreal number system".)
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| Both Conway's original construction and the sign-expansion construction of surreals satisfy these axioms.
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| Given these axioms, Alling<ref name="Alling"/> derives Conway's original definition of ≤ and develops surreal arithmetic.
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| ===Hahn series===
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| Alling<ref>Alling, ''op. cit.'', theorem of §6.55 (p. 246)</ref> also proves that the field of surreal numbers is isomorphic (as an ordered field) to the field of [[Hahn series]] with real coefficients on the value group of surreal numbers themselves (the series representation corresponding to the normal form of a surreal number, as defined above). This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.
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| ==Relation to hyperreals==
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| [[Philip Ehrlich]] has constructed an isomorphism between Conway's maximal surreal number field and the maximal [[hyperreal field|hyperreals]] in [[von Neumann–Bernays–Gödel set theory]].<ref>{{cite journal
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| |author=Philip Ehrlich
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| |authorlink=Philip Ehrlich
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| |year=2012
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| |title=The absolute arithmetic continuum and the unification of all numbers great and small
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| |journal=The Bulletin of Symbolic Logic
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| |volume=18
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| |issue=1
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| |pages=1–45
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| |url=http://www.math.ucla.edu/~asl/bsl/1801-toc.htm
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| |accessdate=2012-01-29 }}</ref>
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| ==See also==
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| {{Infinitesimal navbox}}
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| {{Portal|Mathematics}}
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| * [[Hyperreal number]]
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| * [[Non-standard analysis]]
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| * [[Dehn planes]]
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| ==References==
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| {{Reflist|2}}
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| ==Further reading==
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| * [[Donald Knuth]]'s original exposition: ''Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness''. 1974, ISBN 0-201-03812-9. More information can be found at [http://www-cs-faculty.stanford.edu/~knuth/sn.html the book's official homepage]
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| * An update of the classic 1976 book defining the surreal numbers, and exploring their connections to games: ''On Numbers And Games, 2nd ed.'', John Conway, 2001, ISBN 1-56881-127-6.
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| * An update of the first part of the 1981 book that presented surreal numbers and the analysis of games to a broader audience: ''Winning Ways for Your Mathematical Plays, vol. 1, 2nd ed.'', Berlekamp, Conway, and Guy, 2001, ISBN 1-56881-130-6.
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| * [[Martin Gardner]], ''Penrose Tiles to Trapdoor Ciphers,'' W. H. Freeman & Co., 1989. ISBN 0-7167-1987-8. Chapter 4 — not especially technical overview; reprints the 1976 <cite>Scientific American</cite> article.
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| * Polly Shulman, "[http://discovermagazine.com/1995/dec/infinityplusonea599 Infinity Plus One, and Other Surreal Numbers]". ''[[Discover (magazine)|Discover]],'' December 1995.
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| * A detailed, though somewhat technical, treatment of surreal numbers: ''Foundations of Analysis over Surreal Number Fields'', Alling, Norman L., 1987, ISBN 0-444-70226-1
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| * A treatment of surreals based on the sign-expansion realization: ''An Introduction to the Theory of Surreal Numbers'', Gonshor, Harry, 1986, ISBN 0-521-31205-1
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| * A detailed philosophical development of the concept of surreal numbers as a most general concept of number is [[Alain Badiou]], ''[[Number and Numbers]]'' (New York: Polity Press, 2008): ISBN 0-7456-3879-1 (paperback); ISBN 0-7456-3878-3 (hardcover)
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| ==External links==
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| * [http://www.maths.nottingham.ac.uk/personal/anw/Research/Hack/ Hackenstrings, and the 0.999... ?= 1 FAQ, by A. N. Walker]{{dead link|date=September 2012}}
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| * [http://www.tondering.dk/claus/surreal.html A gentle yet thorough introduction by Claus Tøndering]
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| * {{planetmath reference|id=3352|title=Surreal number}}
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| * [http://scientopia.org/blogs/goodmath/category/surreal-numbers/ Good Math, Bad Math: Surreal Numbers], a series of articles about surreal numbers, their variations, and their applications
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| {{Infinity}}
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| {{Number systems}}
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| {{Infinitesimals}}
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| {{DEFAULTSORT:Surreal Number}}
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| [[Category:Combinatorial game theory]]
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| [[Category:Mathematical logic]]
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| [[Category:Infinity]]
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| [[Category:Real closed field]]
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