
In mathematics, the surreal numbers are a field (mathematics) containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. By limiting the construction to a Grothendieck universe, a set is obtained set, rather than a class (set theory), with an honest field with the cardinality of some strongly inaccessible cardinal. The definition and construction of the surreals is due to John Conway, and exemplifies Conway's characteristic notational cleverness and originality. 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''. == Constructing surreal numbers == The basic idea behind the construction of surreal numbers is similar to Dedekind cuts. We construct new numbers by representing them with two sets of numbers, ''L'' and ''R'', that approximate the new number; the set ''L'' contains a set of numbers below the new number and the set ''R'' contains a set of numbers above the new number. We will write such an approximation as { ''L'' | ''R'' }. We will pose no restrictions upon ''L'' and ''R'' except that each of the numbers in ''L'' should be smaller than any number in ''R''. For example, { {1, 2} | {5, 8} } is a valid construction of a certain number between 2 and 5. (Which number exactly and why will be explained later on.) The sets are explicitly allowed to be empty. The informal interpretation of a pair { ''L'' | {} } will be "a number higher than any number in ''L''", and of { {} | ''R'' } "a number lower than any number in ''R''". This leads to the following construction rule: ;Construction Rule: If ''L'' and ''R'' are two sets of surreal numbers and no member of ''R'' is less than or equal to any member of ''L'' then { ''L'' | ''R'' } is a surreal number. Given a surreal number ''x'' = { ''XL'' | ''XR'' } the sets ''XL'' and ''XR'' are called the ''left set'' of ''x'' and ''right set'' of ''x'' respectively. To avoid lots of brackets we will write { {''a'', ''b'', ... } | { ''x'', ''y'', ... } } simply as { ''a'', ''b'', ... | ''x'', ''y'', ... } and { {''a''} | {} } as { ''a'' | } and { {} | {''a''} } as { | ''a'' }. In order for the generated numbers to actually qualify as numbers there has to be a "less than or equal to" relation (here written as ≤) defined on them. This is supplied by the following rule: ;Comparison Rule: For a surreal number ''x'' = { ''XL'' | ''XR'' } and ''y'' = { ''YL'' | ''YR'' } it holds that ''x'' ≤ ''y'' if and only if ''y'' is less than or equal to no member of ''XL'', and no member of ''YR'' is less than or equal to ''x''. The two rules are recursion, so we need some form of mathematical induction to put them to work. An obvious candidate would be ''finite induction'', i.e., generate all numbers that can be constructed by applying the construction rule a finite number of times, but, as will be explained later on, things get really interesting if we also allow transfinite induction, i.e., apply the rule more often than that. If we want the generated numbers to represent numbers then the ordering that is defined upon them should be a total order. However, the relation ≤ defines only a total preorder, i.e., it is not antisymmetric relation. To remedy this we define the binary relation == over the generated surreal numbers such that : ''x'' == ''y'' iff ''x'' ≤ ''y'' and ''y'' ≤ ''x''. Since this defines an equivalence relation the ordering on the equivalence classes implied by ≤ will be a total order. The interpretation of this will be that if ''x'' and ''y'' are in the same equivalence class then they actually represent the same number. The equivalence classes to which ''x'' and ''y'' belong are denoted as [''x''] and [''y''] respectively. So if ''x'' and ''y'' belong to the same equivalence class then [''x''] = [''y'']. Let us now consider some examples and see how they behave under the ordering. The most simple example is of course : { | } ie: { {} | {} } which can be constructed without any induction at all. We will call this number 0 and the equivalence class [0] will be written as 0. By applying the construction rule we can consider the following three numbers : { 0 | }, { | 0 } and { 0 | 0 } The last number is however not a valid surreal number because 0 ≤ 0. If we now consider the ordering of the valid surreal numbers we will see that : { | 0 } < 0 < { 0 | } where ''x'' < ''y'' denotes that not(''y'' ≤ ''x''). We will refer to { | 0 } and { 0 | } as -1 and 1 respectively, and the corresponding equivalence classes as simply -1 and 1, respectively. Since every equivalence class contains only one element that has so far been defined, we can replace in statements about ordering the surreal numbers with their equivalence classes without the risk of ambiguity. For example, the statement above could also have been written as: : { | 0 } < 0 < { 0 | } or even : -1 < 0 < 1. If we apply the construction rule once more we obtain the following ordered set: : { | -1 } == { | -1, 0 } == { | -1, 1 } == { | -1, 0, 1 } < : { | 0, 1 } == -1 < : { -1 | 0 } == { -1 | 0, 1 } < : { -1 | } == { | 1 } == { -1 | 1 } == 0 < : { 0 | 1 } == { -1, 0 | 1 } < : { -1, 0 | } == 1 < : { 1 | } == { 0, 1 | } == { -1, 1 | } == { -1, 0, 1 | } We can now make three observations: # We have found four new equivalence classes: [{ | -1 }], [{ -1 | 0 }], [{ 0 | 1 }], and [{ 1 | }]. # All equivalence classes now contain more than one element. # The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set. The first observation raises the question of the interpretation of these new equivalence classes. Since the informal interpretation of { | -1 } is "the number just before -1" we will call it number -2 and denote its equivalence class as -2. For a similar reason we will call { 1 | } number 2 and its equivalence class 2. The number { -1 | 0 } is a number between -1 and 0 and we will call it -1/2 and its equivalence class -1/2. Finally we will call { 0 | 1 } the number 1/2 and its equivalence class 1/2. More justification for these names will be given once we have defined addition and multiplication. The second observation raises the question if we can still replace the surreal numbers with their equivalence classes. Fortunately the answer is yes because it can be shown that : if [''XL''] = [''YL''] and [''XR''] = [''YR''] then [{ ''XL'' | ''XR'' }] = [{ ''YL'' | ''YR'' }] where [''X''] denotes { [''x''] | ''x'' in ''X'' }. So the description of the ordered set that was found above can be rewritten to: : { | -1 } == { | -1, 0 } == { | -1, 1 } == { | -1, 0, 1 } < : { |0, 1 } == -1 < : { -1 | 0 } == { -1| 0, 1 } < : { -1 | } == { | 1 } == { -1 | 1 } == 0 < : { 0 | 1 } == { -1, 0 | 1 } < : { -1, 0 | } == 1 < : { 1 | } == { 0, 1 | } == { -1, 1 | } == { -1, 0, 1 | } which in turn can be rewritten as : -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2. The third observation extends to all surreal numbers with finite left and right sets. For infinite left or right set, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element. The number { {1, 2} | {5, 8} } therefore is equivalent to { 2 | 5 }, which will be exactly calculated later. == Computing with surreal numbers == The addition and multiplication of surreal numbers are defined by the following three rules: ;Addition: ''x'' + ''y'' = { ''XL'' + ''y'' ∪ ''x'' + ''YL'' | ''XR'' + ''y'' ∪ ''x'' + ''YR'' } where ''X'' + ''y'' = { ''x'' + ''y'' | ''x'' in ''X'' } and ''x'' + ''Y'' = { ''x'' + ''y'' | ''y'' in ''Y'' }. ;Negation: -''x'' = { -''XR'' | -''XL'' } where -''X'' = { -''x'' | ''x'' in ''X'' } ;Multiplication: ''xy'' = { (''XLy'' + ''xYL'' - ''XLYL'') ∪ (''XRy'' + ''xYR'' - ''XRYR'') | (''XLy'' + ''xYR'' - ''XLYR'') ∪ (''XRy'' + ''xYL'' - ''XRYL'') } where ''XY'' = { ''xy'' | ''x'' in ''X'' and ''y'' in ''Y'' }, ''Xy'' = ''X''{''y''} and ''xY'' = {''x''}''Y''. These operations can be shown to be well-defined for surreal numbers, i.e., if they are applied to well-defined surreal numbers then the result will again be a well-defined surreal number, i.e., the left set of the result will be "smaller" than the right set. With these rules we can now verify that the chosen names of the numbers we found so far are appropriate. It holds for instance that 0 + 0 = 0, 1 + 1 = 2, -(1) = -1 and 1/2 + 1/2 == 1. (Note the use of equality = and equivalence ==!) The operations as defined above are defined for surreal numbers but we would like to generalize them for the equivalence classes we defined on them. This can be done without ambiguity because it holds that : if [''x''] = [''x' ''] and [''y'']=[''y' ''] then [''x'' + ''y''] = [''x' ''+ ''y' ''] and [-''x''] = [-''x' ''] and [''xy''] = [''x'y' ''] Finally it can be shown that the generalized operations on the equivalence classes have the desired algebraic properties, i.e., the equivalence classes plus their ordering and the algebraic operations constitute an ordered field, with the caveat that they do not form a set but a proper mathematical class, see below. 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 numbers and real numbers.) From now on we don't distinguish a surreal number from its equivalence class, and call the equivalence class itself a surreal number. == Generating surreal numbers using finite induction == Until now we have not really looked at what numbers we can and cannot create by applying the construction rule. We will first start with the numbers that can be created by applying the rule a finite number of times. We do this by inductively defining ''Sn'' with ''n'' a natural number as follows: * ''S0'' = {0} * ''S''''i'' + 1 is ''Si'' plus the set of all surreal numbers that are generated by the construction rule from subsets of ''Si''. The set of all surreal numbers that are generated in some ''Si'' is denoted as ''S''ω. The first sets of equivalence classes we will find are as follows: : ''S0'' = { 0 } : ''S1'' = { -1 < 0 < 1 } : ''S2'' = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} : ''S3'' = { -3 < -2 < -1 1/2 < -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } : ''S4'' = ... This leads to the following observations: # In every step the maximum (minimum) is increased (decreased) by 1. # In every step we find the numbers that are in the middle of two consecutive numbers from the previous step. As a consequence all generated numbers are dyadic fractions, i.e., can be written as an irreducible fraction :: ''a'' / 2''b'' where ''a'' and ''b'' are integers and ''b'' ≥ 0. This means that fractions such as 1/3, 2/3, 4/3, 1/5, 5/3, 1/6 et cetera, will not be generated. Note that we can generate numbers that are arbitrarily close to them, but the numbers themselves are never generated. == "To Infinity and Beyond" == The next step consists of taking ''S''ω and continuing to apply the construction rule to it and thus constructing ''S''ω+1, ''S''ω+2 et cetera. Note that the left sets and right sets may now become infinite. In fact, we can define a set ''S''''a'' for any ordinal number ''a'' by transfinite induction. The first time a given surreal number appears in this process is called its ''birthday''. Every surreal number has an ordinal number as its birthday. For example, the birthday of 0 is 0, and the birthday of 1/2 is 2. Already in ''S''ω+1 will we find the fractions that were missing in ''S''ω. For example, the fraction 1/3 can be defined as : 1/3 = { { ''a'' / 2''b'' in ''S''ω | 3''a'' < 2''b'' } | { ''a'' / 2''b'' in ''S''ω | 3''a'' > 2''b'' } }. The correctness of this definition follows from the fact that : 3(1 / 3) == 1. The birthday of 1/3 is ω+1. Not only do all the rest of the rational numbers appear in ''S''ω+1; the remaining finite real numbers do too. For example : Pi = {3, 25/8, 201/64, ... | ..., 101/32, 51/16, 13/4, 7/2, 4}. Another number that is already constructed in ''S''ω+1 is : ε = { 0 | ..., 1/16, 1/8, 1/4, 1/2, 1 }. It is easy to see that this number is larger than zero but less than all positive fractions, and therefore an infinitesimal number. The name for its equivalence class is therefore ε. It is not the only positive infinitesimal because it holds for instance that : 2ε = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 } and : ε / 2 = { 0 | ε }. Note that these numbers are not yet generated in ''S''ω+1. Next to infinitely small numbers also infinitely big numbers are generated such as : ω = { ''S''ω | }. Its value is clearly bigger than any number in ''S''ω and its equivalence class is therefore called ω. This number is equivalent with the ordinal number with the same name. We also have the equality : ω = [{ 1, 2, 3, 4, ... | }] In fact, all ordinal numbers can be expressed as surreal numbers. Since addition and subtraction is defined for all surreal numbers we can use ω like any other number and show for example that : ω + 1 = { ω | } and : ω - 1 = { ''S''ω | ω }. We can also do this for bigger numbers : ω + 2 = { ω + 1 | }, : ω + 3 = { ω + 2 | }, : ω - 2 = { ''S''ω | ω - 1 } and : ω - 3 = { ''S''ω | ω - 2 } and even ω itself : ω + ω = { ω + ''S''ω | } where ''x'' + ''Y'' = { ''x'' + ''y'' | ''y'' in ''Y'' }. Just as 2ω is bigger than ω it can also be shown that ω/2 is smaller than ω because : ω/2 = { ''S''ω | ω - ''S''ω } where ''x'' - ''Y'' = { ''x'' - ''y'' | ''y'' in ''Y'' }. Finally, it can be shown that there is a close relationship between ω and ε because it holds that : 1 / ε = ω Note that addition of ordinals differs from addition of their surreal representations. The sum 1 + ω equals ω as ordinals, but as surreals 1 + ω = ω + 1 > ω. Since every surreal number is constructed from surreal numbers "older" than itself, we can prove many theorems about surreals using transfinite induction: We show that a theorem holds for 0, and then show that it holds for ''x'' = { ''XL'' | ''XR'' } if it holds for all elements of ''XL'' and ''XR''. Lots of numbers can be generated this way; in fact so many that no set can hold them all. The surreal numbers, like the ordinal numbers, form a proper Mathematical class. == Games == 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: ;Construction Rule: If ''L'' and ''R'' are two sets of games then { ''L'' | ''R'' } is a game. Addition, negation, multiplication, and comparison are all defined the same way for both surreal numbers and games. Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 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), 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'', or ''fuzzy game'' (incomparable with zero, such as {1|-1}). 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. 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. == Surreal numbers and combinatorial game theory == The surreal numbers were originally motivated by studies of the game Go (board game), 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. We consider games with these properties: * Two players (named ''Left'' and ''Right'') * Deterministic (no dice or shuffled cards) * No hidden information (such as cards or tiles that a player hides) * Players alternate taking turns * Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row * As soon as there are no legal moves left for a player, the game ends, and that player loses 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. 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''. The winner of the game is determined: * If x>0 then Left will win * If x<0 then Right will win * If x=0 then the player who goes second will win * If x is fuzzy game then the player who goes first will win Sometimes when a game nears the end, it will decompose into several smaller games that do not interact. 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: :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''. In other words, gluing together several non-interacting games is equivalent to simply ''adding'' their Games. Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing yose, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames. 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. == Further reading == * Donald Knuth's original exposition: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. 1974, ISBN 0201038129. More information can be found at [http://www-cs-faculty.stanford.edu/~knuth/sn.html the book's official homepage] * 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 1568811276. * 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 1568811306. * Martin Gardner Penrose Tiles to Trapdoor Ciphers chapter 4 — not especially technical overview; reprints the 1976 Scientific American article ==External links== * [http://www.tondering.dk/claus/surreal.html A gentle yet thorough introduction by Claus Tøndering] * Combinatorial game theory Mathematical logic
Could someone please explain what surreal numbers actually ''are''? -- User:Janet Davis I'll give it a try. See the new paragraph just below the introduction -- MattBrubeck I second that! --User:LMS This is a huge improvement on what was here when I last looked. I even think I understand it. :-) Thanks! --User:Janet Davis Hm. Perhaps I went a bit overboard with my explanation. (I felt that the construction had to be explained to understand the difference between hyperreal numbers and surreal numbers.) Unfortunately I have to get back to work now and the page is not really finished. Perhaps next week. --User:Jan Hidders Jan: Excellent work on editing the introduction. It's now much clearer than I left it after my contributions. -- MattBrubeck Wow, this page is great. Two questions: are the hyperreals embedded in the surreals? How about the ordinals? Ordinal arithmetic is non-commutative, so there must be some problems. --AxelBoldt And two more questions: what about the topology of the surreals? Also, the first paragraph says they don't form a "class" of numbers, but then later it says they form an ordered field. These don't seem to go together. Good questions. I don't know enough about hyperreal numbers to say if they can be embedded into the surreal numbers. Studying they hyperreals is still somewhere on my to-do list. There are some very nice resources on Hyperreals and non-standard analysis on-line: http://online.sfsu.edu/~brian271/nsa.pdf http://www.ugcs.caltech.edu/~shulman/math/nonstandard/node9.html And if you want to find more the magic word is "ultrafilters" :-) Actually I think we should quickly extend the article on hyperreals because it is #1 in Google right now. :-) I think you are right about the Ordinals; hyperreals and surreals both satisfy the algebraic rules of the reals, so there can be an order homomorphism but not an homomorphism that respects the operators. -- JanHidders ---- I don't think it's really correct to say that the surreals form an ordered field, as they are a proper class, not a set. There is no largest ordered field - in fact there are hyperreal fields of arbitrarily large cardinality.
Zundark, 2001-08-17 You are of course correct; in a proper treatment one would distinguish between fields which are sets and "big fields" which are classes, and then we could say that the surreals form a big ordered field and every big ordered field embeds in the surreals. By the way, do you know if the embeddings are unique? --AxelBoldt I don't know if they're unique. In fact I didn't even know every big ordered field embeds in the surreals, although I did know this was true for ordinary ordered fields. (But I don't know much about the surreals anyway.)
Zundark, 2001-08-17 The URL http://www.tondering.dk/claus/surreal.html for the "gentle yet thorough introduction" doesn't (currently) seem to work, nor does any obvious modification of it. As for the non-commutativity of ordinal addition, sure, but there is something called the "natural" or "Hessenberg" sum on ordinals which is commutative. (The definition uses the Cantor normal form, and basically just "sorts" the summands.) Maybe this is what extends to Conway's addition? ----- Have you tried that lately? It works fine for me. user:Koyaanis Qatsi, Monday, July 8, 2002 ----- Why aren't infinitesimals listed on this page at all? What about * and up? Yes those numbers should be mentioned, but you should note they are pseudosurreal numbers(or Games).--SurrealWarrior ----- What's the algebraic closure of surreal numbers? -- Kaol There's some theorem telling you that you get the alg. closure of real-closed fields by adjoining the square root of -1 (Artin and another, I recall). Anyway my guess is that it's the 'obvious' complex number analogue, ie as small as it could be. User:Charles Matthews 11:12, 25 Oct 2003 (UTC) ---- ''Mathematicians have praised the surreal numbers for being simpler, more general, and more cleanly constructed than the more common real number system.'' :Really? Maybe for graduate students and professionals. First of all, I don't understand the comparison, it's apples and oranges. The reals only aim to construct the reals, the surreals are much more ambitious. For dealing with a very abstract notion of "Dedekind cut", comparisons, a system containing lots of other systems, etc. surreals are good. But if you JUST want to get the reals, it's like hitting a fly with frying pan. While I find the subject fascinating and would love to read a more rigorous presentation (i.e. one that explicitly quotes results from set theory, instead of "intuitively" doing things), I would find it difficult to present surreals to say, an undergrad analysis class (at the level of baby Rudin or so). To do them justice (in what supposed to be a rigorous class) would require lots of set theory, a clear presentation of recursive definition, ordinal numbers and arithmetic, and distinction between sets and proper classes. Neither Dedekind cuts nor Cauchy classes require much understand of these. ::I think the point was that for dedekind cuts, you first need to construct the integers, then construct the rationals as ordered pairs of integers, then use those to construct the reals. In doing so you get a second construction of all the rationals and integers (i.e the rational elements of your new real numbers) and it's a bit ugly. The surreal numbers have the advantage that all the numbers used in the construction are surreal numbers themselves. ---- "ω + ω = { ω + Sω | } where x + Y = { x + y | y in Y }" looks confusing since "|" seems to mean "left-right cut" when first used and "such that" in the second use. The same is true for the original definitions of addition, negation and multiplication. --User:Henrygb 10:01, 2 Jun 2004 (UTC) Maybe { ω + Sω | } should be changed to [ ω + Sω | ] (It would have to be changed through out the article). ---- I don't know much about this (yet, I'm learning), but shouldn't :2ε = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 } be :2ε = { ε | ..., 1/16, 1/8, 1/4, 1/2, 1 } ...or are the rwo simply equivalent? User:Roie m 15:27, 9 Jul 2004 (UTC) ---- Would Tic-tac-toe be an example of such a game? Do surreal numbers have any bearing on it? * Two players (named Left and Right) * Deterministic (no dice or shuffled cards) * No hidden information (such as cards or tiles that a player hides) * Players alternate taking turns We get into trouble on the following: * Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row Players must always alternate, no player may skip taking a turn, but game always has at least 5 and at most 9 moves. * As soon as there are no legal moves left for a player, the game ends, and that player loses Tic-tac-toe can end in a draw; the game otherwise ends by meeting a ''winning condition'', not a ''losing condition'' as above, though a player can make two possible wins with one move, thereby guaranteeing his/her win, since opponant cannot block both at once. Comments? == No Mathematics == This article has nothing to do with mathematics: A basic definition is given by itself and some simple calculations lead to an contradiction. This article has to be deleted! :Please explain that comment. I don't see any contradiction referred to in the article, and I think it's fairly well-established that surreal numbers are just as consistent as ZFC mathematics. User:Prumpf 03:42, 30 Aug 2004 (UTC) The relation <= ist defined by <=! In addition, this definition is not precisely defined. :Yes, that's the nature of a recursive definition. I think that section should be improved to make it more clear what actually happens, but deleting the article certainly doesn't seem justified. User:Prumpf 10:09, 30 Aug 2004 (UTC) Ok, you are right. But more and more I think about this definition (and the recursive construction), I find it really not trivial. == Differentiation in Surreals == Shouldn't this page say something about difficulties/advances in differentiation/integration of surreal functions? Anyways to define functions such as ? I'd also like to include some of my own findings but don't know how to.--SurrealWarrior ---- The answers to some of the questions on surreal numbers. Addition and multiplication on surreals do extend the natural sum and product on ordinals, defined via Cantor's normal form. The embedding of the ordinals in the surreals is defined if the following way:
Every ordinal can be identified with the set of ordinals strictly smaller than it, namely . The embbeding is defined inductively by . One can readily verify that this embedding is well-defined, preserves the order, and that the sums and products of ordinals are also ordinals. The restriction of surreal addition and multiplication to the ordinals are the natural addition and multiplication (so, if you do not know what natural + and x are, you can define them as the restriction of surreal + and x). The embedding of an ordered field in the surreal is (in general) not unique: for instance, there are many embeddings of the reals into the surreals. If we avoid the set-theoretic problems, due to the fact that surreal numbers form a proper class, for instance by restricting ourselves to the Grothendieck universe, it is easy to show that the surreal numbers have many automorphisms (as ordered field). This is general model theory: if is the strongly inaccessible cardinal we used to construct the Grothendieck universe, the corresponding set of surreal numbers will be the saturated real closed field of cardinality , and any saturated real closed field has plenty of automorphisms. The functions have been defined on the surreals, and share many of the properties of the corresponding functions on the reals (to be precise, the surreals are an elementary extension of the reals, in the first order language given by the field operations, plus the function). Moreover, every analytic function , defined on the real poly-interval , can be extended to the surreal poly-interval , using the fact that every surreal number can be represented canonically as an infinite sum of powers of (with real coefficients and surreal exponents). Again, the extended analytic functions share many of the properties of the corresponding functions on the reals. However, extending the full funtion on the surreals is problematic, essentially because the 0 set of such function would be an extension of the set of natural numbers, and there is no good candidate for such set (Conway discusses briefly this problem in his book, and shows, for instance, why the set of omnific integers is not a good candidate). As a source, you can consult the following book: Gonshor, Harry "An introduction to the theory of surreal numbers." London Mathematical Society Lecture Note Series, 110. Cambridge University Press, Cambridge, 1986. vi+192 pp. ISBN: 0-521-31205-1 and article: van den Dries, Lou; Ehrlich, Philip "Fields of surreal numbers and exponentiation." Fund. Math. 167 (2001), no. 2, 173--188. 03C64 (12J99) --User:Manta 17:41, 11 Apr 2005 (UTC) What is the definitions for the functions:&?--User:SurrealWarrior 01:46, 13 Apr 2005 (UTC) If you are really interested, you should read the book of Gonshor, where the topic is treated in detail. However, here is the definition for : Given natural number , let be the -truncation of the Taylor expansion of at 0: : If , the recursive definition of is the following: : where varies amomg the natural numbers, and, if , then must be positive. For the logarithm, you can either define it to be the inverse of the exponential, or use a suitable formula, that I am not willing to write down now.--User:Manta 10:11, 13 Apr 2005 (UTC) Thanks!--User:SurrealWarrior 01:56, 15 Apr 2005 (UTC) As a further reference about integration of functions on surreal numbers, you may read the following (it is not an introductory text): :A. Fornasiero "Integration on Surreal Numbers" (PhD. thesis) http://www.dm.unipi.it/~fornasiero/phd_thesis/thesis_fornasiero_linearized.pdf --User:Manta 08:27, 19 Apr 2005 (UTC)
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In mathematics, the surreal numbers are a field (mathematics) containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number, and therefore the surreals are algebraically similar to superreal numbers and hyperreal numbers. By limiting the construction to a Grothendieck universe, a set is obtained set, rather than a class (set theory), with an honest field with the cardinality of some strongly inaccessible cardinal. The definition and construction of the surreals is due to John Conway, and exemplifies Conway's characteristic notational cleverness and originality. 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''. == Constructing surreal numbers == The basic idea behind the construction of surreal numbers is similar to Dedekind cuts. We construct new numbers by representing them with two sets of numbers, ''L'' and ''R'', that approximate the new number; the set ''L'' contains a set of numbers below the new number and the set ''R'' contains a set of numbers above the new number. We will write such an approximation as { ''L'' | ''R'' }. We will pose no restrictions upon ''L'' and ''R'' except that each of the numbers in ''L'' should be smaller than any number in ''R''. For example, { {1, 2} | {5, 8} } is a valid construction of a certain number between 2 and 5. (Which number exactly and why will be explained later on.) The sets are explicitly allowed to be empty. The informal interpretation of a pair { ''L'' | {} } will be "a number higher than any number in ''L''", and of { {} | ''R'' } "a number lower than any number in ''R''". This leads to the following construction rule: ;Construction Rule: If ''L'' and ''R'' are two sets of surreal numbers and no member of ''R'' is less than or equal to any member of ''L'' then { ''L'' | ''R'' } is a surreal number. Given a surreal number ''x'' = { ''XL'' | ''XR'' } the sets ''XL'' and ''XR'' are called the ''left set'' of ''x'' and ''right set'' of ''x'' respectively. To avoid lots of brackets we will write { {''a'', ''b'', ... } | { ''x'', ''y'', ... } } simply as { ''a'', ''b'', ... | ''x'', ''y'', ... } and { {''a''} | {} } as { ''a'' | } and { {} | {''a''} } as { | ''a'' }. In order for the generated numbers to actually qualify as numbers there has to be a "less than or equal to" relation (here written as ≤) defined on them. This is supplied by the following rule: ;Comparison Rule: For a surreal number ''x'' = { ''XL'' | ''XR'' } and ''y'' = { ''YL'' | ''YR'' } it holds that ''x'' ≤ ''y'' if and only if ''y'' is less than or equal to no member of ''XL'', and no member of ''YR'' is less than or equal to ''x''. The two rules are recursion, so we need some form of mathematical induction to put them to work. An obvious candidate would be ''finite induction'', i.e., generate all numbers that can be constructed by applying the construction rule a finite number of times, but, as will be explained later on, things get really interesting if we also allow transfinite induction, i.e., apply the rule more often than that. If we want the generated numbers to represent numbers then the ordering that is defined upon them should be a total order. However, the relation ≤ defines only a total preorder, i.e., it is not antisymmetric relation. To remedy this we define the binary relation == over the generated surreal numbers such that : ''x'' == ''y'' iff ''x'' ≤ ''y'' and ''y'' ≤ ''x''. Since this defines an equivalence relation the ordering on the equivalence classes implied by ≤ will be a total order. The interpretation of this will be that if ''x'' and ''y'' are in the same equivalence class then they actually represent the same number. The equivalence classes to which ''x'' and ''y'' belong are denoted as [''x''] and [''y''] respectively. So if ''x'' and ''y'' belong to the same equivalence class then [''x''] = [''y'']. Let us now consider some examples and see how they behave under the ordering. The most simple example is of course : { | } ie: { {} | {} } which can be constructed without any induction at all. We will call this number 0 and the equivalence class [0] will be written as 0. By applying the construction rule we can consider the following three numbers : { 0 | }, { | 0 } and { 0 | 0 } The last number is however not a valid surreal number because 0 ≤ 0. If we now consider the ordering of the valid surreal numbers we will see that : { | 0 } < 0 < { 0 | } where ''x'' < ''y'' denotes that not(''y'' ≤ ''x''). We will refer to { | 0 } and { 0 | } as -1 and 1 respectively, and the corresponding equivalence classes as simply -1 and 1, respectively. Since every equivalence class contains only one element that has so far been defined, we can replace in statements about ordering the surreal numbers with their equivalence classes without the risk of ambiguity. For example, the statement above could also have been written as: : { | 0 } < 0 < { 0 | } or even : -1 < 0 < 1. If we apply the construction rule once more we obtain the following ordered set: : { | -1 } == { | -1, 0 } == { | -1, 1 } == { | -1, 0, 1 } < : { | 0, 1 } == -1 < : { -1 | 0 } == { -1 | 0, 1 } < : { -1 | } == { | 1 } == { -1 | 1 } == 0 < : { 0 | 1 } == { -1, 0 | 1 } < : { -1, 0 | } == 1 < : { 1 | } == { 0, 1 | } == { -1, 1 | } == { -1, 0, 1 | } We can now make three observations: # We have found four new equivalence classes: [{ | -1 }], [{ -1 | 0 }], [{ 0 | 1 }], and [{ 1 | }]. # All equivalence classes now contain more than one element. # The equivalence class of a number depends only on the maximal element of its left set and the minimal element of the right set. The first observation raises the question of the interpretation of these new equivalence classes. Since the informal interpretation of { | -1 } is "the number just before -1" we will call it number -2 and denote its equivalence class as -2. For a similar reason we will call { 1 | } number 2 and its equivalence class 2. The number { -1 | 0 } is a number between -1 and 0 and we will call it -1/2 and its equivalence class -1/2. Finally we will call { 0 | 1 } the number 1/2 and its equivalence class 1/2. More justification for these names will be given once we have defined addition and multiplication. The second observation raises the question if we can still replace the surreal numbers with their equivalence classes. Fortunately the answer is yes because it can be shown that : if [''XL''] = [''YL''] and [''XR''] = [''YR''] then [{ ''XL'' | ''XR'' }] = [{ ''YL'' | ''YR'' }] where [''X''] denotes { [''x''] | ''x'' in ''X'' }. So the description of the ordered set that was found above can be rewritten to: : { | -1 } == { | -1, 0 } == { | -1, 1 } == { | -1, 0, 1 } < : { |0, 1 } == -1 < : { -1 | 0 } == { -1| 0, 1 } < : { -1 | } == { | 1 } == { -1 | 1 } == 0 < : { 0 | 1 } == { -1, 0 | 1 } < : { -1, 0 | } == 1 < : { 1 | } == { 0, 1 | } == { -1, 1 | } == { -1, 0, 1 | } which in turn can be rewritten as : -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2. The third observation extends to all surreal numbers with finite left and right sets. For infinite left or right set, this is valid in an altered form, since infinite sets might not contain a maximal or minimal element. The number { {1, 2} | {5, 8} } therefore is equivalent to { 2 | 5 }, which will be exactly calculated later. == Computing with surreal numbers == The addition and multiplication of surreal numbers are defined by the following three rules: ;Addition: ''x'' + ''y'' = { ''XL'' + ''y'' ∪ ''x'' + ''YL'' | ''XR'' + ''y'' ∪ ''x'' + ''YR'' } where ''X'' + ''y'' = { ''x'' + ''y'' | ''x'' in ''X'' } and ''x'' + ''Y'' = { ''x'' + ''y'' | ''y'' in ''Y'' }. ;Negation: -''x'' = { -''XR'' | -''XL'' } where -''X'' = { -''x'' | ''x'' in ''X'' } ;Multiplication: ''xy'' = { (''XLy'' + ''xYL'' - ''XLYL'') ∪ (''XRy'' + ''xYR'' - ''XRYR'') | (''XLy'' + ''xYR'' - ''XLYR'') ∪ (''XRy'' + ''xYL'' - ''XRYL'') } where ''XY'' = { ''xy'' | ''x'' in ''X'' and ''y'' in ''Y'' }, ''Xy'' = ''X''{''y''} and ''xY'' = {''x''}''Y''. These operations can be shown to be well-defined for surreal numbers, i.e., if they are applied to well-defined surreal numbers then the result will again be a well-defined surreal number, i.e., the left set of the result will be "smaller" than the right set. With these rules we can now verify that the chosen names of the numbers we found so far are appropriate. It holds for instance that 0 + 0 = 0, 1 + 1 = 2, -(1) = -1 and 1/2 + 1/2 == 1. (Note the use of equality = and equivalence ==!) The operations as defined above are defined for surreal numbers but we would like to generalize them for the equivalence classes we defined on them. This can be done without ambiguity because it holds that : if [''x''] = [''x' ''] and [''y'']=[''y' ''] then [''x'' + ''y''] = [''x' ''+ ''y' ''] and [-''x''] = [-''x' ''] and [''xy''] = [''x'y' ''] Finally it can be shown that the generalized operations on the equivalence classes have the desired algebraic properties, i.e., the equivalence classes plus their ordering and the algebraic operations constitute an ordered field, with the caveat that they do not form a set but a proper mathematical class, see below. 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 numbers and real numbers.) From now on we don't distinguish a surreal number from its equivalence class, and call the equivalence class itself a surreal number. == Generating surreal numbers using finite induction == Until now we have not really looked at what numbers we can and cannot create by applying the construction rule. We will first start with the numbers that can be created by applying the rule a finite number of times. We do this by inductively defining ''Sn'' with ''n'' a natural number as follows: * ''S0'' = {0} * ''S''''i'' + 1 is ''Si'' plus the set of all surreal numbers that are generated by the construction rule from subsets of ''Si''. The set of all surreal numbers that are generated in some ''Si'' is denoted as ''S''ω. The first sets of equivalence classes we will find are as follows: : ''S0'' = { 0 } : ''S1'' = { -1 < 0 < 1 } : ''S2'' = { -2 < -1 < -1/2 < 0 < 1/2 < 1 < 2} : ''S3'' = { -3 < -2 < -1 1/2 < -1 < -3/4 < -1/2 < -1/4 < 0 < 1/4 < 1/2 < 3/4 < 1 < 1 1/2 < 2 < 3 } : ''S4'' = ... This leads to the following observations: # In every step the maximum (minimum) is increased (decreased) by 1. # In every step we find the numbers that are in the middle of two consecutive numbers from the previous step. As a consequence all generated numbers are dyadic fractions, i.e., can be written as an irreducible fraction :: ''a'' / 2''b'' where ''a'' and ''b'' are integers and ''b'' ≥ 0. This means that fractions such as 1/3, 2/3, 4/3, 1/5, 5/3, 1/6 et cetera, will not be generated. Note that we can generate numbers that are arbitrarily close to them, but the numbers themselves are never generated. == "To Infinity and Beyond" == The next step consists of taking ''S''ω and continuing to apply the construction rule to it and thus constructing ''S''ω+1, ''S''ω+2 et cetera. Note that the left sets and right sets may now become infinite. In fact, we can define a set ''S''''a'' for any ordinal number ''a'' by transfinite induction. The first time a given surreal number appears in this process is called its ''birthday''. Every surreal number has an ordinal number as its birthday. For example, the birthday of 0 is 0, and the birthday of 1/2 is 2. Already in ''S''ω+1 will we find the fractions that were missing in ''S''ω. For example, the fraction 1/3 can be defined as : 1/3 = { { ''a'' / 2''b'' in ''S''ω | 3''a'' < 2''b'' } | { ''a'' / 2''b'' in ''S''ω | 3''a'' > 2''b'' } }. The correctness of this definition follows from the fact that : 3(1 / 3) == 1. The birthday of 1/3 is ω+1. Not only do all the rest of the rational numbers appear in ''S''ω+1; the remaining finite real numbers do too. For example : Pi = {3, 25/8, 201/64, ... | ..., 101/32, 51/16, 13/4, 7/2, 4}. Another number that is already constructed in ''S''ω+1 is : ε = { 0 | ..., 1/16, 1/8, 1/4, 1/2, 1 }. It is easy to see that this number is larger than zero but less than all positive fractions, and therefore an infinitesimal number. The name for its equivalence class is therefore ε. It is not the only positive infinitesimal because it holds for instance that : 2ε = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 } and : ε / 2 = { 0 | ε }. Note that these numbers are not yet generated in ''S''ω+1. Next to infinitely small numbers also infinitely big numbers are generated such as : ω = { ''S''ω | }. Its value is clearly bigger than any number in ''S''ω and its equivalence class is therefore called ω. This number is equivalent with the ordinal number with the same name. We also have the equality : ω = [{ 1, 2, 3, 4, ... | }] In fact, all ordinal numbers can be expressed as surreal numbers. Since addition and subtraction is defined for all surreal numbers we can use ω like any other number and show for example that : ω + 1 = { ω | } and : ω - 1 = { ''S''ω | ω }. We can also do this for bigger numbers : ω + 2 = { ω + 1 | }, : ω + 3 = { ω + 2 | }, : ω - 2 = { ''S''ω | ω - 1 } and : ω - 3 = { ''S''ω | ω - 2 } and even ω itself : ω + ω = { ω + ''S''ω | } where ''x'' + ''Y'' = { ''x'' + ''y'' | ''y'' in ''Y'' }. Just as 2ω is bigger than ω it can also be shown that ω/2 is smaller than ω because : ω/2 = { ''S''ω | ω - ''S''ω } where ''x'' - ''Y'' = { ''x'' - ''y'' | ''y'' in ''Y'' }. Finally, it can be shown that there is a close relationship between ω and ε because it holds that : 1 / ε = ω Note that addition of ordinals differs from addition of their surreal representations. The sum 1 + ω equals ω as ordinals, but as surreals 1 + ω = ω + 1 > ω. Since every surreal number is constructed from surreal numbers "older" than itself, we can prove many theorems about surreals using transfinite induction: We show that a theorem holds for 0, and then show that it holds for ''x'' = { ''XL'' | ''XR'' } if it holds for all elements of ''XL'' and ''XR''. Lots of numbers can be generated this way; in fact so many that no set can hold them all. The surreal numbers, like the ordinal numbers, form a proper Mathematical class. == Games == 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: ;Construction Rule: If ''L'' and ''R'' are two sets of games then { ''L'' | ''R'' } is a game. Addition, negation, multiplication, and comparison are all defined the same way for both surreal numbers and games. Every surreal number is a game, but not all games are surreal numbers, e.g. the game { 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), 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'', or ''fuzzy game'' (incomparable with zero, such as {1|-1}). 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. 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. == Surreal numbers and combinatorial game theory == The surreal numbers were originally motivated by studies of the game Go (board game), 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. We consider games with these properties: * Two players (named ''Left'' and ''Right'') * Deterministic (no dice or shuffled cards) * No hidden information (such as cards or tiles that a player hides) * Players alternate taking turns * Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row * As soon as there are no legal moves left for a player, the game ends, and that player loses 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. 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''. The winner of the game is determined: * If x>0 then Left will win * If x<0 then Right will win * If x=0 then the player who goes second will win * If x is fuzzy game then the player who goes first will win Sometimes when a game nears the end, it will decompose into several smaller games that do not interact. 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: :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''. In other words, gluing together several non-interacting games is equivalent to simply ''adding'' their Games. Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing yose, and realized that it would be useful to have some way to combine the analyses of non-interacting subgames. 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. == Further reading == * Donald Knuth's original exposition: Surreal Numbers: How Two Ex-Students Turned on to Pure Mathematics and Found Total Happiness. 1974, ISBN 0201038129. More information can be found at [http://www-cs-faculty.stanford.edu/~knuth/sn.html the book's official homepage] * 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 1568811276. * 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 1568811306. * Martin Gardner Penrose Tiles to Trapdoor Ciphers chapter 4 — not especially technical overview; reprints the 1976 Scientific American article ==External links== * [http://www.tondering.dk/claus/surreal.html A gentle yet thorough introduction by Claus Tøndering] * Combinatorial game theory Mathematical logic
Surreal number
Could someone please explain what surreal numbers actually ''are''? -- User:Janet Davis I'll give it a try. See the new paragraph just below the introduction -- MattBrubeck I second that! --User:LMS This is a huge improvement on what was here when I last looked. I even think I understand it. :-) Thanks! --User:Janet Davis Hm. Perhaps I went a bit overboard with my explanation. (I felt that the construction had to be explained to understand the difference between hyperreal numbers and surreal numbers.) Unfortunately I have to get back to work now and the page is not really finished. Perhaps next week. --User:Jan Hidders Jan: Excellent work on editing the introduction. It's now much clearer than I left it after my contributions. -- MattBrubeck Wow, this page is great. Two questions: are the hyperreals embedded in the surreals? How about the ordinals? Ordinal arithmetic is non-commutative, so there must be some problems. --AxelBoldt And two more questions: what about the topology of the surreals? Also, the first paragraph says they don't form a "class" of numbers, but then later it says they form an ordered field. These don't seem to go together. Good questions. I don't know enough about hyperreal numbers to say if they can be embedded into the surreal numbers. Studying they hyperreals is still somewhere on my to-do list. There are some very nice resources on Hyperreals and non-standard analysis on-line: http://online.sfsu.edu/~brian271/nsa.pdf http://www.ugcs.caltech.edu/~shulman/math/nonstandard/node9.html And if you want to find more the magic word is "ultrafilters" :-) Actually I think we should quickly extend the article on hyperreals because it is #1 in Google right now. :-) I think you are right about the Ordinals; hyperreals and surreals both satisfy the algebraic rules of the reals, so there can be an order homomorphism but not an homomorphism that respects the operators. -- JanHidders ---- I don't think it's really correct to say that the surreals form an ordered field, as they are a proper class, not a set. There is no largest ordered field - in fact there are hyperreal fields of arbitrarily large cardinality.
Zundark, 2001-08-17 You are of course correct; in a proper treatment one would distinguish between fields which are sets and "big fields" which are classes, and then we could say that the surreals form a big ordered field and every big ordered field embeds in the surreals. By the way, do you know if the embeddings are unique? --AxelBoldt I don't know if they're unique. In fact I didn't even know every big ordered field embeds in the surreals, although I did know this was true for ordinary ordered fields. (But I don't know much about the surreals anyway.)
Zundark, 2001-08-17 The URL http://www.tondering.dk/claus/surreal.html for the "gentle yet thorough introduction" doesn't (currently) seem to work, nor does any obvious modification of it. As for the non-commutativity of ordinal addition, sure, but there is something called the "natural" or "Hessenberg" sum on ordinals which is commutative. (The definition uses the Cantor normal form, and basically just "sorts" the summands.) Maybe this is what extends to Conway's addition? ----- Have you tried that lately? It works fine for me. user:Koyaanis Qatsi, Monday, July 8, 2002 ----- Why aren't infinitesimals listed on this page at all? What about * and up? Yes those numbers should be mentioned, but you should note they are pseudosurreal numbers(or Games).--SurrealWarrior ----- What's the algebraic closure of surreal numbers? -- Kaol There's some theorem telling you that you get the alg. closure of real-closed fields by adjoining the square root of -1 (Artin and another, I recall). Anyway my guess is that it's the 'obvious' complex number analogue, ie as small as it could be. User:Charles Matthews 11:12, 25 Oct 2003 (UTC) ---- ''Mathematicians have praised the surreal numbers for being simpler, more general, and more cleanly constructed than the more common real number system.'' :Really? Maybe for graduate students and professionals. First of all, I don't understand the comparison, it's apples and oranges. The reals only aim to construct the reals, the surreals are much more ambitious. For dealing with a very abstract notion of "Dedekind cut", comparisons, a system containing lots of other systems, etc. surreals are good. But if you JUST want to get the reals, it's like hitting a fly with frying pan. While I find the subject fascinating and would love to read a more rigorous presentation (i.e. one that explicitly quotes results from set theory, instead of "intuitively" doing things), I would find it difficult to present surreals to say, an undergrad analysis class (at the level of baby Rudin or so). To do them justice (in what supposed to be a rigorous class) would require lots of set theory, a clear presentation of recursive definition, ordinal numbers and arithmetic, and distinction between sets and proper classes. Neither Dedekind cuts nor Cauchy classes require much understand of these. ::I think the point was that for dedekind cuts, you first need to construct the integers, then construct the rationals as ordered pairs of integers, then use those to construct the reals. In doing so you get a second construction of all the rationals and integers (i.e the rational elements of your new real numbers) and it's a bit ugly. The surreal numbers have the advantage that all the numbers used in the construction are surreal numbers themselves. ---- "ω + ω = { ω + Sω | } where x + Y = { x + y | y in Y }" looks confusing since "|" seems to mean "left-right cut" when first used and "such that" in the second use. The same is true for the original definitions of addition, negation and multiplication. --User:Henrygb 10:01, 2 Jun 2004 (UTC) Maybe { ω + Sω | } should be changed to [ ω + Sω | ] (It would have to be changed through out the article). ---- I don't know much about this (yet, I'm learning), but shouldn't :2ε = { ε | ..., ε + 1/16, ε + 1/8, ε + 1/4, ε + 1/2, ε + 1 } be :2ε = { ε | ..., 1/16, 1/8, 1/4, 1/2, 1 } ...or are the rwo simply equivalent? User:Roie m 15:27, 9 Jul 2004 (UTC) ---- Would Tic-tac-toe be an example of such a game? Do surreal numbers have any bearing on it? * Two players (named Left and Right) * Deterministic (no dice or shuffled cards) * No hidden information (such as cards or tiles that a player hides) * Players alternate taking turns We get into trouble on the following: * Every game must end in a finite number of moves, even when the players don't alternate, and one player can move multiple times in a row Players must always alternate, no player may skip taking a turn, but game always has at least 5 and at most 9 moves. * As soon as there are no legal moves left for a player, the game ends, and that player loses Tic-tac-toe can end in a draw; the game otherwise ends by meeting a ''winning condition'', not a ''losing condition'' as above, though a player can make two possible wins with one move, thereby guaranteeing his/her win, since opponant cannot block both at once. Comments? == No Mathematics == This article has nothing to do with mathematics: A basic definition is given by itself and some simple calculations lead to an contradiction. This article has to be deleted! :Please explain that comment. I don't see any contradiction referred to in the article, and I think it's fairly well-established that surreal numbers are just as consistent as ZFC mathematics. User:Prumpf 03:42, 30 Aug 2004 (UTC) The relation <= ist defined by <=! In addition, this definition is not precisely defined. :Yes, that's the nature of a recursive definition. I think that section should be improved to make it more clear what actually happens, but deleting the article certainly doesn't seem justified. User:Prumpf 10:09, 30 Aug 2004 (UTC) Ok, you are right. But more and more I think about this definition (and the recursive construction), I find it really not trivial. == Differentiation in Surreals == Shouldn't this page say something about difficulties/advances in differentiation/integration of surreal functions? Anyways to define functions such as ? I'd also like to include some of my own findings but don't know how to.--SurrealWarrior ---- The answers to some of the questions on surreal numbers. Addition and multiplication on surreals do extend the natural sum and product on ordinals, defined via Cantor's normal form. The embedding of the ordinals in the surreals is defined if the following way:
Every ordinal can be identified with the set of ordinals strictly smaller than it, namely . The embbeding is defined inductively by . One can readily verify that this embedding is well-defined, preserves the order, and that the sums and products of ordinals are also ordinals. The restriction of surreal addition and multiplication to the ordinals are the natural addition and multiplication (so, if you do not know what natural + and x are, you can define them as the restriction of surreal + and x). The embedding of an ordered field in the surreal is (in general) not unique: for instance, there are many embeddings of the reals into the surreals. If we avoid the set-theoretic problems, due to the fact that surreal numbers form a proper class, for instance by restricting ourselves to the Grothendieck universe, it is easy to show that the surreal numbers have many automorphisms (as ordered field). This is general model theory: if is the strongly inaccessible cardinal we used to construct the Grothendieck universe, the corresponding set of surreal numbers will be the saturated real closed field of cardinality , and any saturated real closed field has plenty of automorphisms. The functions have been defined on the surreals, and share many of the properties of the corresponding functions on the reals (to be precise, the surreals are an elementary extension of the reals, in the first order language given by the field operations, plus the function). Moreover, every analytic function , defined on the real poly-interval , can be extended to the surreal poly-interval , using the fact that every surreal number can be represented canonically as an infinite sum of powers of (with real coefficients and surreal exponents). Again, the extended analytic functions share many of the properties of the corresponding functions on the reals. However, extending the full funtion on the surreals is problematic, essentially because the 0 set of such function would be an extension of the set of natural numbers, and there is no good candidate for such set (Conway discusses briefly this problem in his book, and shows, for instance, why the set of omnific integers is not a good candidate). As a source, you can consult the following book: Gonshor, Harry "An introduction to the theory of surreal numbers." London Mathematical Society Lecture Note Series, 110. Cambridge University Press, Cambridge, 1986. vi+192 pp. ISBN: 0-521-31205-1 and article: van den Dries, Lou; Ehrlich, Philip "Fields of surreal numbers and exponentiation." Fund. Math. 167 (2001), no. 2, 173--188. 03C64 (12J99) --User:Manta 17:41, 11 Apr 2005 (UTC) What is the definitions for the functions:&?--User:SurrealWarrior 01:46, 13 Apr 2005 (UTC) If you are really interested, you should read the book of Gonshor, where the topic is treated in detail. However, here is the definition for : Given natural number , let be the -truncation of the Taylor expansion of at 0: : If , the recursive definition of is the following: : where varies amomg the natural numbers, and, if , then must be positive. For the logarithm, you can either define it to be the inverse of the exponential, or use a suitable formula, that I am not willing to write down now.--User:Manta 10:11, 13 Apr 2005 (UTC) Thanks!--User:SurrealWarrior 01:56, 15 Apr 2005 (UTC) As a further reference about integration of functions on surreal numbers, you may read the following (it is not an introductory text): :A. Fornasiero "Integration on Surreal Numbers" (PhD. thesis) http://www.dm.unipi.it/~fornasiero/phd_thesis/thesis_fornasiero_linearized.pdf --User:Manta 08:27, 19 Apr 2005 (UTC)
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