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Surreal Number

Surreal number

In mathematics, the surreal numbers are a field 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, rather than a class, with an honest field with the cardinality of some strongly inaccessible cardinal. The definition and construction of the surreals is due to John Horton 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 . 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, 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 will be "a number higher than any number in L", and of "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 is a surreal number. Given a surreal number x = the sets X_L and X_R are called the left set of x and right set of x respectively. To avoid lots of brackets we will write simply as and as and as . 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 = and y = it holds that x ≤ y if and only if y is less than or equal to no member of X_L, and no member of Y_R is less than or equal to x. The two rules are recursive, so we need some form of 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. 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 : , and 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 < where x < y denotes that not(y ≤ x). We will refer to and 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 < or even : -1 < 0 < 1. If we apply the construction rule once more we obtain the following ordered set: :

< :

-1 < :

< :

0 < :

< :

1 < :

We can now make three observations: # We have found four new equivalence classes: [], [], [], and []. # 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 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 number 2 and its equivalence class 2. The number is a number between -1 and 0 and we will call it -1/2 and its equivalence class -1/2. Finally we will call 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 [X_L] = [Y_L] and [X_R] = [Y_R] then [] = [] where [X] denotes . So the description of the ordered set that was found above can be rewritten to: :

< :

-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 therefore is equivalent to , 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 = where X + y = and x + Y = . ;Negation: -x = where -X = ;Multiplication: xy = where XY = , Xy = X and xY = 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 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 S_n with n a natural number as follows:
- S₀ =
- S_{i + 1} is S_i plus the set of all surreal numbers that are generated by the construction rule from subsets of S_i. The set of all surreal numbers that are generated in some S_i is denoted as S_ω. The first sets of equivalence classes we will find are as follows: : S₀ = : S₁ = : S₂ = : S₃ = : 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. A number is equivalent to the simplest number between L and R, i.e., the number between L and R with the smallest ordinal as its birthday. , therefore, is equivalent to 3, because the birthday of 3 is less than the birthday of any other number between 2 and 5. 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 = . 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 : π = . Another number that is already constructed in S_ω+1 is : ε = . 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 = and : ε / 2 = . 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 = . 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 = [] 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 = and : ω - 1 = . We can also do this for bigger numbers : ω + 2 = , : ω + 3 = , : ω - 2 = and : ω - 3 = and even ω itself : ω + ω = where x + Y = . Just as 2ω is bigger than ω it can also be shown that ω/2 is smaller than ω because : ω/2 = where x - 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 = if it holds for all elements of X_L and X_R. 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 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 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 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, 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, or fuzzy (incomparable with zero, such as ). 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, and there are numerous connections between popular games and the surreals. In this section, we will use a capitalized Game for the mathematical object , 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 , 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 || 0 then the player who goes first will win. The notation G || H means that G and H are incomparable. G || H is equivalent to G-H || 0. 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, as opposed to positive, negative, or zero. An example of a fuzzy game is star (
- ). 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: :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. 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. Historically, Conway developed the theory of surreal numbers in the reverse order of how it has been presented here. He was analyzing 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.

Alternative realization

Definitions

In an alternative realization, called the sign-expansion or sign-sequence of a surreal number, a surreal number is a function whose domain is an ordinal and range is a subset of . Define the binary predicate "simpler than" on numbers by x is simpler than y if x is a proper subset of y, i.e. if dom(x) < dom(y) and x(α) = y(α) for all α < dom(x). For numbers x and y, define x < y if one of the following holds:
- x is simpler than y and y(dom(x)) = + 1;
- y is simpler than x and x(dom(y)) = - 1;
- 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. Equavalently, let δ(x,y) = min( ∪ ), so that x = y iff δ(x,y) = dom(x) = dom(y). Then, for numbers x and y, x < y iff one of the following holds:
- δ(x,y) = dom(x) ∧ δ(x,y) < dom(y) ∧ y(δ(x,y)) = + 1;
- δ(x,y) < dom(x) ∧ δ(x,y) = dom(y) ∧ x(δ(x,y)) = - 1;
- δ(x,y) < dom(x) ∧ δ(x,y) < dom(y) ∧ x(δ(x,y)) = - 1 ∧ y(δ(x,y)) = + 1. For numbers x and y, x ≤ y iff x < y ∨ x = y, x > y iff y < x, and x ≥ y iff y ≤ x. < is transitive, and for all numbers x and y, exactly one of x < y, x = y, x > y, holds (law of trichotomy). This means that < is a linear order (except that < is a proper class). For sets of numbers, L and R such that ∀x ∈ L ∀y ∈ R (x < y), there exists a unique number z such that
- ∀x ∈ L (x < z) ∧ ∀y ∈ R (z < y),
- For any number w such that ∀x ∈ L (x < w) ∧ ∀y ∈ R (w < y), w = z or z is simpler than w. 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 deonted by σ(L,R). For a number x, define its left set L(x) and right set R(x) by
- L(x) = ;
- R(x) = , then σ(L(x),R(x)) = x. 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.

Addition and Multiplication

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
- L = ∪,
- R = ∪. 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. It follows that a number x is positive iff 0 < dom(x) and x(0) = + 1, and x is negative iff 0 < dom(x) and x(0) = - 1. 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
- L = ∪ ,
- R = ∪ . The multiplicative identity is given by the number 1 = , i.e. the number 1 has domain equal to the ordinal 1, and 1(0) = + 1.

Correspondence between realizations

The map from Conway's realization to the alternative realization is given by f() = σ(M,S), where M = and S = . The inverse map from the alternative realization to Conway's realization is given by g(x) = , where L = and R = . 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]
- Category:Combinatorial game theory Category:Mathematical logic

Mathematics

Mathematics is often defined as the study of topics such as quantity, structure, space, and change. Another view, held by many mathematicians, is that mathematics is the body of knowledge justified by deductive reasoning, starting from axioms and definitions. Practical mathematics, in nearly every society, is used for such purposes as accounting, measuring land, or predicting astronomical events. Mathematical discovery or research often involves discovering and cataloging patterns, without regard for application. The remarkable fact that the "purest" mathematics often turns out to have practical applications is what Eugene Wigner has called "the unreasonable effectiveness of mathematics." Today, the natural sciences, engineering, economics, and medicine depend heavily on new mathematical discoveries. The word "mathematics" comes from the Greek μάθημα (máthema) meaning "science, knowledge, or learning" and μαθηματικός (mathematikós) meaning "fond of learning". It is often abbreviated maths in Commonwealth English and math in North American English.

History

:Main article: History of mathematics The evolution of mathematics might be seen to be an ever-increasing series of abstractions, or alternatively an expansion of subject matter. The first abstraction was probably that of numbers. The realization that two apples and two oranges do have something in common, namely that they fill the hands of exactly one person, was a breakthrough in human thought. In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. Arithmetic (e.g. addition, subtraction, multiplication and division), naturally followed. Monolithic monuments testify to a knowledge of geometry. Further steps need writing or some other system for recording numbers such as tallies or the knotted strings called khipu used by the Inca empire to store numerical data. Numeral systems have been many and diverse. Historically, the major disciplines within mathematics arose, from the start of recorded history, out of the need to do calculations on taxation and commerce, to understand the relationships among numbers, to measure land, and to predict astronomical events. These needs can be roughly related to the broad subdivision of mathematics, into the studies of quantity, structure, space, and change. Mathematics since has been much extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries have been made throughout history and continue to be made today.

Inspiration, pure and applied mathematics, and aesthetics

Mathematics arises wherever there are difficult problems that involve quantity, structure, space, or change. At first these were found in commerce, land measurement and later astronomy; nowadays, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. Newton invented infinitesimal calculus and Feynman his Feynman path integral using a combination of reasoning and physical insight, and today's string theory also inspires new mathematics. Some mathematics is only relevant in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. As in most areas of study, the explosion of knowledge in the scientific age has led to specialization in mathematics. One major distinction is between pure mathematics and applied mathematics. Within applied mathematics, two major areas have split off and become disciplines in their own right, statistics and computer science. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty also in a clever proof, such as Euclid's proof that there are infinitely many prime numbers, and in a numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in "A Mathematicians Apology" expressed the belief that these esthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. Main article: Mathematical beauty.

Notation, language, and rigor

Mathematical writing is not easily accessible to the layperson. A Brief History of Time, Stephen Hawking's 1988 bestseller, contained a single mathematical equation. This was the author's compromise with the publisher's advice, that each equation would halve the sales. The reasons for the inaccessibility even of carefully-expressed mathematics can be partially explained. Contemporary mathematicians strive to be as clear as possible in the things they say and especially in the things they write (this they have in common with lawyers). They refer to rigor. To accomplish rigor, mathematicians have extended natural language. There is precisely-defined vocabulary for referring to mathematical objects, and stating certain common relations. There is an accompanying mathematical notation which, like musical notation, has a definite content and also has a strict grammar (under the influence of computer science, more often now called syntax). Some of the terms used in mathematics are also common outside mathematics, such as ring, group and category; but are not such that one can infer the meanings. Some are specific to mathematics, such as homotopy and Hilbert space. It was said that Henri Poincaré was only elected to the Académie Française so that he could tell them how to define automorphe in their dictionary. Rigor is fundamentally a matter of mathematical proof. Mathematicians want their theorems to follow mechanically from axioms by means of formal axiomatic reasoning. This is to avoid mistaken 'theorems', based on fallible intuitions; of which plenty of examples have occurred in the history of the subject (for example, in mathematical analysis). Axioms in traditional thought were 'self-evident truths', but that conception turns out not to be workable in pushing the mathematical boundaries. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (strong enough) axiom system has undecidable formulas; and so a final axiomatization of mathematics is unavailable. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.

Is mathematics a science?

Carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. The mathematician-physicist Leon M. Lederman has quipped: "The physicists defer only to mathematicians, and the mathematicians defer only to God (though you may be hard pressed to find a mathematician that modest)." If one considers science to be strictly about the physical world, then mathematics, or at least pure mathematics, is not a science. An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. In fact, the theoretical physicist, J. M. Ziman, proposed that science is public knowledge and thus includes mathematics. [http://info.med.yale.edu/therarad/summers/ziman.htm] In any case, mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences.

Overview of fields of mathematics

As noted above, the major disciplines within mathematics first arose out of the need to do calculations in commerce, to understand the relationships between numbers, to measure land, and to predict astronomical events. These four needs can be roughly related to the broad subdivision of mathematics into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics). The study of quantity starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are characterized in arithmetic. The deeper properties of whole numbers are studied in number theory. The study of structure began with investigations of Pythagorean triples. Neolithic monuments on the British Isles are constructed using Pythagorean triples. Eventually, this led to the invention of more abstract numbers, such as the square root of two. The deeper structural properties of numbers are studied in abstract algebra and the investigation of groups, rings, fields and other abstract number systems. Included is the important concept of vectors, generalized to vector spaces and studied in linear algebra. The study of vectors combines three of the fundamental areas of mathematics, quantity, structure, and space. The study of space originates with geometry, beginning with Euclidean geometry. Trigonometry combines space and number. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Within differential geometry are the concepts of fiber bundles, calculus on manifolds. Within algebraic geometry is the description of geometric objects as solution sets of polynomal equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may be the greatest growth area in 20th century mathematics. Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a most useful tool. The central concept used to describe a changing quantity is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. These have been generalized, with the inclusion of the square root of negative one, to the complex numbers, which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many phenomena in nature can be described by dynamical systems; chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior. Beyond quantity, structure, space, and change are areas of pure mathematics that can be approached only by deductive reasoning. In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic, which divides into recursion theory, model theory, and proof theory, is now closely linked to computer science. When electronic computers were first conceived, several essential theoretical concepts in computer science were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, and information theory. Many of those topics are now investigated in theoretical computer science. Discrete mathematics is the common name for the fields of mathematics most generally useful in computer science. An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a part. It is used in all the sciences. Numerical analysis investigates methods for using computers to efficiently solve a broad range of mathematical problems that are typically beyond human capacity, and taking rounding errors or other sources of error into account to obtain credible answers.

Major themes in mathematics

An alphabetical and subclassified list of mathematical topics is available. The following list of themes and links gives just one possible view. For a fuller treatment, see Areas of mathematics or the list of lists of mathematical topics.

Change

:Ways to express and handle change in mathematical functions, and changes between numbers. : :Arithmetic – Calculus – Vector calculus – Analysis – Differential equations – Dynamical systems – Chaos theory – List of functions

Foundations and methods

:Approaches to understanding the nature of mathematics. :philosophy of mathematics – mathematical intuitionism – mathematical constructivism – foundations of mathematics – set theory – symbolic logic – model theory – category theory – Logic – reverse mathematics – table of mathematical symbols

Discrete mathematics

:Discrete mathematics involves techniques that apply to objects that can only take on specific, separated values. : :Combinatorics – Naive set theory – Theory of computation– Cryptography – Graph theory

Applied mathematics

:Applied mathematics uses the full knowledge of mathematics to solve real-world problems. :Mathematical physics – Mechanics – Fluid mechanics – Numerical analysis – Optimization – Probability – Statistics – Mathematical economics – Financial mathematics – Game theory – Mathematical biology – Cryptography – Information theory

Important theorems

:These theorems have interested mathematicians and non-mathematicians alike. :See list of theorems for more :Pythagorean theorem – Fermat's last theorem – Gödel's incompleteness theorems – Fundamental theorem of arithmetic – Fundamental theorem of algebra – Fundamental theorem of calculus – Cantor's diagonal argument – Four color theorem – Zorn's lemma – Euler's identity – classification theorems of surfaces – Gauss-Bonnet theorem – Quadratic reciprocity – Riemann-Roch theorem.

Important conjectures

See list of conjectures for more :These are some of the major unsolved problems in mathematics. :Goldbach's conjecture – Twin Prime Conjecture – Riemann hypothesis – Poincaré conjecture – Collatz conjecture – P=NP? – open Hilbert problems.

History and the world of mathematicians

Mathematics and other fields

:Mathematics and architecture – Mathematics and education – Mathematics of musical scales

Common misconceptions

Mathematics is not a closed intellectual system, in which everything has already been worked out. There is no shortage of open problems. Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. It often consists of determined attacks on famous questions, consisting of proof-attempts made in an isolated way (that is, long papers not supported by previously published theory). The relationship to generally-accepted mathematics is similar to that between pseudoscience and real science. The misconceptions involved are normally based on:
- misunderstanding of the implications of mathematical rigour;
- attempts to circumvent the usual criteria for publication of mathematical papers in a learned journal after peer review, with assumptions of bias;
- lack of familiarity with, and therefore underestimation of, the existing literature. The case of Kurt Heegner's work shows that the mathematical establishment is neither infallible, nor unwilling to admit error in assessing 'amateur' work. And like astronomy, mathematics owes much to amateur contributors such as Fermat and Mersenne. Mathematics is not accountancy. Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks. Advances in abstract mathematics are mostly irrelevant to the efficiency of concrete bookkeeping, but the use of computers clearly does matter. Mathematics is not numerology. Numerology uses modular arithmetic to reduce names and dates down to numbers, but assigns emotions or traits to these numbers intuitively or on the basis of traditions. Mathematical concepts and theorems need not correspond to anything in the physical world. In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system. While these axioms are derived from our perceptions and experience, they are not dependent on them. And yet, mathematics is extremely useful for solving real-world problems. It is this fact that led Eugene Wigner to write an essay on The Unreasonable Effectiveness of Mathematics in the Natural Sciences. Mathematics is not about unrestricted theorem proving, any more than literature is about the construction of grammatically correct sentences. However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers. These techniques have proven useful in formal verification of programs and hardware designs. However, they are unlikely to generate (in the near term, at least) mathematics with any widely recognized aesthetic value.

Bibliography

- Benson, Donald C., The Moment Of Proof: Mathematical Epiphanies (1999).
- Courant, R. and H. Robbins, What Is Mathematics? (1941);
- Davis, Philip J. and Hersh, Reuben, The Mathematical Experience. Birkhäuser, Boston, Mass., 1980. A gentle introduction to the world of mathematics.
- Boyer, Carl B., History of Mathematics, Wiley, 2nd edition 1998 available, 1st edition 1968 . A concise history of mathematics from the Concept of Number to contemporary Mathematics.
- Gullberg, Jan, Mathematics--From the Birth of Numbers. W.W. Norton, 1996. An encyclopedic overview of mathematics presented in clear, simple language.
- Hazewinkel, Michiel (ed.), Encyclopaedia of Mathematics. Kluwer Academic Publishers 2000. A translated and expanded version of a Soviet math encyclopedia, in ten (expensive) volumes, the most complete and authoritative work available. Also in paperback and on CD-ROM.
- Kline, M., Mathematical Thought from Ancient to Modern Times (1973).
- Pappas, Theoni, The Joy Of Mathematics (1989).

External links

- [http://www.cut-the-knot.org/ Interactive Mathematics Miscellany and Puzzles] — A collection of articles on various math topics, with interactive Java illustrations at cut-the-knot
- Rusin, Dave: [http://www.math-atlas.org/ The Mathematical Atlas]. A guided tour through the various branches of modern mathematics.
- Stefanov, Alexandre: [http://us.geocities.com/alex_stef/mylist.html Textbooks in Mathematics]. A list of free online textbooks and lecture notes in mathematics.
- Weisstein, Eric et al.: [http://www.mathworld.com/ MathWorld: World of Mathematics]. An online encyclopedia of mathematics.
- Polyanin, Andrei: [http://eqworld.ipmnet.ru/ EqWorld: The World of Mathematical Equations]. An online resource focusing on algebraic, ordinary differential, partial differential (mathematical physics), integral, and other mathematical equations.
- A mathematical thesaurus maintained by the [http://nrich.maths.org/ NRICH] project at the University of Cambridge (UK), [http://thesaurus.maths.org/ Connecting Mathematics]
- [http://planetmath.org/ Planet Math]. An online math encyclopedia under construction, focusing on modern mathematics. Uses the GFDL, allowing article exchange with Wikipedia. Uses TeX markup.
- [http://www.mathforge.net/ Mathforge]. A news-blog with topics ranging from popular mathematics to popular physics to computer science and education.
- [http://www.youngmath.net/concerns Young Mathematicians Network (YMN)]. A math-blog "Serving the Community of Young Mathematicians". Topics include: Math News, Grad and Undergrad Life, Job Search, Career, Work & Family, Teaching, Research, Misc...
- [http://metamath.org/ Metamath]. A site and a language, that formalize math from its foundations.
- [http://world.std.com/~reinhold/dir/mathmovies.html Math in the Movies]. A guide to major motion pictures with scenes of real mathematics
- [http://math.cofc.edu/faculty/kasman/MATHFICT/default.html Mathematics in fiction]. Links to works of fiction that refer to mathematics or mathematicians.
- [http://www.mathhelpforum.com/math-help Math Help Forum]. A forum, for math help, math discussion and debate.
- [http://www.sosmath.com/CBB S.O.S. Mathematics Cyberboard] a math help forum which incorporates a LaTeX extension, making it easier for members to write and display math formulae.
- [http://www-history.mcs.st-and.ac.uk/~history/ Mathematician Bibliography]. Extensive history and quotes from all famous mathematicians.
- [http://www.physicsmathforums.com/ Physics Math Forums]
- Category:School subjects fiu-vro:Matõmaatiga zh-min-nan:Sò·-ha̍k ko:수학 ms:Matematik ja:数学 simple:Mathematics th:คณิตศาสตร์

Real numbers

In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. The term "real number" is a retronym coined in response to "imaginary number". Real numbers may be rational or irrational; algebraic or transcendental; and positive, negative, or zero. Real numbers measure continuous quantities. They may in theory be expressed by decimal fractions that have an infinite sequence of digits to the right of the decimal point; these are often (mis-)represented in the same form as 324.823211247… The three dots indicate that there would still be more digits to come, no matter how many more might be added at the end. Measurements in the physical sciences are almost always conceived as approximations to real numbers. Writing them as decimal fractions (which are rational numbers that could be written as ratios, with an explicit denominator) is not only more compact, but to some extent conveys the sense of an underlying real number. The real numbers are the central object of study in real analysis. A real number is said to be computable if there exists an algorithm that yields its digits. Because there are only countably many algorithms, but an uncountable number of reals, most real numbers are not computable. Some constructivists accept the existence of only those reals that are computable. The set of definable numbers is broader, but still only countable. Computers can only approximate most real numbers with rational numbers; these approximations are known as floating point numbers or fixed-point numbers; see real data type. Computer algebra systems are able to treat some real numbers exactly by storing an algebraic description (such as "sqrt(2)") rather than their decimal approximation. Mathematicians use the symbol R (or alternatively,

\Bbb

, the letter "R" in blackboard bold) to represent the set of all real numbers. The notation Rⁿ refers to an n-dimensional space of real numbers; for example, a value from R³ consists of three real numbers and specifies a location in 3-dimensional space. In mathematics, real is used as an adjective, meaning that the underlying field is the field of real numbers. For example real matrix, real polynomial and real Lie algebra.

History

Vulgar fractions had been used by the Egyptians around 1000 BC; around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers. Negative numbers were invented by Indian mathematicians around 600 AD, and then possibly reinvented in China shortly after. They were not used in Europe until the 1600s, but even in the late 1700s, Leonhard Euler discarded negative solutions to equations as unrealistic. The development of calculus in the 1700s used the entire set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871.

Definition

Construction from the rational numbers

The real numbers can be constructed as a completion of the rational numbers. For details and other construction of real numbers, see construction of real numbers.

Axiomatic approach

Let R denote the set of all real numbers. Then:
- The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
- The field R is ordered, meaning that there is a total order ≥ such that, for all real numbers x, y and z:
- if x ≥ y then x + z ≥ y + z;
- if x ≥ 0 and y ≥ 0 then xy ≥ 0.
- The order is Dedekind-complete, i.e., every non-empty subset S of R with an upper bound in R has a least upper bound (also called supremum) in R. The last property is what differentiates the reals from the rationals. For example, the set of rationals with square less than 2 has a rational upper bound (e.g., 1.5) but no rational least upper bound, because the square root of 2 is not rational. The real numbers are uniquely specified by the above properties. More precisely, given any two Dedekind complete ordered fields R₁ and R₂, there exists a unique field isomorphism from R₁ to R₂, allowing us to think of them as essentially the same mathematical object.

Properties

Completeness

The main reason for introducing the reals is that the reals contain all limits. More technically, the reals are complete (in the sense of metric spaces or uniform spaces, which is a different sense than the Dedekind completeness of the order in the previous section). This means the following: A sequence (x_n) of real numbers is called a Cauchy sequence if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x_m| is less than ε provided that n and m are both greater than N. In other words, a sequence is a Cauchy sequence if its elements x_n eventually come and remain arbitrarily close to each other. A sequence (x_n) converges to the limit x if for any ε > 0 there exists an integer N (possibly depending on ε) such that the distance |x_n − x| is less than ε provided that n is greater than N. In other words, a sequence has limit x if its elements eventually come and remain arbitrarily close to x. It is easy to see that every convergent sequence is a Cauchy sequence. An important fact about the real numbers is that the converse is also true: :Every Cauchy sequence of real numbers is convergent. That is, the reals are complete. Note that the rationals are not complete. For example, the sequence (1, 1.4, 1.41, 1.414, 1.4142, 1.41421, ...) is Cauchy but it does not converge to a rational number. (In the real numbers, in contrast, it converges to the square root of 2.) The existence of limits of Cauchy sequences is what makes calculus work and is of great practical use. The standard numerical test to determine if a sequence has a limit is to test if it is a Cauchy sequence, as the limit is typically not known in advance. For example, the standard series of the exponential function :

\mathrm^x = \sum_^ \frac

converges to a real number because for every x the sums :

\sum_^ \frac

can be made arbitrarily small by choosing N sufficiently large. This proves that the sequence is Cauchy, so we know that the sequence converges even if we do not know ahead of time what the limit is.

"The complete ordered field"

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways. First, an order can be lattice-complete. It is easy to see that no ordered field can be lattice-complete, because it can have no largest element (given any element z, z + 1 is larger), so this is not the sense that is meant. Additionally, an order can be Dedekind-complete, as defined in the section Axioms. The uniqueness result at the end of that section justifies using the word "the" in the phrase "complete ordered field" when this is the sense of "complete" that is meant. This sense of completeness is most closely related to the construction of the reals from Dedekind cuts, since that construction starts from an ordered field (the rationals) and then forms the Dedekind-completion of it in a standard way. These two notions of completeness ignore the field structure. However, an ordered group (and a field is a group under the operations of addition and subtraction) defines a uniform structure, and uniform structures have a notion of completeness (topology); the description in the section Completeness above is a special case. (We refer to the notion of completeness in uniform spaces rather than the related and better known notion for metric spaces, since the definition of metric space relies on already having a characterisation of the real numbers.) It is not true that R is the only uniformly complete ordered field, but it is the only uniformly complete Archimedean field, and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field". Since it can be proved that any uniformly complete Archimedean field must also be Dedekind complete (and vice versa, of course), this justifies using "the" in the phrase "the complete Archimedean field". This sense of completeness is most closely related to the construction of the reals from Cauchy sequences (the construction carried out in full in this article), since it starts with an Archimedean field (the rationals) and forms the uniform completion of it in a standard way. But the original use of the phrase "complete Archimedean field" was by David Hilbert, who meant still something else by it. He meant that the real numbers form the largest Archimedean field in the sense that every other Archimedean field is a subfield of R. Thus R is "complete" in the sense that nothing further can be added to it without making it no longer an Archimedean field. This sense of completeness is most closely related to the construction of the reals from surreal numbers, since that construction starts with a proper class that contains every ordered field (the surreals) and then selects from it the largest Archimedean subfield.

Advanced properties

The reals are uncountable; that is, there are strictly more real numbers than natural numbers, even though both sets are infinite. This is proved with Cantor's diagonal argument. In fact, the cardinality of the reals is 2^ω, i.e., the cardinality of the set of subsets of the natural numbers. Since only a countable set of real numbers can be algebraic, almost all real numbers are transcendental. The non-existence of a subset of the reals with cardinality strictly between that of the integers and the reals is known as the continuum hypothesis. The continuum hypothesis can neither be proved nor be disproved; it is independent from the axioms of set theory. The real numbers form a metric space: the distance between x and y is defined to be the absolute value |x − y|. By virtue of being a totally ordered set, they also carry an order topology; the topology arising from the metric and the one arising from the order are identical. The reals are a contractible (hence connected and simply connected), separable metric space of dimension 1, and are everywhere dense. The real numbers are locally compact but not compact. There are various properties that uniquely specify them; for instance, all unbounded, continuous, and separable order topologies are necessarily homeomorphic to the reals. Every nonnegative real number has a square root in R, and no negative number does. This shows that the order on R is determined by its algebraic structure. Also, every polynomial of odd degree admits at least one root: these two properties make R the premier example of a real closed field. Proving this is the first half of one proof of the fundamental theorem of algebra. The reals carry a canonical measure, the Lebesgue measure, which is the Haar measure on their structure as a topological group normalised such that the unit interval [0,1] has measure 1. The supremum axiom of the reals refers to subsets of the reals and is therefore a second-order logical statement. It is not possible to characterize the reals with first-order logic alone: the Löwenheim-Skolem theorem implies that there exists a countable dense subset of the real numbers satisfying exactly the same sentences in first order logic as the real numbers themselves. The set of hyperreal numbers is much bigger than R but also satisfies the same first order sentences as R. Ordered fields that satisfy the same first-order sentences as R are called nonstandard models of R. This is what makes nonstandard analysis work; by proving a first-order statement in some nonstandard model (which may be easier than proving it in R), we know that the same statement must also be true of R.

Generalizations and extensions

The real numbers can be generalized and extended in several different directions. Perhaps the most natural extension are the complex numbers which contain solutions to all polynomial equations. However, the complex numbers are not an ordered field. Ordered fields extending the reals are the hyperreal numbers and the surreal numbers; both of them contain infinitesimal and infinitely large numbers and thus are not Archimedean. Occasionally, the two formal elements +∞ and −∞ are added to the reals to form the extended real number line, a compact space which is not a field but retains many of the properties of the real numbers. Self-adjoint operators on a Hilbert space (for example, self-adjoint square complex matrices) generalize the reals in many respects: they can be ordered (though not totally ordered), they are complete, all their eigenvalues are real and they form a real associative algebra. Positive-definite operators correspond to the positive reals and normal operators correspond to the complex numbers. Category:Elementary mathematics Category:Real numbers Category:Set theory ko:실수 ja:実数 th:จำนวนจริง

Absolute value

In mathematics, the absolute value (or modulus¹) of a real number is its numerical value without regard to its sign. So, for example, 3 is the absolute value of both 3 and −3. In computers, the mathematical function used to perform this calculation is usually given the name abs(). Generalizations of the absolute value for real numbers, occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts. norm

Real numbers

For any real number

a,

the absolute value or modulus of

a,

is denoted ²

|a|,

and is defined as :

|a| := \begin a, & \mbox  a \ge 0  \\ -a,  & \mbox a < 0. \end

As can be seen from the above definition, the absolute value of

a

is always either positive or zero, never negative. From a geometric point of view, the absolute value of a real number is the distance along the real number line of that number from zero, and more generally the absolute value of the difference of two real numbers is the distance between them. Indeed the notion of an abstract distance function in mathematics can be seen to be a generalization of the properties of the absolute value (see "Distance" below). The following proposition, gives an identity which is sometimes used as an alternative (and equivalent) definition of the absolute value: PROPOSITION 1: :

|a| = \sqrt

The absolute value has the following four fundamental properties: PROPOSITION 2: : Other important properties of the absolute value include: PROPOSITION 3: : Two other useful inequalities are: :

|a| \le b \iff -b \le a \le b

|a| \ge b \iff a \le -b \mbox b \le a

The above are often used in solving inequalities; for example: :

Complex numbers

image:complex.png

Since the complex numbers are not ordered, the definition given above for the real absolute value, can not be directly generalized for a complex number. However the identity given in Proposition 1: :

|a| = \sqrt

can be seen as motivating the following definition. For any complex number :

z = x + iy\,

the absolute value or modulus of

z

is denoted

|z|,

and is defined as :

|z| :=  \sqrt.

It follows that the absolute value of a real number x is equal to its absolute value considered as a complex number since: :

|x + i0| = \sqrt = \sqrt = |x|.

Similar to the geometric interpretation of the absolute value for real numbers, it follows from the Pythagorean theorem that the absolute value of a complex number is the distance in the complex plane of that complex number from the origin, and more generally, that the absolute value of the difference of two complex numbers is equal to the distance between those two complex numbers. The complex absolute value shares all the properties of the real absolute value given in Propositions 2 and 3 above. In addition, If :

z = x + \mathrmy = r (\cos \phi + \mathrm\sin \phi ) \,

and :

\bar = x - iy

is the complex conjugate of

z

, then it is easily seen that :

|z| = r\,

|z|=|\bar|

|z| = \sqrt

Absolute value functions

The real absolute value function is continuous everywhere. It is differentiable everywhere except for x = 0. It is monotonically decreasing on the interval (-∞, 0] and monotonically increasing on the interval [0, ∞). Since a real number and its negative have the same absolute value, it is an even function, and is hence not invertible. The complex absolute value function is continuous everywhere but differentiable nowhere (One way to see this is to show that it does not obey the Cauchy-Riemann equations). Both the real and complex functions are idempotent.

Ordered rings

The definition of absolute value given for real numbers above can easily be extended to any ordered ring. That is, if

a

is an element of an ordered ring

R

, then the absolute value of

a

, denoted by

|a|

, is defined to be: :

|a| := \begin a, & \mbox  a \ge 0  \\ -a,  & \mbox a < 0, \end

where

-a

is the additive inverse of

a

, and

0

is the additive identity element.

Distance

The absolute value is closely related to the idea of distance. As noted above, the absolute value of a real or complex number is the distance from that number to the origin, along the real number line, for real numbers, or in the complex plane, for complex numbers, and more generally, the absolute value of the difference of two real or complex numbers is the distance between them. The standard Euclidean distance between two points :

a = (a_1, a_2, \cdots , a_n)

and :

b = (b_1, b_2, \cdots , b_n)

in Euclidean n-space is defined as: :

\sqrt.

This can be seen to be a generalization of

|a - b|,

since if

a,

b

are real, then by Proposition 1, :

|a - b| = \sqrt

while if :

a = a_1 + i a_2 \,

and :

b = b_1 + i b_2 \,

are complex numbers, then : The above shows that the "absolute value" distance for the real numbers or the complex numbers, agrees with the standard Euclidean distance they inherit as a result of considering them as the one and two-dimensional Euclidean spaces respectively. The properties of the absolute value of the difference of two real or complex numbers: non-negativity, identity of indiscernibles, symmetry and the triangle inequality given in Propositions 2 and 3 above, can be seen to motivate the more general notion of a distance function as follows: A real valued function

d

on a set

X \times X

is called a distance function (or a metric) for

X

, if it satisfies the following four axioms: :

Fields

The fundamental properties of the absolute value for real numbers given in Proposition 2 above, can be used to generalize the notion of absolute value to an arbitrary field, as follows. A real-valued function

v

on a field

F

is called an absolute value (also a modulus, magnitude, value, or valuation) if it satisfies the following four axioms: : It follows from the above that

v(1) = 1

, where

1

denotes the multiplicative identity element of

F

. The real and complex absolute values defined above are examples of absolute values for an arbitrary field. If

v

is an absolute value on

F

, then the function

d

F \times F

, defined by

d(a, b) = v(a - b)

, is a metric, and if

e

is the multiplicative identity in

F

, then the following are equivalent:
-

d

satisfies the ultrametric inequality

d(x, y) \le \mathrm\.

\big\

is bounded in R.
-

v\Big(\sum_^n e\Big) \le 1

for every

n \in \mathbb.

v(a + b) \le \mathrm\

for all

a, b \in F.

An absolute value which satisfies any (hence all) of the above conditions is said to be non-Archimedean, otherwise it is said to be Archimedean.³

Vector spaces

Again the fundamental properties of the absolute value for real numbers, can be used, with a slight modification, to generalize the notion to an arbitrary vector space. A real valued function ||·|| on a vector space

V

a over a field

F

, is called an absolute value (or more usually a norm) if it satisfies the following axioms: For all

a

F

, and

\mathbf

\mathbf

V

, : The norm of a vector is also called its length or magnitude. In the case of Euclidean space Rⁿ, the function :

\|(x_1, x_2, \cdots , x_n) \| = \sqrt

is a norm called the Euclidean norm. When the real numbers R are considered as the one-dimensional vector space R¹, the absolute value is a norm, and is the p-norm for any p. In fact the absolute value is the "only" norm in R¹, in the sense that, for every norm ||·|| in R¹, ||x||=||1||·|x|. The complex absolute value is a special case of the norm in an inner product space. It is identical to the Euclidean norm, if the complex plane is identified with the Euclidean plane R².

Algorithms

In the C programming language, the abs(), labs(), llabs() (in C99), fabs(), fabsf(), and fabsl() functions compute the absolute value of an operand. Coding the integer version of the function is trivial, ignoring the boundary case where the largest negative integer is input: int abs(int i) The floating-point versions are trickier, as they have to contend with special codes for infinity and not-a-numbers. Using assembly language, it is possible to take the absolute value of a register in just three instructions (example shown for a 32-bit register on an x86 architecture, Intel syntax): cdq xor eax, edx sub eax, edx cdq extends the sign bit of eax into edx. If eax is nonnegative, then edx becomes zero, and the latter two instructions have no effect, leaving eax unchanged. If eax is negative, then edx becomes 0xFFFFFFFF, or -1. The next two instructions then become a two's complement inversion, giving the absolute value of the negative value in eax.

References

- Nahin, Paul J.; [http://www.amazon.com/gp/reader/0691027951/ref=sib_dp_pt/103-5443484-7306247#reader-link An Imaginary Tale]; Princeton University Press; (hardcover, 1998). ISBN 0691027951
- O'Connor, J.J. and Robertson, E.F.; [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html "Jean Robert Argand"]
- Schechter, Eric; Handbook of Analysis and Its Foundations, pp 259-263, [http://www.amazon.com/gp/reader/0126227608/103-5443484-7306247?v=search-inside&keywords=absolute%20value "Absolute Values"], Academic Press (1997) ISBN 0126227608
- Weisstein, Eric W.; MathWorld: [http://mathworld.wolfram.com/AbsoluteValue.html "Absolute Value"]

Notes

¹ Jean-Robert Argand, is credited with introducing the term "modulus" in 1806, see: [http://www.amazon.com/gp/reader/0691027951/ref=sib_vae_pg_73/103-5443484-7306247?%5Fencoding=UTF8&keywords=modulus&p=S02K&twc=4&checkSum=0BsRgLAMFNMXnqArYGxr33gLjR56d%2Bc2nsSoQnGOEKE%3D#reader-page Nahin], [http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Argand.html O'Connor and Robertson], and [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolram.com].

² [http://functions.wolfram.com/ComplexComponents/Abs/35/ functions.Wolram.com] credits Karl Weierstrass with introducing the notation |x| in 1841.

³ [http://www.amazon.com/gp/reader/0126227608/103-5443484-7306247?v=search-inside&keywords=absolute%20value Schechter, p 260-261].

Category:Numeration ja:絶対値 th:ค่าสัมบูรณ์

Superreal number

The superreal numbers compose a more inclusive category than hyperreal number. Suppose X is a Tychonoff space, also called a T_3.5 space, and C(X) is the algebra of continuous real-valued functions on X. Suppose P is a prime ideal in C(X). Then the factor algebra A = C(X)/P is by definition an integral domain which is a real algebra and which can be seen to be totally ordered. The quotient field F of A is a superreal field if F strictly contains the real numbers

\Bbb

, so that F is not order isomorphic to

\Bbb

, though they may be isomorphic as fields. If the prime ideal P is a maximal ideal, then F is a field of hyperreal numbers. The terminology is due to Dales and Woodin.

References

- H. Garth Dales and W. Hugh Woodin: Super-Real Fields, Clarendon Press, 1996.
- L. Gillman and M. Jerison: Rings of Continuous Functions, Van Nostrand, 1960. category: field theory category: real closed field category: formally real field

Grothendieck universe

In mathematics, a Grothendieck universe is a set with the following properties: # If x ∈ U and if y ∈ x, then y ∈ U. # If x,y ∈ U, then ∈ U. # If x ∈ U, then P(x) ∈ U. (P(x) is the power set of x.) # If

\_

is a family of elements of U, and if I ∈U, then the union

\cup_ x_\alpha

is an element of U. A Grothendieck universe is meant to provide a set in which all of mathematics can be performed. (In fact, it provides a model for set theory.) As an example, we will prove an easy proposition. ;Proposition 1. :If x ∈ U and y ⊆ x, then y ∈ U. ;Proof. :y ∈ P(x) because y ⊆ x. P(x) ∈ U because x ∈ U, so y ∈ U. It is similarly easy to prove that any Grothendieck universe U contains:
- All singletons of each of its elements,
- All products of all families of elements of U indexed by an element of U,
- All disjoint unions of all families of elements of U indexed by an element of U,
- All intersections of all families of elements of U indexed by an element of U,
- All functions between any two elements of U, and
- All subsets of U whose cardinal is an element of U. In particular, it follows from the last axiom that if U is non-empty, it must contain all of its finite subsets and a subset of each finite cardinality. One can also prove immediately from the definitions that the intersection of any class of universes is a universe. Grothendieck universes are equivalent to strongly inaccessible cardinals. More formally, the following two axioms are equivalent: : (U) For all sets x, there exists a Grothendieck universe U such that x ∈ U. : (C) For all cardinals κ, there is a strongly inaccessible cardinal λ which is strictly larger than κ. To prove this fact, we give explicit constructions. Let κ be a strongly inaccessible cardinal. Say that a set S is strictly of type κ if for any sequence s_n ∈ ... ∈ s₀ ∈ S, |s_n| < κ. (S itself corresponds to the empty sequence.) Then the set u(κ) of all sets strictly of type κ is a Grothendieck universe of cardinality κ. The proof of this fact is long, so for details, we refer to Bourbaki's article, listed in the references. To show that the large cardinal axiom (C) implies the universe axiom (U), choose a set x. Let x₀ = x, and for all n, let x_n = ∪x be the union of the elements of x. Let y = ∪_nx_n. By (C), there is a strongly inaccessible cardinal κ such that |y| < κ. Let u(κ) be the universe of the previous paragraph. x is strictly of type κ, so x ∈ u(κ). To show that the universe axiom (U) implies the large cardinal axiom (C), choose a strongly inaccessible cardinal κ. κ is the cardinality of the Grothendieck universe u(κ). By (U), there is a Grothendieck universe V such that U ∈ V. Then κ < 2^κ ≤ |V|. In fact, any Grothendieck universe is of the form u(κ) for some κ. This gives another form of the equivalence between Grothendieck universes and strongly inaccessible cardinals: :For any Grothendieck universe U, |U| is a strongly inaccessible cardinal, and for any strongly inaccessible cardinal κ, there is a Grothendieck universe u(κ). Furthermore, u(|U|)=U, and |u(κ)|=κ. Since the existence of strongly inaccessible cardinals cannot be proved from the axioms of Zermelo-Fraenkel set theory, the existence of universes cannot be proved from Zermelo-Fraenkel set theory either. The idea of universes is due to Alexander Grothendieck, who used them as a way of avoiding proper classes in algebraic geometry.

References

Bourbaki, N., Univers, appendix to Exposé I of Artin, M., Grothendieck, A., Verdier, J. L., eds., Théorie des Topos et Cohomologie Étale des Schémas (SGA 4), second edition, Springer-Verlag, Heidelberg, 1972. An electronic copy is available [http://modular.fas.harvard.edu/sga/sga/index.html here] (starting on page 185 of IV-I). Category:Set theory Category:Category theory

Class (set theory)

In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. Some classes are sets (for instance, the class of all integers that are even), but others are not (for instance, the class of all ordinal numbers or the class of all sets). A class that is not a set is called a proper class. A proper class cannot be an element of a set or a class and is not subject to the Zermelo-Fraenkel axioms of set theory; thereby a number of paradoxes of naive set theory are avoided. Instead, these paradoxes become proofs that a certain class is proper. For example, Russell's paradox becomes a proof that the class of all sets is proper, and the Burali-Forti paradox becomes a proof that the class of all ordinal numbers is proper. The standard Zermelo-Fraenkel set theory axioms do not talk about classes; classes exist only in the metalanguage as equivalence classes of logical formulas. Another approach is taken by the von Neumann-Bernays-Gödel axioms; classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class. The proper classes, then, are those classes that are not elements of any other class. Several objects in mathematics are too big for sets and need to be described with classes, for instance large categories or the class-field of surreal numbers. The word "class" is sometimes used synonymously with "set," most notably in the term "equivalence class." This usage dates from a historical period where classes and sets were not distinguished as they are in modern terminology. Many discussions of "classes" in the 19th century and earlier are really referring to sets, or perhaps to a more ambiguous concept. Category:Category theory Category:Set theory

Strongly inaccessible cardinal

In set theory, an uncountable cardinal number κ is called weakly inaccessible if the following two conditions hold. # cf(κ) = κ, where cf denotes the cofinality. Such a cardinal is called a regular cardinal. # There is no next smaller cardinal number; i.e., for every cardinal γ < κ, there is another cardinal number δ between γ and κ. Such a cardinal number κ is called a limit cardinal. Every transfinite cardinal number, except for aleph-null which meets those two conditions (but is not weakly inaccessible because it is countable), is either regular or a limit; however, only a rather large cardinal number can be both. If condition 2. above is replaced by :2'. For every cardinal γ < κ, 2^γ < κ (that is, κ is a strong limit cardinal). then κ is called strongly inaccessible, or just inaccessible. Again,

\aleph_0

meets this condition, but is not inaccessible because it is countable. Assuming that ZFC is consistent, the existence of (strongly or weakly) inaccessible cardinals provably cannot be proved in ZFC; inaccessible cardinals are therefore a type of large cardinal. In fact, ZFC cannot even prove that the existence of inaccessible cardinals is consistent with ZFC (because ZFC+"there exists an inaccessible cardinal" proves the consistency of ZFC); however, the assumption that there is no inaccessib

Surreal number

Constructing surreal numbers

over the generated surreal numbers such that : x

< :

< :

< :

-1 < :

0 < :

1 < :

Computing with surreal numbers

1. (Note the use of equality = and equivalence

Generating surreal numbers using finite induction

"To Infinity and Beyond"

Definitions

Addition and Multiplication

Correspondence between realizations

- [http://www.tondering.dk/claus/surreal.html A gentle yet thorough introduction by Claus Tøndering] - Category:Combinatorial game theory Category:Mathematical logic

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References

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References

Class (set theory)

Strongly inaccessible cardinal

- [http://www.tondering.dk/claus/surreal.html A gentle yet thorough introduction by Claus Tøndering]
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