Surreal number
The surreal numbers form a class of numbers that includes all real numbers, as well as " infinitely large " numbers that are greater than any real number. Each real number is surrounded by surreal numbers that you are closer than any other real number, in particular, there are " infinitesimal " numbers that are closer to zero than any positive real number. In this they agree with the hyper- real numbers, but they are constructed in a completely different manner and contain the hyper- real numbers as a subset.
The word " surreal " comes from the French, meaning " the reality ". It is also used for the style of surrealism.
Surreal numbers were first introduced by John Conway in 1974 and described in detail in Donald E. Knuth's book Surreal Numbers: How Two Ex - Students Turned on to Pure Mathematics and Found Total Happiness. This book is really a reference book, but a short story, and it's one of the few cases in which a new mathematical idea was first presented in a literary work. In his book, which is kept in dialogue form, Knuth coined the term surreal numbers for what Conway originally called just numbers. Conway liked the new name, so he took it over later. He described the surreal numbers and used them for the analysis of games (including Go) in his book On Numbers and Games (1976).
- 5.1 Theoretical amount Comment
Motivation
Surreal numbers are a number of reasons interesting structures. For one, they are created by two simple rules " out of nothing" and still have similar characteristics as the real numbers. We are forced to prove any statement that we believe in the real numbers for granted, such as that in force or that follows from always. The surreal numbers are therefore a good opportunity to practice methods of abstract algebra. From Conway some applications have been in game theory presented (see below the section on game theory).
Finally, the surreal numbers form as the hyper- real numbers a model of non-standard analysis, exist in, for example, infinitesimal numbers. Before we begin to define surreal numbers, we should realize that we are starting purely set-theoretically with the construction of the surreal numbers and know about number properties, such as "small", zero, one, addition and multiplication at first.
Construction
The basic idea behind the construction of the surreal numbers is similar to Dedekind cuts. We generate a new number, by specifying two sets L and R of numbers that approximate the new number. The quantity L is composed of numbers that are smaller than the new number, and R is a set of numbers that are greater than the new number. We write such a representation as { L | R}. On the sets L and R, we provides the condition that every element of L is supposed to be smaller than every element of R. For example, { {1, 2} | { 5, 8} } is a valid ( " well-formed " ) construction a certain number of 2 to 5 ( which is will be explained later. ) It is expressly permits the quantities are empty. Interpreting the representation { L | {}} is ' a number which is greater than each number in L ", and { } { | r } is ' a number which is smaller than each number in R". The design principle of surreal numbers, however hochrekursiv; next to a construction rule we also need a comparison rule for recursively declared order relation ( less than or equal ), which must be used for the application of the rule of construction.
Design and comparison rule
Is a surreal number given, then we call the left and the right amount of quantity.
To simplify the notation we let the amount of brackets and the empty set at the left and right amount away where no misunderstandings are to be feared, that is, we write, for example, to and for. An object { L | R} that satisfies the Less than or equal condition, also called well-formed, in order to distinguish it from later considered objects without this condition (known as the Games).
These two rules are recursive, so we need some form of induction to work with them. One candidate would be the complete induction, but as we will see later, only makes the transfinite induction, the thing really interesting, that is if you " more than finitely many " apply these rules.
Relations
In order for the generated objects can be usefully called numbers, they should have a total order. However, the relation is only a total quasi-ordering, that is, it is reflexive and transitive, but not antisymmetric ( and does not necessarily follow that x = y). To remedy this, we define a relation == on the surreal numbers:
This is an equivalence relation and the equivalence classes are totally ordered. X and y are in the same equivalence class, then they ask the same surreal number represents the equivalence class of x we will write as [ x], where x is a representative of the class [ x] is. So it is equivalent to x == y [x ] = [y ]. This procedure corresponds to the construction of the rational numbers as equivalence classes of fractions of integers, or the construction of the real numbers as equivalence classes of Cauchy sequences.
Examples
Let us now consider some examples of surreal numbers. Since we still do not know any surreal numbers is the only quantity that we can take for L and R, the empty set. Our first number is so
This number satisfies the rule of construction, since the empty set contains no element which could violate any condition. We call them 0 and we write its equivalence class than 0 After the comparison rule
By applying the rule of construction, we obtain the numbers
The last figure, however, since 0 ≤ 0, no well-formed surreal numbers. The numbers found so far can be sorted as follows:
Where x is
If we apply the rule of construction one more time in every possible way in, we get next to some invalid numbers, the well-formed surreal numbers
We make three observations:
The first observation leads us to the question of how these new equivalence classes are to be interpreted.
Since { | -1 } is less than -1, we call them -2 and their equivalence class -2.
The number {1 |} we call 2, {-1 | 0 } is between -1 and 0 and we call -1 / 2, and {0 | 1} we call 1/2.
We write the equivalence classes than -2, 2, -1 / 2 and 1/2.
Another reason for these names we get when we have defined addition and multiplication.
The second observation leads us to the question of whether we can still identify a surreal number with its equivalence class.
The answer is positive, because you can show:
Here is the set of equivalence classes of elements of X. Thus, we can rewrite the list from above
Or shorter
The third observation can be generalized to arbitrary surreal numbers with finite left and right quantities.
The above-mentioned number of {1, 2 | 5, 8} {2 is therefore equal to | 5}, and is later shown in greater detail.
Infinite sets must contain no greatest element, therefore applies to numbers with infinite sets only a modified statement.
The addition and multiplication of surreal numbers is defined by the following three rules:
Here, we use the set-theoretic extension of the operators, ie it mean, for example
And
These links are well defined in the sense that the combination of well-formed surreal numbers is again a well-formed surreal numbers, that is a number whose left set of "small " as the right amount is.
With these calculation rules, we can now see that fit the previously assigned name, because it is, for example,
( Note the distinction between the equality " =" and the equivalence " =="! )
The links can be transferred to the equivalence classes, because due to
They are also well-defined links of the equivalence classes.
Finally, one can show that the links between the equivalence classes have nice algebraic properties:
There is even a special orderly body: namely, the largest.
One can show that every ordered field can be embedded in the surreal numbers.
We no longer distinguish in the following between a surreal number and its equivalence class, so call the equivalence class itself a surreal number.
So far we have not systematically looked after, what numbers we obtained by the construction rule and which are not.
We start with the numbers that we can achieve in a finite number of steps.
We do this inductively, by amounts of Sn for every natural number n defined as follows:
The set of all surreal numbers, which are in any Si, we call Sω.
The first sets of surreal numbers are
We observe two things:
One implication of this is that we can accurately generate all dyadic fractions in this way, that is, every rational number of the form
By an integer A and B is a natural number in Sω.
However, we are Other fractions such as 1 /3, 2 /3, 1 /5, 1/6 found in any Sn, as long as n is a natural number.
What you could instead take for n, we'll get to in the next section.
Now that we have a further amount of Sω of surreal numbers, nothing prevents us from applying the design rule on them and sets Sω 1, 2 Sω to construct, etc..
The quantities on the left and right side of surreal numbers can now be infinitely large.
In fact, we can define for each ordinal a is an amount of Sa surreal numbers by transfinite induction.
The smallest ordinal a such that a surreal number x is contained in the set Sa, we call the birthday of x.
For example, 0 is the birthday of 0 and 2 is the birthday of 1/2.
One can show that by the expression { { a} | { b} } for surreal numbers a
Even in Sω 1 we find the fractions which were missing us in Sω.
For example,
The correctness of this definition follows from the equivalence
The birthday of 1/3 is ω 1.
In Sω 1 all real numbers are already included.
Man himself will illustrate that with the fact that each of intervals exactly defines a real number and every real number can be represented by nested intervals.
Now all numbers with the representation ( k, j are any integers ) are already contained in Sω.
Moreover, one can find with these numbers as limits already for all real numbers of intervals.
Now you take the lower limits of these nested intervals in the left-hand quantity and the upper limits of nested intervals in the right amount and you get the desired real number as part of Sω 1.
A different number that is in Sω 1,
One can easily see that this number is greater than 0 but less than any positive fraction.
It is therefore an infinitesimal number.
We denote its equivalence class therefore with ε.
This is not the only infinitesimal number, because it is, for example,
These figures are, however, only in Sω 2.
In addition to infinitely small numbers infinitely large numbers can be found in Sω 1, such as
This number is greater than every number in Sω, in particular greater than any natural number, we denote its equivalence class, therefore, with ω.
This number corresponds to the ordinal ω.
It is also
One can even represent any ordinal number as surreal.
Since addition and subtraction are declared on all surreal numbers, we can expect with ω as any other number and, for example, calculate the following:
This goes also for larger addend:
And even with ω itself:
Wherein x y x y = { | y Y } above.
Just as 2 · ω = ω ω is greater than ω, ω is / 2 is smaller than ω, because
Where x - y = { x - y | y in Y}.
Finally, we find a strong correlation between ω and ε, because it is
Note, however, that the calculation rules for ordinals differ from those of the surreal representations: In the ordinal ω = ω 1 is true < ω 1, as a surreal number, however, is 1 ω = ω 1> ω.
Many numbers can be generated in this way, even so many that no amount of it all can absorb.
The surreal numbers form as the ordinals a real class.
Since each surreal number is composed of surreal numbers, which have a smaller birthday, one can prove almost all statements about surreal numbers by transfinite induction.
Man pointing to the fact that a statement for x = { XL | XR } is valid if it is true for every element of XL and XR.
As mentioned in the section Calculating with surreal numbers, the surreal numbers form a proper class, so not a lot in terms of a standard set theory, such as the Zermelo -Fraenkel set theory.
The reason is that each ordinal is represented as a surreal number and already form the ordinals a real class.
The definition of a Surrealzahl as a pair of left and right amount amount of Surrealzahlen now closes initially not exclude that one of these quantities could be a real class.
To avoid this, an upper bound for the birthdays of members of their left and right quantities ( their representatives ) must specify at each statement about a lot of Surrealzahlen.
For the considerations in this article to get for example, the first uncountable ordinal, as a barrier from.
If one the other hand, the representatives of each older figures (as the be displayed ) limit, although the set-theoretic problem would be solved, but you would need a lot more purely technical considerations in all transformations.
The definition of a surreal number contained a restriction that each element of L must be less than every element of R. If you drop this restriction, we obtain a larger class of objects, which are called games (English for games).
A Game is produced by application of this rule:
Comparison, equivalence, addition, negation, and multiplication are defined the same as for surreal numbers.
Every surreal number is a game, but there are also not well-formed games, such as {0 | 0}.
The class of games is more general than that of surreal numbers and has a simpler definition, but it lacks some of the nice properties of the surreal numbers.
The games do not form a body.
You only have a partial order, that is, there are games that are not comparable.
Every surreal number is either greater than, equal to or less than 0, but a game is either greater than, equal to, less than 0 or incomparable with 0 ( engl. fuzzy).
If x, y, z surreal numbers and x == y, then x · z == y · z
If x, y, z with x == y games, however, it is not always true x == y · z · z
The surreal numbers were originally motivated by the study of the game Go, and there are many connections between well-established games and surreal numbers.
We consider games with the following properties:
Such games are chess, checkers, mill and go, but not the most card games, memory or " Ludo ".
In most games, initially none of the players has a great advantage.
Falls the game but ahead, then occur sometime situations in which one of the players has a clear advantage.
To analyze a game, you now assigns to each possible game situation to a game.
The value of a game position is the game { L | R }, where L contains the values of all positions that can be achieved by a train of links, and R the values of all positions that can be achieved by a train of law.
This simple way to link games with game positions, yields interesting results.
Suppose two perfect players are in a game situation, the game is x.
Then the winner of the game is fixed:
Sometimes, when a game is near the end, it breaks down into smaller parts that are completely independent.
In the game of Go, for example, the game board is divided gradually between the parties until only small, isolated islands remain neutral space in which players can bet.
Each island acts like your own Go game on a very small board.
It would be useful to analyze each consignment separately and then to combine the results to get to an analysis of the whole lot.
This is no easy task.
For example, you could have two sublots, in each of which the first player wins, but one considers both Parties, the other player wins.
However, there is a way to carry out this analysis, by the following theorem:
In other words, summarize several independent parts corresponding to the addition of their games.
Conway developed the theory of surreal numbers in the reverse order to the selected presentation here.
He analyzed the playoffs and go looking for a way to combine the analysis of independent sublots.
So he developed the also called Combinatorial game theory concept of games with the possibilities of addition, negation, and a size comparison.
Finally, he noted that a certain class of games ( which later were called surreal numbers) has interesting properties and found for them a multiplication with the class of surreal numbers, a body was, which contains both the reals and the ordinals.
Calculating with surreal numbers
Production by complete (finite) induction
"To the infinity and beyond"
Theoretical amount Comment
Generalization: Games
Surreal numbers and game theory
Historical Development