Choice functionFrom Wikipedia, the free encyclopedia

Contents

1 Algebraic number 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 The eld of algebraic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Related elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4.1 Numbers dened by radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4.2 Closed-form number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Algebraic integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.6 Special classes of algebraic number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Choice function 62.1 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 History and importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Renement of the notion of choice function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.1 Bourbaki tau function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Rational number 83.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2.1 Embedding of integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.2 Equality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.3 Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.4 Addition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.5 Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.6 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.7 Division . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.8 Inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.9 Exponentiation to integer power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

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3.3 Continued fraction representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.4 Formal construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.5 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.6 Real numbers and topological properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.7 p-adic numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Zhegalkin polynomial 154.1 Boolean equivalent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.2 Formal properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Method of Computation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.3.1 The method of Indeterminate Coecents . . . . . . . . . . . . . . . . . . . . . . . . . . 164.3.2 Using PDCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4.4 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.7 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.7.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.7.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 1

Algebraic number

An algebraic number is a possibly complex number that is a root of a nite,[1] non-zero polynomial in one variablewith rational coecients (or equivalently by clearing denominators with integer coecients). Numbers suchas that are not algebraic are said to be transcendental. Almost all real and complex numbers are transcendental.(Here almost all has the sense all but a countable set"; see Properties.)

1.1 Examples The rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the abovedenition because x = a/b is the root of bx a .[2]

The quadratic surds (irrational roots of a quadratic polynomial ax2+bx+c with integer coecients a , b , andc ) are algebraic numbers. If the quadratic polynomial is monic (a = 1) then the roots are quadratic integers.

The constructible numbers are those numbers that can be constructed from a given unit length using straightedgeand compass and their opposites. These include all quadratic surds, all rational numbers, and all numbers thatcan be formed from these using the basic arithmetic operations and the extraction of square roots. (Note thatby designating cardinal directions for 1, 1, i , and i , complex numbers such as 3 + p2i are consideredconstructible.)

Any expression formed from algebraic numbers using any combination of the basic arithmetic operations andextraction of nth roots gives another algebraic number.

Polynomial roots that cannot be expressed in terms of the basic arithmetic operations and extraction of nthroots (such as the roots of x5 x + 1 ). This happens with many, but not all, polynomials of degree 5 orhigher.

Gaussian integers: those complex numbers a+ bi where both a and b are integers are also quadratic integers.

Trigonometric functions of rational multiples of (except when undened): that is, the trigonometric numbers.For example, each of cos(/7) , cos(3/7) , cos(5/7) satises 8x34x24x+1 = 0 . This polynomial isirreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan(3/16), tan(7/16) , tan(11/16) , tan(15/16) all satisfy the irreducible polynomial x4 4x3 6x2 + 4x+ 1 ,and so are conjugate algebraic integers.

Some irrational numbers are algebraic and some are not: The numbers p2 and 3p3/2 are algebraic since they are roots of polynomials x2 2 and 8x3 3 ,respectively.

The golden ratio is algebraic since it is a root of the polynomial x2 x 1 . The numbers and e are not algebraic numbers (see the LindemannWeierstrass theorem);[3] hence theyare transcendental.

1

2 CHAPTER 1. ALGEBRAIC NUMBER

1.2 Properties

Algebraic numbers on the complex plane colored by degree. (red=1, green=2, blue=3, yellow=4)

The set of algebraic numbers is countable (enumerable).[4][5]

Hence, the set of algebraic numbers has Lebesgue measure zero (as a subset of the complex numbers), i.e."almost all" complex numbers are not algebraic.

Given an algebraic number, there is a unique monic polynomial (with rational coecients) of least degree thathas the number as a root. This polynomial is called its minimal polynomial. If its minimal polynomial hasdegree n , then the algebraic number is said to be of degree n . An algebraic number of degree 1 is a rationalnumber. A real algebraic number of degree 2 is a quadratic irrational.

All algebraic numbers are computable and therefore denable and arithmetical.

The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without rst or lastelement, so is order-isomorphic to the set of rational numbers.

For real numbers a and b, the complex number a + bi is algebraic if and only if both a and b are algebraic.[6]

1.3 The eld of algebraic numbersThe sum, dierence, product and quotient of two algebraic numbers is again algebraic (this fact can be demonstratedusing the resultant), and the algebraic numbers therefore form a eld Q (sometimes denoted by A, though this usu-ally denotes the adele ring). Every root of a polynomial equation whose coecients are algebraic numbers is againalgebraic. This can be rephrased by saying that the eld of algebraic numbers is algebraically closed. In fact, it is thesmallest algebraically closed eld containing the rationals, and is therefore called the algebraic closure of the rationals.The set of real algebraic numbers itself forms a eld.[7]

1.4 Related elds

1.5. ALGEBRAIC INTEGERS 3

Algebraic numbers colored by degree (blue=4, cyan=3, red=2, green=1). The unit circle in black.

1.4.1 Numbers dened by radicals

All numbers that can be obtained from the integers using a nite number of integer additions, subtractions, multiplications,divisions, and taking nth roots where n is a positive integer (i.e, radical expressions) are algebraic. The converse,however, is not true: there are algebraic numbers that cannot be obtained in this manner. All of these numbers aresolutions to polynomials of degree 5. This is a result of Galois theory (see Quintic equations and the AbelRunitheorem). An example of such a number is the unique real root of the polynomial x5 x 1 (which is approximately1.167304).

1.4.2 Closed-form number

Main article: Closed-form number

Algebraic numbers are all numbers that can be dened explicitly or implicitly in terms of polynomials, starting fromthe rational numbers. Onemay generalize this to "closed-form numbers", whichmay be dened in various ways. Mostbroadly, all numbers that can be dened explicitly or implicitly in terms of polynomials, exponentials, and logarithmsare called elementary numbers, and these include the algebraic numbers, plus some transcendental numbers. Mostnarrowly, one may consider numbers explicitly dened in terms of polynomials, exponentials, and logarithms thisdoes not include algebraic numbers, but does include some simple transcendental numbers such as e or log(2).

1.5 Algebraic integersMain article: Algebraic integerAn algebraic integer is an algebraic number that is a root of a polynomial with integer coecients with leadingcoecient 1 (a monic polynomial). Examples of algebraic integers are 5 + 132, 2 6i, and 12(1 + i3). Note,therefore, that the algebraic integers constitute a proper superset of the integers, as the latter are the roots of monicpolynomials x k for all k Z. In this sense, algebraic integers are to algebraic numbers what integers are to rationalnumbers.

4 CHAPTER 1. ALGEBRAIC NUMBER

Algebraic numbers colored by leading coecient (red signies 1 for an algebraic integer).

The sum, dierence and product of algebraic integers are again algebraic integers, which means that the algebraicintegers form a ring. The name algebraic integer comes from the fact that the only rational numbers that are algebraicintegers are the integers, and because the algebraic integers in any number eld are in many ways analogous to theintegers. If K is a number eld, its ring of integers is the subring of algebraic integers in K, and is frequently denotedas OK. These are the prototypical examples of Dedekind domains.

1.6 Special classes of algebraic number Gaussian integer Eisenstein integer Quadratic irrational Fundamental unit Root of unity Gaussian period PisotVijayaraghavan number Salem number

1.7 Notes[1] In order for a number to be algebraic, it has to be the root of a nite, non-zero polynomial. Pi, commonly known to be

transcendental, is a root of sin(x) which is analytic (meaning that it is equal to its innite Taylor series). Thus transcendentalnumbers can be roots of polynomials, but only if those polynomials are innite.

[2] Some of the following examples come from Hardy and Wright 1972:159160 and pp. 178179

[3] Also Liouvilles theorem can be used to produce as many examples of transcendentals numbers as we please, cf Hardyand Wright p. 161

[4] Hardy and Wright 1972:160 / 2008:205

1.8. REFERENCES 5

[5] Niven 1956, Theorem 7.5.

[6] Niven 1956, Corollary 7.3.

[7] Niven 1956, p. 92.

1.8 References Artin, Michael (1991), Algebra, Prentice Hall, ISBN 0-13-004763-5, MR 1129886 Hardy, G.H. and Wright, E.M. 1978, 2000 (with general index) An Introduction to the Theory of Numbers: 5th

Edition, Clarendon Press, Oxford UK, ISBN 0-19-853171-0 Ireland, Kenneth; Rosen, Michael (1990), A Classical Introduction to Modern Number Theory, Graduate Textsin Mathematics 84 (Second ed.), Berlin, New York: Springer-Verlag, ISBN 0-387-97329-X, MR 1070716

Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556

Niven, Ivan 1956. Irrational Numbers, Carus Mathematical Monograph no. 11, Mathematical Association ofAmerica.

Ore, ystein 1948, 1988, Number Theory and Its History, Dover Publications, Inc. New York, ISBN 0-486-65620-9 (pbk.)

Chapter 2

Choice function

For the combinatorial choice function C(n, k), see Combination and Binomial coecient.

A choice function (selector, selection) is a mathematical function f that is dened on some collectionX of nonemptysets and assigns to each set S in that collection some element f(S) of S. In other words, f is a choice function for X ifand only if it belongs to the direct product of X.

2.1 An exampleLet X = { {1,4,7}, {9}, {2,7} }. Then the function that assigns 7 to the set {1,4,7}, 9 to {9}, and 2 to {2,7} is achoice function on X.

2.2 History and importanceErnst Zermelo (1904) introduced choice functions as well as the axiom of choice (AC) and proved the well-orderingtheorem,[1] which states that every set can be well-ordered. AC states that every set of nonempty sets has a choicefunction. A weaker form of AC, the axiom of countable choice (AC) states that every countable set of nonemptysets has a choice function. However, in the absence of either AC or AC, some sets can still be shown to have achoice function.

If X is a nite set of nonempty sets, then one can construct a choice function for X by picking one elementfrom each member of X: This requires only nitely many choices, so neither AC or AC is needed.

If every member of X is a nonempty set, and the union SX is well-ordered, then one may choose the leastelement of each member ofX . In this case, it was possible to simultaneously well-order every member ofXby making just one choice of a well-order of the union, so neither AC nor AC was needed. (This exampleshows that the well-ordering theorem implies AC. The converse is also true, but less trivial.)

2.3 Renement of the notion of choice functionA function f : A ! B is said to be a selection of a multivalued map :A B (that is, a function ' : A ! P(B)from A to the power set P(B) ), if

8a 2 A (f(a) 2 '(a)) :

The existence of more regular choice functions, namely continuous or measurable selections is important in the theoryof dierential inclusions, optimal control, and mathematical economics.[2]

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2.4. SEE ALSO 7

2.3.1 Bourbaki tau functionNicolas Bourbaki used epsilon calculus for their foundations that had a symbol that could be interpreted as choosingan object (if one existed) that satises a given proposition. So if P (x) is a predicate, then x(P ) is the object thatsatises P (if one exists, otherwise it returns an arbitrary object). Hence we may obtain quantiers from the choicefunction, for example P (x(P )) was equivalent to (9x)(P (x)) .[3]However, Bourbakis choice operator is stronger than usual: its a global choice operator. That is, it implies the axiomof global choice.[4] Hilbert realized this when introducing epsilon calculus.[5]

2.4 See also Axiom of countable choice Hausdor paradox Hemicontinuity

2.5 Notes[1] Zermelo, Ernst (1904). Beweis, dass jede Menge wohlgeordnet werden kann. Mathematische Annalen 59 (4): 51416.

doi:10.1007/BF01445300.

[2] Border, Kim C. (1989). Fixed Point Theorems with Applications to Economics and Game Theory. Cambridge UniversityPress. ISBN 0-521-26564-9.

[3] Bourbaki, Nicolas. Elements of Mathematics: Theory of Sets. ISBN 0-201-00634-0.

[4] John Harrison, The Bourbaki View eprint.

[5] Here, moreover, we come upon a very remarkable circumstance, namely, that all of these transnite axioms are derivablefrom a single axiom, one that also contains the core of one of the most attacked axioms in the literature of mathematics,namely, the axiom of choice: A(a) ! A("(A)) , where " is the transnite logical choice function. Hilbert (1925), Onthe Innite, excerpted in Jean van Heijenoort, From Frege to Gdel, p. 382. From nCatLab.

2.6 ReferencesThis article incorporates material from Choice function on PlanetMath, which is licensed under the Creative CommonsAttribution/Share-Alike License.

Chapter 3

Rational number

Rationals redirects here. For other uses, see Rational (disambiguation).

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers,p and q, with the denominator q not equal to zero.[1] Since q may be equal to 1, every integer is a rational number.The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold Q , Unicode );[2] it was thusdenoted in 1895 by Peano after quoziente, Italian for "quotient".The decimal expansion of a rational number always either terminates after a nite number of digits or begins torepeat the same nite sequence of digits over and over. Moreover, any repeating or terminating decimal representsa rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary,hexadecimal).A real number that is not rational is called irrational. Irrational numbers include 2, , e, and . The decimalexpansion of an irrational number continues without repeating. Since the set of rational numbers is countable, andthe set of real numbers is uncountable, almost all real numbers are irrational.[1]

The rational numbers can be formally dened as the equivalence classes of the quotient set (Z (Z \ {0})) / ~, wherethe cartesian product Z (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not 0 (n 0),and "~" is the equivalence relation dened by (m1,n1) ~ (m2,n2) if, and only if, m1n2 m2n1 = 0.In abstract algebra, the rational numbers together with certain operations of addition and multiplication form thearchetypical eld of characteristic zero. As such, it is characterized as having no proper subeld or, alternatively,being the eld of fractions for the ring of integers. Finite extensions of Q are called algebraic number elds, and thealgebraic closure of Q is the eld of algebraic numbers.[3]

In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can beconstructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or innite decimals.Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undened).

3.1 TerminologyThe term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers.In mathematics, the adjective rational often means that the underlying eld considered is the eld Q of rationalnumbers. Rational polynomial usually, and most correctly, means a polynomial with rational coecients, also calleda polynomial over the rationals. However, rational function does not mean the underlying eld is the rationalnumbers, and a rational algebraic curve is not an algebraic curve with rational coecients.

3.2 ArithmeticSee also: Fraction (mathematics) Arithmetic with fractions

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3.2. ARITHMETIC 9

3.2.1 Embedding of integersAny integer n can be expressed as the rational number n/1.

3.2.2 Equalityab =

cd if and only if ad = bc:

3.2.3 OrderingWhere both denominators are positive:

ab 0 ^ m1n2 n1m2) _ (n1n2 < 0 ^ m1n2 n1m2):The integers may be considered to be rational numbers by the embedding that maps m to [(m,1)].

12 CHAPTER 3. RATIONAL NUMBER

3.5 Properties

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A diagram illustrating the countability of the rationals

The set Q, together with the addition and multiplication operations shown above, forms a eld, the eld of fractionsof the integers Z.The rationals are the smallest eld with characteristic zero: every other eld of characteristic zero contains a copy ofQ. The rational numbers are therefore the prime eld for characteristic zero.The algebraic closure of Q, i.e. the eld of roots of rational polynomials, is the algebraic numbers.The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost allreal numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, innitelymany other ones. For example, for any two fractions such that

a

b