Ten Ways of Looking at Real Numbers

Robert Mayans

Department of Math/CSci/Physics

Fairleigh Dickinson University

Varieties of Mathematical Text

• Books– Reference books– Text books– Lecture Notes– Handbooks and encyclopedias

Varieties of Mathematical Text

• Papers– Research papers– Survey articles– Collected works

• Online Resources– MathSciNet– Online databases– E-journals

The Mathematics Hypertext Project (MHP)

• A Web-based hypertext of mathematics• A design paper describes goals, organization,

technology issues, etc.http://jodi.ecs.soton.ac.uk/Articles/v05/i01/Mayans/jodi/

• First release in June, 2005• This presentation discusses some of the pages

on the real numbers

Text on Real Numbers

• We aim for a “comprehensive introduction" to the real numbers.

• Real numbers are everywhere dense in mathematics.

• Real numbers have different meanings in different contexts.

Text on Real Numbers

• What to include on a text on real numbers?– Foundations and construction of real numbers– Characterizations of the real numbers by

structure.– Extensions and substructures of the real

numbers– Real numbers classified in different ways.– Real numbers are a system related to other

systems.

#1. Foundations of the Real Number System

• How to define real numbers and their basic operations from rationals or integers

• Presentation of three methods– Dedekind cuts of rational numbers– Equivalence classes of Cauchy sequences of

rational numbers– Base-10 digit sequences

#1. Foundations of the Real Numbers

• Discussion of foundations from John Conway, "On Numbers and Games"

N

ZQ+

QR+

R

#2. Real Numbers as a Linear Order

• The real numbers form the unique linear order that is:– dense– without endpoints– Dedekind-complete– separable (countable dense subset)

#2. Real Numbers as a Linear Order

• Suslin Problem: Replace separability with the countable chain condition:– every collection of disjoint nontrivial closed

intervals is at most countable.• Does this characterize the real numbers?• A counterexample is called a Suslin line• The existence of Suslin lines are independent of

the axioms of ZFC set theory

#2. Real Numbers as a Linear Order

• The real numbers may be viewed as a space of branches of an infinite tree.

• Trees are partial orders whose initial segments {x : X<p} are well-ordered. The branches are the maximal chains in the partial order.

• Different infinite trees (Aronzsajn tree, Kurepa tree) give rise to different linear orders (Aronszajn line, Kurepa line).

#3. Real Numbers as a Topological Space

• A characterization of the usual topology of the real line:

• If X is a regular, separable, connected, locally connected space, in which every point is a cut point, then X is homeomorphic to the real line.– A point p in a connected space X is a cut-

point if X\{p} is disconnected.

• Another characterization: Replace “regular” with “metrizable”.

#3. Real Numbers as a Topological Space

• The real numbers form a complete separable metric space, a “Polish space”. Also, it is perfect.

• Other examples of perfect Polish spaces:– Cantor space: all sequences of 0’s and 1’s– Baire space: all sequences of natural numbers– Finite/countable products of perfect Polish

spaces• Every perfect Polish space is Borel-isomorphic to

the real numbers.

#3. Real Numbers as a Topological Space

• The real line is a one-dimensional topological manifold.

• Classification of connected Hausdorff one-dimensional manifolds– the real line– the circle– the long line– the open long ray

#4. Real Numbers as a Completion of the Rational Numbers

• A valuation is a function from a field to the nonnegative real numbers with properties analogous to a norm or absolute value:

baCba

baba

aa

,max

0 iff 0

#4. Real Numbers as a Completion of the Rational Numbers

• Two valuations are equivalent if one is a power of the other.

• Every valuation is equivalent to one which satisfies:

• Such a valuation defines a metric on a field: the distance between a and b is

baba

abba

#4. Real Numbers as a Completion of the Rational Numbers

• Ostrowski’s theorem: The inequivalent valuations on the rational numbers are absolute value, the trivial valuation, and the p-adic valuations for every prime p.

#4. Real Numbers as a Completion of the Rational Numbers

• The metric completions of the rationals defined by a valuation are:– discrete topology on Q (with the trivial valuation)– R (with absolute value)

– Rp (the p-adic reals, with the p-adic valuation).

#5. The Real Numbers as a Field

• The real numbers form an ordered field.• Subfields of the real numbers:

– rational numbers– real algebraic number fields– computable real numbers– constructible real numbers

• The algebraic completion of the real numbers is the field of complex numbers

#5. The Real Numbers as a Field

• The field of real numbers is the prototypical real-closed field: its algebraic closure is a finite extension.

• The Artin-Schreier theorem characterizes a real-closed field:– it has characteristic 0– algebraic closure by adjoining i, where i2 = -1– it has a linear order– every positive number has a square root– -1 is not a sum of squares

#5. The Real Numbers as a Field

• Any field is a vector space over a subfield.• The real numbers form a vector space over the

rational numbers.• A basis for this vector space is called a Hamel

space.

#6. The Real Numbers as an Algebra

• To what extent can the operations on the reals extend to finite-dimensional algebras over the reals?

• Here we list a few results.

#6. The Real Numbers as an Algebra

• The finite-dimensional associative real division algebras are the real numbers, complex numbers, and the quaternions. (Frobenius)

• The finite-dimensional real commutative division algebras with unit are the real numbers and the complex numbers. (Hopf)

• The finite-dimensional real division algebras have dimension 1, 2, 4, or 8. (Kervaire, Milnor)

#7. The Cardinal of the Real Numbers

• Cantor showed that the real numbers are not equinumerous with the integers.

• Write as the cardinal of the set of real numbers, the cardinal of the continuum.

• The Continuum Hypothesis: Does ?1c

c

#7. The Cardinal of the Real Numbers

• The continuum must satisfy:

• The second condition guarantees that:• Not much else restricts the possible values of the

continuum.

00 )(cf , cc

c

#7. The Cardinal of the Real Numbers

• Easton’s theorem: Let be any regular cardinal in the ground model of ZFC with cofinality

• Then there is a generic extension which preserves cardinalities, in which

• For example, the continuum could bec

0

.not but ,,,, 21 n

#7. The Cardinal of the Real Numbers

• A variety of “cardinal invariants” of the continuum: cardinals between .

• We give two examples: the bounding number b, and the dominating number d.

• Let f, g: N N. We say f dominates g iff f(n)≥g(n) for sufficiently large n.

c and 1

#7. The Cardinal of the Real Numbers

• The bounding number b: the minimum number of functions f such that no g dominates every f.

• The dominating number d: the minimum number of functions f such that every g is dominated by a function f.

• c db1

#8. Number Theoretic Classification of Real Numbers

• Rational numbers, algebraic numbers, transcendental numbers.

• Liouville’s theorem: numbers that can be very well approximated by rationals must be transcendental.

• If, for infinitely many n, there is a rational such that ,then α is transcendental.nqqp /

#8. Number Theoretic Classification of Real Numbers

• Mahler's classification of real numbers– A: algebraic numbers– S, T, U: classes of transcendental numbers

• Roughly speaking, it measures how well can a number be approximated by algebraic numbers.

• If x, y are algebraically dependent, then x and y belong to the same Mahler class.

• Most real numbers are S-numbers by measure, U-numbers by category.

#9. The Real Numbers as a First-Order Theory

• Tarski's decidability theorem: The first-order theory of real-closed fields is decidable.

• There is an algorithmic procedure to determine if a first-order sentence about the real numbers in the language of ordered fields is true or false.

#9. The Real Numbers as a First-Order Theory

• Nonstandard real numbers extend the real number system with infinitesimal numbers.

• One construction is with an ultrapower of an first-order model of the real numbers, with all possible constants, predicates, and functions.

• Every nonstandard real number may be written uniquely as a sum of a standard real number and an infinitesimal.

#10. Surreal Numbers

• Surreal numbers are a subclass of a class of finitely-move two-person games.

• One development: a surreal is an ordinal-length sequence of +’s and –’s.

• Surreals are lexicographically ordered by -, (empty), +.

• The surreal numbers, as a proper class, form an ordered field.

• The real numbers are a subfield of the surreals of order .

#10. Surreal Numbers

Examples of surreal numbers in order:• -- -2• - -1• -+ -1/2• 0• +-+ ¾• ++++ 4

#10. Surreal Numbers

Surreals of order :all dyadic fractions

Surreals of order :all real numbersall dyadic fractions

1

Many more views of the real numbers

• Geometry axioms for the real line• Real numbers as infinite continued fractions• Numeration schemes for real numbers• Alternative foundations: constructivism,

intuitionism, nonstandard set theory• Computational approximations to real numbers:

floating point numbers, interval arithmetic, and so on.

Many more views of the real numbers

• Complexity and randomness measures on real numbers (for example, Turing degrees)

• Historical and philosophical perspectives: the real numbers as an idealization of a measurement, the meaning and use of infinitesimals, and so on.

• Real numbers as a representation of an infinite sequence of Bernoulli trials

• Real numbers generated by formal languages.

Many more views of the real numbers

• Digit patterns in real numbers, such as normal numbers.

• Real numbers as set-theoretic codes. A real number may code:– a cardinal collapse– a Borel set– a countable model of set theory– a strategy for an infinite two-person game.

Organizing Multiple Theories

• How should the hypertext on real numbers be organized?

• Less than a grand all-encompassing architecture• More that a simple listing of topics in unrelated

slots. • The goal is a readable, searchable, general

introduction to the real number system.

Organizing Multiple Theories

• It must also show relationships across categories.

• It must lead to more in-depth text• It must be in a form that is easy to update and

extend.• It must help readers searching for a topic.

Hypertext Structures

• Most text in this system is in one of two forms: “book text” and “core text”.

Book Text

• Book text gives an orderly development of mathematical ideas.

• Shorter and more narrowly focused than most math books.

• Theorems, proof, definitions, examples• Other books attached in a tree-like structure.

Core Text

• Short, highly-linked texts, organized around a concept or method

• Discursive, condensed discussions of a mathematical topic

• Previews, surveys, summaries, leading to other text.

• Helps the user navigate to other topics.• The same topic may reappear in several core

texts.

Core TextReal

Numbers

main essay

Core Text

Real

Numbers

Complex

Numbers

Vector

Spaces

Sets of Real

Numbers

Functions of a Real

Variable

Core Text and Book Text

Mathematics Hypertext Project

• First step of a very long term project.• Need for contributors and collaboration.• Goal is the building of large-scale structures of

mathematical text.