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"
R
R+
Q
Q+
Z
N
#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 cutpoint 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 onedimensional 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:
  a   0 iff a  0
  a  b     a    b 
  a  b   C  max   a ,  b 
#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:   a  b     a    b 
• Such a valuation defines a metric on a field: the
distance between a and b is   a  b    b  a 
#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 realclosed field: its algebraic closure is a finite
extension.
• The Artin-Schreier theorem characterizes a realclosed 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 c as the cardinal of the set of real
numbers, the cardinal of the continuum.
• The Continuum Hypothesis: Does c   1 ?
#7. The Cardinal of the Real Numbers
• The continuum must satisfy:
c   0 , cf ( c )   0

• The second condition guarantees that: c  
• Not much else restricts the possible values of the
continuum.
#7. The Cardinal of the Real Numbers
• Easton’s theorem: Let  be any regular cardinal
in the ground model of ZFC with cofinality   0
• Then there is a generic extension which
preserves cardinalities, in which c  
• For example, the continuum could be
 1 ,  2 ,  ,  n ,  but not   .
#7. The Cardinal of the Real Numbers
• A variety of “cardinal invariants” of the
continuum: cardinals between  1 and c .
• 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.
#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.
•   b  d  c
1
#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   p / q  q  n ,then α is transcendental.
#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 FirstOrder 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 FirstOrder 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 numbers
all 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 Text
Real
Numbers
main
essay
Core Text
Sets of Real
Numbers
Real
Numbers
Vector
Spaces
Functions of a Real
Variable
Complex
Numbers
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.
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