Subject: sci.math FAQ: Numbers
Summary: Part 10 of many, New version,
Nntp-Posting-Host: neumann.uwaterloo.ca
Date: Tue, 25 Apr 1995 17:41:53 GMT


Archive-Name: sci-math-faq/numbers
Version: 6.2


   
   

                              WHAT ARE NUMBERS?




     _________________________________________________________________

     * Introduction
     * Construction of the Number System
     * Construction of N
     * Construction of Z
     * Construction of Q
     * Construction of R
     * Construction of C
     * Rounding things up
     * What's next?


     _________________________________________________________________




Introduction



   Informally:
     * \N = { 0,1,... } or \N = { 1,2,... }
       Wether 0 is in \N depends on where you live and what is your field
       of interest. At the informal level it is a religious topic.
     * \Z = { ..., - 1,0,1,... }
     * \Q = { p/q | p, q in \Z and q != 0 }
     * \R = { d_0.d_1d_2... | d_0 in \Z and 0 <= d_i <= 9 for i > 0 }
     * \C = { a + b o i | a, b in \R and i^2 = -1 }




     _________________________________________________________________



Construction of the Number System



   Formally (following the main stream in math) the numbers are
   constructed from scratch out of the axioms of Zermelo Fraenkel set
   theory (a.k.a. ZF set theory) [Enderton77, Henle86, Hrbacek84]. The
   only things that can be derived from the axioms are sets with the
   empty set at the bottom of the hierarchy. This will mean that any
   number is a set (it is the only thing you can derive from the axioms).
   It doesn't mean that you always have to use set notation when you use
   numbers: just introduce the ``informal'' numbers as an abbreviation of
   the formal counterparts.

   The construction starts with \N and algebraically speaking, \N with
   its operations and order is quite a weak structure. In the following
   constructions the structures will be strengthen one step at the time:
   \Z will be an integral domain, \Q will be a field, for the field \R
   the order will be made complete, and field \C will be made
   algebraically complete.

   Before we start, first some notational stuff:
     * a pair (a,b) = { { a } , { a,b } } ,
     * an equivalence class [a] = { b | a == b } ,
     * the successor of a is s(a) = a U { a } .




     _________________________________________________________________


Construction of N



     * { } in \N
     * if a in \N then s(a) in \N
     * \N is the smallest possible set such that the preceding rules
       hold.

   Informally n = { 0,...,n - 1 } (thus 0 = { } , 1 = { 0 } , 2 = { 0,1 }
   , 3 = { 0,1,2 } ). We will refer to the elements of \N by giving them
   a subscript _n . The relation <_n on \N is defined as: a_n <_n b_n iff
   a_n in b_n . We can define +_n as follows:
     * a_n +_n 0_n = a_n
     * a_n +_n s(b_n) = s(a_n +_n b_n)

   Define *_n as:
     * a_n *_n 0_n = 0_n
     * a_n *_n s(b_n) = (a_n *_n b_n) +_n a_n




     _________________________________________________________________


Construction of Z



   We define an equivalence relation on \N x \N : (a_n,b_n) ==_z(c_n,d_n)
   iff a_n +_n d_n = c_n +_n b_n . Note that ==_z ``simulates'' a
   subtraction in \N . \Z = { [(a_n,b_n)]_z | a_n, b_n in \N } . We will
   refer to the elements of \Z by giving them a subscript _z . The
   elements of \N can be embedded as follows: embed_n : \N --> \Z such
   that embed_n(a_n) = [(a_n,0_n)]_z . Furthermore we can define:
     * [(a_n,b_n)]_z <_z [(c_n,d_n)]_z iff a_n +_n d_n <_n c_n +_n b_n
     * [(a_n,b_n)]_z +_z [(c_n,d_n)]_z = [(a_n +_n c_n, b_n +_n d_n)]_z
     * [(a_n,b_n)]_z *_z [(c_n,d_n)]_z =
       [((a_n *_n c_n) +_n (b_n *_n d_n), (a_n *_n d_n) +_n (c_n *_n
       b_n))]_z




     _________________________________________________________________



Construction of Q



   We define an equivalence relation on \Z x (\Z \ { 0_z }) : (a_z,b_z)
   ==_q (c_z,d_z) iff a_z *_z d_z = c_z *_z b_z . Note that ==_q
   ``simulates'' a division in \Z . \Q = { [(a_z,b_z)]_q | a_z in \Z and
   b_z in \Z \ { 0_z } } . We will refer to the elements of \Q by giving
   them a subscript _q . The elements of \Z can be embedded as follows:
   embed_z : \Z --> \Q such that embed_z(a_z) = [(a_z,1_z)]_q .
   Furthermore we can define:
     * [(a_z,b_z)]_q <_q [(c_z,d_z)]_q iff a_z *_z d_z <_z c_z *_z b_z
       when 0_z <_z b_z and 0_z <_z d_z
     * [(a_z,b_z)]_q +_q [(c_z,d_z)]_q = [((a_z *_z d_z) +_z (c_z *_z
       b_z), b_z *_z d_z)]_q
     * [(a_z,b_z)]_q *_q [(c_z,d_z)]_q = [(a_z *_z c_z, b_z *_z d_z)]_q




     _________________________________________________________________


Construction of R



   The construction of \R is different (and more awkward to understand)
   because we must ensure that the cardinality of \R is greater than that
   of \Q .
   Set c is a Dedekind cut iff
     * { } subset c subset \Q (strict inclusions!)
     * c is closed downward:
       if a_q in c and b_q <_q a_q then b_q in c
     * c has no largest element:
       there isn't an element a_q in c such that b_q <_q a_q for all b_q
       != a_q in c

   You can think of a cut as taking a pair of scissors and cutting \Q in
   two parts such that one part contains all the small numbers and the
   other part contains all large numbers. If the part with the small
   numbers was cut in such a way that it doesn't have a largest element,
   it is called a Dedekind cut. \R = { c | c is a Dedekind cut } . We
   will refer to the elements of \R by giving them a subscript _r . The
   elements of \Q can be embedded as follows: embed_q : \Q --> \R such
   that embed_q(a_q) = { b_q | b_q <_q a_q } . Furthermore we can define:
     * a_r <_r b_r iff a_r subset b_r (strict inclusion!)
     * a_r +_r b_r = { c_q +_q d_q | c_q in a_r and d_q in b_r }
     * -_r a_r = ; { b_q | there exists an c_q in \Q such that b_q <_q
       c_q and (-1)_q *_q c_q \not in a_r }
     * |a_r|_r = a_r U -_r a_r
     * *_r is defined as:
          + if not a_r <_r 0_r and not b_r <_r 0_r
            then a_r *_r b_r = 0_r U { c_q *_q d_q | c_q in a_r and d_q
            in b_r }
          + if a_r <_r 0_r and b_r <_r 0_r then a_r *_r b_r = |a_r|_r *_r
            |b_r|_r
          + otherwise a_r *_r b_r = -_r (|a_r|_r *_r |b_r|_r)



   There exists an alternative definition of \R using Cauchy sequences: a
   Cauchy sequence is a s : \N --> \Q such that s(i_n) +_q((-1)_q *_q
   s(j_n)) can be made arbitrary near to 0_q for all sufficiently large
   i_n and j_n . We will define an equivalence relation ==_r on the set
   of Cauchy sequences as: r ==_r s iff r(m_n) +_q((-1)_q *_q s(m_n)) can
   be made arbitrary close to 0_q for all sufficiently large m_n . \R = {
   [s]_r | s is a Cauchy sequence } . Note that this definition is close
   to ``decimal'' expansions.


     _________________________________________________________________




Construction of C



   \C = \R x \R . We will refer to the elements of \C by giving them a
   subscript _c . The elements of \R can be embedded as follows: embed_r
   : \R --> \C such that embed_r(a_r) = (a_r,0_r) . Furthermore we can
   define:
     * (a_r,b_r) +_c (c_r,d_r) = (a_r +_r c_r, b_r +_r d_r )
     * (a_r,b_r) *_c (c_r,d_r) = ((a_r *_r c_r) +_r -_r (b_r * d_r), (a_r
       *_r d_r) +_r (b_r *_r c_r))



   There exists an elegant alternative definition using ideals. To be a
   bit sloppy: \C = \R[x]/< (x *_r x) +_r 1_r > , i.e. \C is the
   resulting quotient ring of factoring ideal < (x *_r x) +_r 1_r > out
   of the ring \R[x] of polynomials over \R . The sloppy part is that we
   need to define concepts like quotient ring, ideal, and ring of
   polynomials. Note that this definition is close to working with i^2 =
   -1 : (x *_r x) +_r 1_r = 0_r can be rewritten as (x *_r x) = (-1)_r .


     _________________________________________________________________




Rounding things up



   At this moment we don't have that \N is a subset of \Z , \Z of \Q ,
   etc. But we can get the inclusions if we look at the embedded copies
   of \N , \Z , etc. Let
     * \N' = ran embed_r o embed_q o embed_z o embed_n
     * \Z' = ran embed_r o embed_q o embed_z
     * \Q' = ran embed_r o embed_q
     * \R' = ran embed_r

   For these sets we have \N' subseteq \Z' subseteq \Q' subseteq \R'
   subseteq \C . Furthermore these sets have all the properties that the
   ``informal'' numbers have.


     _________________________________________________________________





What's next?



   Well, for some of the more alien parts of math we can extend this
   standard number system with some exotic types of numbers. To name a
   few:
     * Cardinals and ordinals
       Both are numbers in ZF set theory [Enderton77, Henle86, Hrbacek84]
       and so they are sets as well. Cardinals are numbers that represent
       the sizes of sets, and ordinals are numbers that represent well
       ordered sets. Finite cardinals and ordinals are the same as the
       natural numbers. Cardinals, ordinals, and their arithmetic get
       interesting and ``tricky'' in the case of infinite sets.
     * Hyperreals
       These numbers are constructed by means of ultrafilters [Henle86]
       and they are used in non-standard analysis. With hyperreals you
       can treat numbers like Leibnitz and Newton did by using
       infinitesimals.
     * Quaternions and octonions
       Normally these are constructed by algebraic means (like the
       alternative \C definition that uses ideals) [Shapiro75].
       Quaternions are used to model rotations in 3 dimensions. Octonions
       are just esoteric artifacts :-). Well, if you know where they are
       used for, feel free to contribute to the FAQ.

   Cardinals and ordinals are commonly used in math. Most mortals won't
   encounter (let alone use) hyperreals, quaternions, and octonions.



   References

   H.B. Enderton. Elements of Set Theory. Academic Press, 1977.



   J.M. Henle. An Outline of Set Theory. Springer Verlag, 1986.



   K. Hrbacek and T. Jech. Introduction to Set Theory. M. Dekker Inc.,
   1984.



   L. Shapiro. Introduction to Abstract Algebra. McGraw-Hill, 1975.

     _________________________________________________________________


This section of the FAQ is (C) Hans de Vreught. Send comments and or
corrections relating to this part to:

hdev@cp.tn.tudelft.nl (Hans de Vreught)



 
   


