Constructions of Real Numbers
ad...@numericable.fr
I know different constructions of real numbers, one with Dedekind's
cuts, another one with Cauchy's sequences...
I understand them and they are "pretty", BUT these constructions are
not natural for me, I mean they are not intuitive...
I have heard that a construction of real numbers by decimal expansions
is possible, and it seems very intuitive then...
But the problem is that arithmetical operations (+,x) are difficult to
define formally, and show that R is a field too...
Could everyone give me links or complete proofs with construction of
real numbers by decimal expansions please?
Thank you in advance, this subject is wonderful and thinking about
constructions of "basic" sets is very interesting :)
Best regards.
PS: I am french, and my english is poor!
gow...@hotmail.com
ad...@numericable.fr wrote:
> Hello!
>
> I know different constructions of real numbers, one with Dedekind's
> cuts, another one with Cauchy's sequences...
>
> I understand them and they are "pretty", BUT these constructions are
> not natural for me, I mean they are not intuitive...
>
> I have heard that a construction of real numbers by decimal expansions
> is possible, and it seems very intuitive then...
>
> But the problem is that arithmetical operations (+,x) are difficult to
> define formally, and show that R is a field too...
>
> Could everyone give me links or complete proofs with construction of
> real numbers by decimal expansions please?
>
>
There was a paper by Gian-Carlo Rota and, I think, N. Metropolis, quite
a while ago which showed how to do this.
ADmax
F. Faltin, N. Metropolis, B. Ross, G.-C. Rota, The Real Numbers as a
Wreath Product
But I can't find it on the web... do you have .pdf file?
Dave L. Renfro
> I know different constructions of real numbers, one with
> Dedekind's cuts, another one with Cauchy's sequences...
>
> I understand them and they are "pretty", BUT these
> constructions are not natural for me, I mean they are
> not intuitive...
>
> I have heard that a construction of real numbers by decimal
> expansions is possible, and it seems very intuitive then...
[snip]
I know this doesn't specifically answer your question
(how one verifies the field properties of the reals when
the reals are constructed from decimal expansions), but
you and some others might be interested in the following
excerpt from an in-progress manuscript of mine.
There are many ways to construct the real numbers. Among
the more common methods are the use of Cauchy sequences,
Dedekind cuts, decimal expansions, and nested intervals with
rational endpoints. For some other ways to construct the real
numbers, see Dhombres [1], Knopfmacher/Knopfmacher [4] [5] [6],
Maier/Maier [7], Rieger [10], and Shiu [11]. For historical
issues related to various constructions of the real numbers,
see Ferreirós [2] (Chapter IV), Manheim [8] (Sections 4.6-4.13,
pp. 76-95), Pellicer [9], and Simsa [12]. Fowler [3] proves
that sqrt(2) times sqrt(3) equals sqrt(6) for several ways of
constructing the real numbers, after a brief discussion about
the significance of this identity.
[1] Jean G. Dhombres, "Real numbers from Cauchy to Robinson",
Southeast Asian Bulletin of Mathematics 1 (1977), 9-20.
[MR 58 #21199]
[2] José Ferreirós, LABYRINTH OF THOUGHT: A HISTORY OF SET THEORY
AND ITS ROLE IN MODERN MATHEMATICS, Science Networks /
Historical Studies #23, Birkhäuser Verlag, 1999,
xxi + 440 pages. [MR 2000m:03005; Zbl 934.03058]
http://www.emis.de/cgi-bin/MATH-item?0934.03058
[3] David Fowler, "Dedekind's theorem: sqrt(2) x sqrt(3)
= sqrt(6)", American Mathematical Monthly 99 #8
(October 1992), 725-733. [MR 93h:01022; Zbl 766.01016]
http://www.emis.de/cgi-bin/MATH-item?0766.01016
[4] Arnold Knopfmacher and John Knopfmacher, "A new construction
of the real numbers (via infinite products)", Nieuw Archief
voor Wiskunde (4) 5 (1987), 19-31.
[MR 88i:11007; Zbl 624.10007]
http://www.emis.de/cgi-bin/MATH-item?0624.10007
[5] Arnold Knopfmacher and John Knopfmacher, "Two concrete new
constructions of the real numbers", Rocky Mountain Journal
of Mathematics 18 (1988), 813-824.
[MR 90k:26003a; Zbl 677.10006]
http://www.emis.de/cgi-bin/MATH-item?0677.10006
[6] Arnold Knopfmacher and John Knopfmacher, "Two constructions
of the real numbers via alternating series", International
Journal of Mathematics and Mathematical Sciences 12 (1989),
603-613. [MR 90k:26003b; Zbl 683.10008]
http://www.emis.de/cgi-bin/MATH-item?0683.10008
[7] David E. Maier and Eugene A. Maier, "Construction of the
real numbers", (Two-Year) College Mathematics Journal
4 #1 (Winter 1973), 31-35.
[8] Jerome H. Manheim, THE GENESIS OF POINT SET TOPOLOGY,
Pergamon Press, 1964, xiii + 166 pages.
[MR 37 #2561; Zbl 119.17702]
http://www.emis.de/cgi-bin/Zarchive?an=0119.17702
[9] Manuel López Pellicer, "Las construcciones de los números
reales" [Constructions of real numbers], pp. 11-33 in
HISTORIA DE LA MATEMÁTICA EN EL SIGLO XIX (PARTE 2),
Real Academia de Ciencias Exactas, Físicas y Naturales
(Madrid), 1994. [MR 98f:01030; Zbl 952.00024]
http://www.emis.de/cgi-bin/MATH-item?0952.00024
[10] Georg Johann Rieger, "A new approach to the real numbers
(motivated by continued fractions)", Abhandlungen der
Braunschweigischen Wissenschaftlichen Gesellschaft 33
(1982), 205-217. [MR 84j:26002; Zbl 513.10009]
http://www.emis.de/cgi-bin/MATH-item?0513.10009
[11] Peter Shiu, "A new construction of the real numbers",
Mathematical Gazette 58 #403 (March 1974), 39-46.
[12] Jaromír Simsa, "Development of the concept of real numbers"
(Czech), pp. 259-282 in MATHEMATICS IN THE 16TH AND 17TH
CENTURIES (Czech) (Jevícko, 1997), Dej. Mat./Hist. Math.
#12, Prometheus, Prague, 1999. [MR 2003g:01001]
Dave L. Renfro
gow...@hotmail.com
ADmax wrote:
> Is it this paper?
>
> F. Faltin, N. Metropolis, B. Ross, G.-C. Rota, The Real Numbers as a
> Wreath Product
>
> But I can't find it on the web... do you have .pdf file?
>
Here is a complete citation:
Metropolis, N.; Rota, Gian-Carlo
Significance arithmetic---on the algebra of binary strings.
Studies in numerical analysis (papers in honour of Cornelius Lanczos on
the occasion of his 80th birthday), pp. 241--251.
Academic Press, London, 1974.
I don't know whether the complete paper is available on the web. Hope
this helps.
ADmax
article :(
Thanks for your help
A N Niel
<ad...@numericable.fr> wrote:
> [...] they are not intuitive [...]
>
> I have heard that a construction of real numbers by decimal expansions
> is possible, and it seems very intuitive then...
>
You think base ten is intuitive only because your ancestors had ten
fingers.
Ryan Reich
ADmax
Ryan Reich
On 17 Jun 2006 13:19:17 -0700, ADmax <ad...@numericable.fr> wrote:
> Ryan Reich wrote:
>> On Sat, 17 Jun 2006 15:27:08 -0400, A N Niel <ann...@nym.alias.net.invalid> wrote:
>> > You think base ten is intuitive only because your ancestors had ten
>> > fingers.
>>
>> What better source of intuition could there be?
>>
>
> You can change base ten with another base, I dont care about it :)
But then what's the excuse for it being intuitive?
Timothy Golden BandTechnology.com
Hi. I have devoloped an algebraic multidimensional system built around
sign and magnitude that i call polysign numbers:
http://www.bandtechnology.com/PolySigned/PolySigned.html
I use magnitude and sign as components of elementary values and when
combined an n-signed system becomes n-1 dimensional. Hence two-signed
numbers are one-dimensional. The two-signed numbers in this sytem are
the real numbers and the three-signed numbers are complex numbers. Yet
people insist that the system be built from the real numbers. I am
having a difficult time addressing this. Certainly my sytem builds the
real numbers. I need magnitude as a starting point though. Do you see a
conflict there? Do I have to claim a new definition of the reals? It is
not acceptable to build this system from the real numbers because there
are some basic parts of the geometry that will not be manifested,
particularly directed lattices:
http://www.bandtechnology.com/PolySigned/Lattice/Lattice.html
I could use some advice. I believe that I have the right to construct
whatever I wish mathematically and that it's value becomes apparent
through it's consequences. However working with others or convincing
others of the construction is broken at this juncture. For some reason
people can't see the real numbers as two-signed numbers. If you have
any judgement or attack on the system it is most welcome. Even without
judgement advice on this technicality would be helpful.
-Tim
Gene Ward Smith
Timothy Golden BandTechnology.com wrote:
> The two-signed numbers in this sytem are
> the real numbers and the three-signed numbers are complex numbers.
Only because you used real numbers to define them. If the componets are
rational numbers, you do *not* get the real numbers.
> Yet people insist that the system be built from the real numbers. I am
> having a difficult time addressing this. Certainly my sytem builds the
> real numbers.
How? Can you show how, starting from rational polysigns, you construct
the reals?
As for people insisting, so far as I can see it is you who put the real
numbers in there. If you don't want real number components, then what
is your alternative?
I need magnitude as a starting point though. Do you see a
> conflict there?
Yes. The absolute value of a real is a non-negative real. You *can*
construct the reals starting from these and adding a sign. Is that what
you propose?
>Do I have to claim a new definition of the reals?
Only if
(1) You actually give a definition, and
(2) It is new.
So far that hasn't happened.
> http://www.bandtechnology.com/PolySigned/Lattice/Lattice.html
The word "lattice" in mathematics already has two unrelated defintions;
I doubt there is room for a third.
> For some reason
> people can't see the real numbers as two-signed numbers.
They can see it just fine. Start from the positive reals, add times -1
to get negative reals, and then add zero, which is unsigned.
mvonka...@yahoo.com
ad...@numericable.fr wrote:
> Hello!
>
> I know different constructions of real numbers, one with Dedekind's
> cuts, another one with Cauchy's sequences...
>
> I understand them and they are "pretty", BUT these constructions are
> not natural for me, I mean they are not intuitive...
It is quite ironic that you intuitively dislike Cauchy sequences and
seek a ``more intuitive" construction based on decimal expansions.
Decimal expansions ARE Cauchysequences. When I say that pi =
3.141592653... I am referring to a certain Cauchy sequence of rational
numbers (the sequence 3, 3.1, 3.14, 3.1415 etc.) and am calling its
equivalence class pi. The base 10 construction of decimal expansions
is logically nothing more than singling out a particular subset of
Cauchy sequences (that we happen to find convenient or ``intuitive")
and noting that that subset is sufficient to construct all natural
numbers.
> I have heard that a construction of real numbers by decimal expansions
> is possible, and it seems very intuitive then...
It only seems intuitive because you (as have I and most everyone) have
been brainwashed by years of education to find base 10 intuitive. Your
first exposure to ``calculus" was not in highschool or college but in
the second grade (or whenever you first understood the statement 1/3 =
.3333...). Granted, base 10 has its conveniences (not to mention that
nature gave us 10 convenient counting devices), but prior to whenever
it was invented in ancient India, other geniuses (i.e. ancient Greeks
and others) did not find it so intuitive. On convenience of decimal
expansions is that by singling out the base 10 Cauchy sequences one
generally gets a unique sequence (rather than an equivalence class of
sequences) to represent each real number, with the slight caveat that
we get duplication of decimal expansions for certain real numbers due
to rounding up (i.e. 243.64999... = 243.65000...).
> But the problem is that arithmetical operations (+,x) are difficult to
> define formally, and show that R is a field too...
Quite so! If one defines reals by decimal expansions, a truely
rigorous explanation of addition and multiplication is long, tedious,
boring, and, dare I say, not so intuitive. General Cauchy sequences
are much more convenient.
>
> Could everyone give me links or complete proofs with construction of
> real numbers by decimal expansions please?
>
>
> Thank you in advance, this subject is wonderful and thinking about
> constructions of "basic" sets is very interesting :)
>
> Best regards.
>
> PS: I am french, and my english is poor!
Best regards,
Mike
ADmax
[ It is quite ironic that you intuitively dislike Cauchy sequences and
[ seek a ``more intuitive" construction based on decimal expansions.
[ Decimal expansions ARE Cauchysequences. When I say that pi =
[ 3.141592653... I am referring to a certain Cauchy sequence of
rational
[ numbers (the sequence 3, 3.1, 3.14, 3.1415 etc.) and am calling its
[ equivalence class pi. The base 10 construction of decimal expansions
[ is logically nothing more than singling out a particular subset of
[ Cauchy sequences (that we happen to find convenient or ``intuitive")
[ and noting that that subset is sufficient to construct all natural
[ numbers.
I am sorry but Cauchy sequences are NOT intuitive!!! This is only
because you learn that
real numbers form a *complete* field that you know that Cauchy
sequences are important!
Moreover, if you look at Cauchy sequences ' definition, it is not an
intuitive definition
and i doubt that someone that never studied them can discover them.
In contrary, someone that just knows decimal expansions can study and
understand the problem of constructing real numbers!
Best Regards.
mvonka...@yahoo.com
How can anyone ``just know" decimal expansions? They have to be
learned and understood before they can be ``just known". I insist that
to ``just know" decimal expansions one must at some rudimentary level
understand, however naively, on some conceptual level what decimal
expansions actually are. What they are is a certain subclass of CAUCHY
SEQUENCES. I agree that one does not have to start with the full
general understanding of Cauchy sequences in a metric space and achieve
in elementary school the depth of understanding that one hopes to
achieve later in life after a thorough study of calculus. Decimal
expansions are a perfectly appropriate way to introduce a student to
real numbers in grade school. I just thought it would be helpful to
anyone struggling with Cauchy sequences to realize that the theory is
just a logically rigorized and thoroughly generalized extension of the
overly simplified special case that we first study in elementary
school. In other words, when struggling with a complicated general
mathematical theory, it helps to realize that one is already familiar
with certain simple special cases. When I teach calculus and get to
convergence of sequences and series, I like to point out that their 7
year old minds seemed to make some sense out of the statement that 1/3
= .33333... and that the calculus theory of convergent sequences is
just a detailed, logically rigorized and generalized version of
something that a young child can intuitively grasp to some extent.
best regards
ADmax
intuitive or not... Everyone can think about it and give his feelings
... But I am only interested to find proofs and complete work on
construction of real numbers by decimal expansions. So I have to
reformulate my question:
I want to see complete proofs about construction of real numbers by
decimal expansions
because this is what I want to see, and dont try to tell me that Cauchy
sequences or Dedekind cuts are better or not, this is not my purpose to
talk about this :)
Thank you all !
William Elliot
> I want to see complete proofs about construction of real numbers by
> decimal expansions because this is what I want to see, and dont try to
> tell me that Cauchy sequences or Dedekind cuts are better or not, this
> is not my purpose to talk about this :)
>
I want it seen done for real numbers in [0,1] and don't tell
me that's not all of the reals. Tell me how you do or don't
distinguish between 1 & 0.999... and others of like ilk. ;-)
ADmax
while 1.0000... is a normal decimal expansion, so my idea was to define
real numbers to be the set of all normal decimal expansion, I mean all
decimal expansions whose end is not 9999999999..
Problem is to define arithmetical operations (+,x) on this set and show
that one gets a field
William Elliot
The sum two normal decimals is abnormal.
Top posting sucks
http://oakroadsystems.com/genl/unice.htm
Timothy Golden BandTechnology.com
I appreciate your persistence.
I haven't heard back from you on the product results:
http://bandtechnology.com/PolySigned/Deformation/P4T3Comparison.html
http://groups.google.com/group/sci.math/msg/85d1d12d26c714c9
For you it's a 'done deal', but perhaps we should both maintain an open
position on this. The images are slightly different size so you might
jump to the conclusion that a scaling factor will resolve the
difference. However, I have tried this and the scaling factor will
reduce the error in one dimension but leave it in another. I have to be
open to a bug in my code. Though I am careful about verification this
one has a lot of steps so the probability of error is higher.
I would really appreciate you staying on with that one since you have
expertise in these things.
Now back to this discussion on the reals. How can it be valid to
construct the real numbers from the real numbers? That is the argument
that you are making above here. The polysigned numbers require that in
P2 for any magnitude x:
- x + x = 0.
This is a sign definition. It is extensible. So in this regard any part
of the real numbers that includes this aspect can be left out of the
basis x. I already identify x as a magnitude and Wikipedia does NOT
require that magnitude come from the real numbers. If I had to define
magnitude it would be something like:
A continuous natural value
It is very fundamental, much more so than the real numbers. It is so
slim that it does not come with operators. These have to be defined on
top of them and the atachment of sign is the way that these operators
manifest themselves. There are two fundamental means of combining
operators. Superposition plays a basic role and a notion of
multiplication is necessary so that things like:
(2.3)(1.1) = 2.53
can be done, where all values are magnitudes. In that multiplication is
supersuperposition even this product can be broken into a counting
algorithm when scaling is implemented. I suppose I could work out some
limit theorem for that but why should we even need to go this far down?
Ditch sign and you have magnitude. Magnitude is a simple concept.
Therefor it need not be defined by a more compicated concept. If modern
mathematics fails in this regard then that is just alll the more
support for the polysign construction.
But Gene you are dodging the direct question on whether or not you
accept:
http://en.wikipedia.org/wiki/Magnitude_%28mathematics%29
which I posed to you on:
http://groups.google.com/group/sci.math/msg/f3f48ff392d77f1b
There is no restriction on magnitude being defined via the reals.
Magnitude can be defined on the reals, but the concept is that
magnitude already exists and the notion is congruent with the absolute
value of the difference of two real values. With a piece of hair in
your hands you can demonstrate distance without the real numbers. It is
a farce to split this hair. But in the interest of making a compelling
and thorough argument I have to go here to address your denial. I deny
that I build polysigned numbers from the reals. You attempt to build
them that way and I refute your interpretation. The pro forma methods
that you work in are not necessary for this construction.
Mathematicians and physicists are minimalists by nature and the usage
of Cartesian product is not necessary for this dimensional
construction. The algebraic form is the best definition. The concepts
are so fundamental as to be at the base of mathematics.
-Tim
ADmax
William Elliot wrote:
> 0.9090909... + 0.090909... = 0.999999...
> The sum two normal decimals is abnormal.
NO!! this is just that your rule for + is not good !!!
0.9090909...+.090909... cannot be 0.999999... !!!
that is all the difficulty to define +,x :)
but thank you for this nice example, you pointed an interesting problem
:)
Dave L. Renfro
> There are many ways to construct the real numbers. Among
> the more common methods are the use of Cauchy sequences,
> Dedekind cuts, decimal expansions, and nested intervals
> with rational endpoints.
In addition to the references I posted yesterday, the following
book seems very worthwhile (for many things, not just the
present topic -- look through its on-line table of contents):
J. K. Truss, "Foundations of Mathematical Analysis", 1997.
http://books.google.com/books?vid=ISBN0198533756
Chapter 6.3 (pp. 132-136): "Comparison of different constructions of R"
(pp. 132-133) from Cauchy sequences and from Dedekind cuts
(pp. 133-134) from decimal expansions
(pp. 134-135) from (topological) compactification
(pp. 135-136) from nested intervals
(pp. 136-137) order-completeness versus sequential completeness
A phrase search for "comparison of different" in this book brings
up p. 132. From there you can advance to pp. 133 & 134.
A phrase search for "nest of intervals" brings up p. 135. From
there you can advance to p. 136.
For much more on this topic, a good place to begin is:
http://books.google.com/books?q=construction-of-real-numbers
http://books.google.com/books?q=construction-of-the-real-numbers
Take notes, then visit a university library . . .
Dave L. Renfro
Virgil
"ADmax" <ad...@numericable.fr> wrote:
But if you accept decimal representataions, you must equally accept
other basal representations, such a binary, trinary, and so on, which
gives you infinitely many rational sequences 'converging' to any given
real.
It is hardly a stretch to consider all rational sequence converging to a
given real,and then you have Cauchy sequences.
ADmax
numbers by decimal expansions!!! :)
ADmax
numbers by decimal expansions!!! :)
Robert Israel
<mvonka...@yahoo.com> wrote:
> When I teach calculus and get to
>convergence of sequences and series, I like to point out that their 7
>year old minds seemed to make some sense out of the statement that 1/3
>= .33333...
... although this newsgroup has seen lots of minds considerably older
than 7 not making sense out of the statement that 1 = .99999...
Robert Israel isr...@math.MyUniversity'sInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Virgil
"ADmax" <ad...@numericable.fr> wrote:
> Maybe, but the topic is to find proof(s) of construction of real
> numbers by decimal expansions!!! :)
One need only show that every equivalence class of Cauchy sequences
contains a sequence of decimal expansions converging to the usual
decimal representation, and all the Cauchy proofs carry over.
Shmuel (Seymour J.) Metz
06/17/2006
at 05:53 AM, ad...@numericable.fr said:
>I have heard that a construction of real numbers by decimal
>expansions is possible, and it seems very intuitive then...
It's possible, it's ugly, it's more work than the other approaches and
it's a lot *less* intuitive than the other approaches once you deal
with all of the nits.
>But the problem is that arithmetical operations (+,x) are difficult
>to define formally, and show that R is a field too...
Whereas that's easy with cuts and with equivalence classes of Cauchy
Sequences.
>PS: I am french, and my english is poor!
Your English seems perfectly adequate.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
Shmuel (Seymour J.) Metz
06/18/2006
at 02:32 AM, mvonka...@yahoo.com said:
>Decimal expansions are a perfectly appropriate way to introduce a
>student to real numbers in grade school.
I disagree. It would be much better to introduce real numbers
geometrically.
William Elliot
> <mvonka...@yahoo.com> wrote:
>
> > When I teach calculus and get to
> >convergence of sequences and series, I like to point out that their 7
> >year old minds seemed to make some sense out of the statement that 1/3
> >= .33333...
>
> ... although this newsgroup has seen lots of minds considerably older
> than 7 not making sense out of the statement that 1 = .99999...
>
ADmax
Shmuel (Seymour J.) Metz wrote:
>
> I disagree. It would be much better to introduce real numbers
> geometrically.
>
What do you mean with geometrically? Do you want to tell your students
that real numbers are X-coordinates of points on a line?
Gene Ward Smith
Why do that? Let AB be a line segment, and extend it out to a ray in
one direction. Then for
C a point on the ray, the ratio AB:AC we may associate with the point
C, and hence equate C to a positive real number. Then extending the ray
in the other direction, if AC' is congruent to AC, C' we equate to -C.
We equate A to zero and B to one, and there's your real number line.