Natural numbers?
May T. Young
Some authors include zero in the set of natural
numbers, and some do not. Does this usage vary over time, by
geography, or between disciplines of math? What are the
differences between including and excluding zero?
Thanks,
M
Allen Knutson
> Some authors include zero in the set of natural
>numbers, and some do not. Does this usage vary over time, by
>geography, or between disciplines of math? What are the
>differences between including and excluding zero?
Over time, more have included zero, I'd say. Between disciplines, yes;
those fields that use N a lot (rather than Z or R) tend to include zero -
particularly set theorists, for example. Geography I dunno.
Differences: either way, there are going to be some theorems that become
more obnoxious to state. Personally I'm a big fan of including 0,
because it's more pleasant to write Z^+ (or if you're feeling silly, N^+)
to exclude zero than N U {0} include it. Allen K.
Gerald Edgar
I find, as a general rule, with some exceptions, that texts in algebra
include 0 and texts in analysis exclude it.
--
Gerald A. Edgar Internet: ed...@math.ohio-state.edu
Department of Mathematics Bitnet: EDGAR@OHSTPY
The Ohio State University telephone: 614-292-0395 (Office)
Columbus, OH 43210 -292-4975 (Math. Dept.) -292-1479 (Dept. Fax)
herb rabinowitz
: Silly undergraduate question:
can you list one author that includes zero in sthe seet of natural numbers?
from what i understand if one tries to make a one to one correspondance
between a set and the n.n. one always starts with one
--
Herb
Jay Kangel
writes:
>i do not believe that zero is a natural number.
>can you list one author that includes zero in sthe seet of natural
>numbers?
The first sentence in Gerald E. Sacks "Degrees Of Unsolvability" [second
edition] is:
"The natural numbers are 0, 1, 2, ....; let f and g be functions from the
natural numbers into the natural numbers."
In section 0 of Joseph R. Shoenfield's "Degrees Of Unsolvability" [VERY
popular title] it states:
"A natural number is a non-negative integer."
Philip Gibbs
The "natural numbers" do not include zero or negative integers.
The "whole numbers" include zero, and of course, the "integers"
include zero and negative integers. Many mathematicians pefer
to say "positive integers", "non-negative integers" etc, to
avoid any missunderstanding.
Rasmus Borup Hansen
dealing with fundamental set-theory. From the ZF (Zernulo-Frankel
(spelling?)) axioms its easy to prove that the empty set exists
and is unique. Usually a natural number is defined to be the set of
its precessors, and since intuitively zero must be identified with the
empty set, we have that
0=\emptyset, 1={0}={\emptyset}, 2={0,1}={\emptyset, {\emptyset}},
3={0,1,2}={\emptyset, {\emptyset}, {\emptyset, {\emptyset}}}
etc. Thus we have that the idea of cardinality can be defined so it
behaves like we want it to. A set A has cardinality n if and only if
there exists a bijective mapping from A to the natural number n.
Some say that God created nothing (i.e. the empty set), man created
the rest...
Traditionally zero is not included in the set of natural numbers if we
are dealing with analysis or some other part of mathematics that is
"high level". The answer to the question must be: It depends on what
you're doing.
--
---------------------------------------------------------------------
| Rasmus Borup Hansen (rbhf...@math.ku.dk) $e^{i\pi}+1=0$|
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|TeXnician; `Hacker' on `The Hacker Test'; Master of dirty TeX-code; |
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|theoretic problems: Fermat's Theorem? That's trivial...; Lover of |
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|---------------------------------------------------------------------|
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|way to do something that the first way that pops into your head. |
| (From The \TeX{}book by Donald E. Knuth (The Almighty))|
|---------------------------------------------------------------------|
|Did you know that |
|$$\int{x{p-1}\,dx\over x^{2m+1}-a^{2m+1}}={-2\over(2m+1)a^{2m-p+1}}\s|
|um_{k=1}^m\sin\left(2kp\pi\over2m+1\right)\arctan\left(x-a\cos\left(2|
|k\pi\over2m+1\right)\over a\sin\left(2k\pi\over2m+1\right)\right)+{1\|
|over(2m+1)a^{2m-p+1}}\sum_{k=1}^m\cos\left(2kp\pi\over2m+1\right)\log|
|\left(x^2-2ax\cos\left(2k\pi\over2m+1\right)+a^2\right)+{\log(x-a)\ov|
|er(2m+1)a^{2m-p+1}},$$ where $0<p\le2m+1$? |
---------------------------------------------------------------------
Jerry Grossman
It more varies over level at which the topic is being discussed. Obviously,
it's an arbitrary decision (although I'll argue in a minute for my
choice), but people like elementary school mathematics educators need to
view it as some big important TRUTH that students can memorize. So the
math ed biz (as Tom Lehrer called it) decreed (probably starting back in
the 1960s) that the natural numbers were {1,2,...}, and if one wanted to
include 0, then one referred to the whole numbers. Maybe this makes
sense pedagogically and historically, since 0 is perhaps a more
difficult concept to understand than 1, 2, 3, ... .
There are many arguments for including 0 as a natural number. The one I
find most persuasive is that if we do, then the natural numbers arise
very very naturally (pun intended) as the cardinalities of finite sets
(since the empty set has cardinality 0). So if you're tabulating votes,
put mine down in the "include it" column.
Philip Gibbs
> Zero is usually included in the set of natural numbers if you are
> dealing with fundamental set-theory. From the ZF (Zernulo-Frankel
> (spelling?)) axioms its easy to prove that the empty set exists
> and is unique. Usually a natural number is defined to be the set of
> its precessors, and since intuitively zero must be identified with the
> empty set, we have that
> 0=\emptyset, 1={0}={\emptyset}, 2={0,1}={\emptyset, {\emptyset}},
> 3={0,1,2}={\emptyset, {\emptyset}, {\emptyset, {\emptyset}}}
> etc.
Are you sure you don't mean the ordinal numbers?
Marc Whitaker
: Silly undergraduate question:
Not so silly. Definitions are important in math (and elsewhere, they
just don't realize it.)
Natural or Counting numbers -> {1, 2, 3, 4, . . . }
Whole numbers -> {0, 1, 2, 3, 4, . . . }
Integers -> { . . . -3, -2, -1, 0, 1, 2, 3, . . . }
Rational numbers -> I expect you know all this.
Zero is an important number. Things are true for all natural numbers
that are not true of all whole numbers because they are not true for zero.
When you have to do proofs in Modern Algebra or Advanced Calculus, you
will find most of your errors in definitions.
Jan Willem Nienhuys
#May T. Young (myo...@farad.elee.calpoly.edu) wrote:
#
#: Silly undergraduate question:
#: Some authors include zero in the set of natural
#: numbers, and some do not. Does this usage vary over time, by
#: geography, or between disciplines of math? What are the
#: differences between including and excluding zero?
#
#Not so silly. Definitions are important in math (and elsewhere, they
#just don't realize it.)
#
#Natural or Counting numbers -> {1, 2, 3, 4, . . . }
#Whole numbers -> {0, 1, 2, 3, 4, . . . }
#Integers -> { . . . -3, -2, -1, 0, 1, 2, 3, . . . }
#Rational numbers -> I expect you know all this.
#
This is typical Anglo-Saxon: "Whole number" is just the same
as "Integers", except the latter is Latin-derived. I guess if
you try to translate it into any other language, you wouldn't be
able to find two corresponding words (example: Ganze Zahlen in German,
Entiers in French, en Gehele Getallen in Dutch).
N.G. de Bruijn has suggested to use the self-explanatory Zeropositives
for 0, 1, 2, etc. His suggestion was not taken up. Too many
associations with something else, I guess.
JWN
Milo Gardner
First, the history of zero is not well reported --- in that Islamic
scholars are commonly given 100% credit for diffusing a Hindu zero
symbol -- as if Hindu scholars were first to develop zero ---
Note that Egyptian Middle Kingdom accountants used nfr --- a
hieroglypth --- as well as the RMP and Plimpton Tablet using a
space --- in 1650 BC and 1800 BC respectively. In this sense Greeks
that did not accept zero -- as I'll show Aztecs did not use a Mayan
zero -- are given political cover by those that exclude zero from
various fundamental sets of numbers.
As noted, Mayans were provided a zero by the Olmecs probably as
early as 1,000 BC in a positional number system -- a feature that
was marked counting-wise within base 20 as 0, 1, ..., 19. Yet, for
politcal or pedagogical reasons Aztecs later on, after 1200 AD
decided to begin counting 1, 2, 3, ... , 20 and generally omit
zero from early childhood education ---
So from a historical sense, for Greek to not accept Egyptians
ideas on zero, Aztecs to not accept Mayan/Olmecs ideas of zero
and modern exclusion of zero --- early on in our "modern" schools
may be an issue that needs to be re-visited--- rather than allow
teachers to decide the elements of numbers, numeracy and arithmetic
that are easy for them to present. Children should deserve the
best --- show Babylonian, Egyptian, Olmec and Mayan zero early
on in oru schools --- and reduce the European stress on Greeks --
and plane geometry --- right?
--
: Fidonet: Milo Gardner 1:203/289 .. speaking for only myself.
: Internet: Milo.G...@ubik.wmeonlin.sacbbx.com
herb rabinowitz
: May T. Young (myo...@farad.elee.calpoly.edu) wrote:
: : Silly undergraduate question:
: : Some authors include zero in the set of natural
: : numbers, and some do not. Does this usage vary over time, by
: : geography, or between disciplines of math? What are the
: : differences between including and excluding zero?
I must be getting old because i have heard many responses to the above
question.. when i asked for a reference to the ans. 0 is a n.n. i got a reply
many books state that 0 is a n.n. (new type of reference...many)
however when checking the foundations of the real number system starting
with Peano's axioms i seemes to have read
1. 1 is a nn.
2. 1 is not the succesor of a nn
3. every nn has a successor
4. the induction principle.
if one assumes his axioms can those who say 0 is a nn tell me it's
successor.
herb@ dorsai.dorsai.org
: Not so silly. Definitions are important in math (and elsewhere, they
: just don't realize it.)
: Natural or Counting numbers -> {1, 2, 3, 4, . . . }
: Whole numbers -> {0, 1, 2, 3, 4, . . . }
: Integers -> { . . . -3, -2, -1, 0, 1, 2, 3, . . . }
: Rational numbers -> I expect you know all this.
: Zero is an important number. Things are true for all natural numbers
: that are not true of all whole numbers because they are not true for zero.
: When you have to do proofs in Modern Algebra or Advanced Calculus, you
: will find most of your errors in definitions.
: ma...@crash.cts.com
--
Herb
Jay Kangel
writes:
>I must be getting old because i have heard many responses to the above
>question.. when i asked for a reference to the ans. 0 is a n.n. i got a
reply
>many books state that 0 is a n.n. (new type of reference...many)
>however when checking the foundations of the real number system >starting
>with Peano's axioms i seemes to have read
>1. 1 is a nn.
>2. 1 is not the succesor of a nn
>3. every nn has a successor
>4. the induction principle.
>if one assumes his axioms can those who say 0 is a nn tell me it's
>successor.
An author who states 0 is a natural number will likely have a slightly
different set of Peano axioms.
1'. 0 is a nn
2'. 0 is not the succesor of a nn
3'. every nn has a successor
4'. the induction principle.
Someone mentioned that there are theorems which are true for the
(positive) natural numbers but are not true if 0 were included in the
natural numbers. Consider the theorem by someone who uses 1 as the least
natural number. "Every natural number is greater than zero." Someone who
considers zero a natural number might restate this theorem as "Every
nonzero natural number is greater than zero."
This is something that can vary from author to author. To discover what a
particular author is thinking read the preliminary or introductory
sections.
franklin
Allen Knutson (all...@frappe.ugcs.caltech.edu) wrote:
: Personally I'm a big fan of including 0,
: because it's more pleasant to write Z^+ (or if you're feeling silly, N^+)
: to exclude zero than N U {0} include it. Allen K.
It's funny you say this, because after exposure to many processors where
BPL (Branch if plus) *includes* zero, I always considered that 0 \in Z^+,
and so it makes more sense to exclude 0 from N. (exactly same reasoning,
different definition of Z^+. I used this notation throughout 4 years at
Cambridge, and only a couple of supervisors ever objected).
Personally, I figure that if I count {1,2,3,4,5}, the number I end up with
should be the number of items in the set. None of these "I have 0,1,2,3,4
suitcases! Arghh! I had 5 suitcases at the start of the journey!" jokes
typical of set theorists. (I've always wondered about this - that joke
is one I've heard told by set theorists of set theorists, so it appears
that 0 \in N is confusing to them as well as mortals. But they still claim
it's "natural")
Dave
Helmut Richter
>In article <Cstww...@dorsai.org>, he...@dorsai.org (herb rabinowitz)
>writes:
>>with Peano's axioms i seemes to have read
>>1. 1 is a nn.
>>2. 1 is not the succesor of a nn
>>3. every nn has a successor
>>4. the induction principle.
>>if one assumes his axioms can those who say 0 is a nn tell me it's
>>successor.
>An author who states 0 is a natural number will likely have a slightly
>different set of Peano axioms.
>1'. 0 is a nn
>2'. 0 is not the succesor of a nn
>3'. every nn has a successor
>4'. the induction principle.
If n.n. are defined by the Peano axioms, we have to ask first why the
Peano axioms are sufficient to define n.n. They are just the axioms
of an algebraic theory with an 0-ary function (i.e. constant) "unique
constant that is no successor" and an unary function "successor".
Such axioms are obviously not sufficient as a definition (e.g. one
could not define any particular set by the group axioms). The Peano
axioms do, however, define somethings because, fortunately, all
structures satisfying the axioms are isomorphic. In particlar, the two
structures "n.n. starting with 0" and "n.n. starting with 1" are also
isomorphic. As the Peamo axioms do not define more than an "isomorphy
class", the two definitions are actually the same; their difference
being beyond the scope of the Peano axioms.
William Boshuck
> I must be getting old because i have heard many responses to the above
>question.. when i asked for a reference to the ans. 0 is a n.n. i got a reply
>many books state that 0 is a n.n. (new type of reference...many)
Here's a few "old fashioned" references:
C. Chang & H. J. Keisler's _Model Theory_,
H. Enderton's _A Mathematical Introduction to Logic_,
N. Jacobson's _Basic Algebra I_,
S. Kleene's _Introduction to Metamathematics_,
K. Kunen's _Set Theory_ (I bet that T. Jech does too, but I don't
have the book at hand),
S. Mac Lane & G. Birkoff's _Algebra_ (and _Survey of Modern Algebra_),
J. Dugundji's topology book talks of ordinals instead and, of course,
includes 0; if I remember correctly, S. Willard's topology book
includes 0 as well. Also, I. Kaplansky is a little famous for
induction proofs which begin with 0, and have the advantage that the
base step is utterly trivial (I have this second hand, but if my
memory serves me, this is the style of _Set Theory and Metric
Spaces_).
Of the few (zero-less) exceptions I ran across: L. Gillman & M.
Jerison's _Rings of Continuous Functions_, Boolos & Jeffery's
_Computability and Logic_, and T. Apostol's _Mathematical Analysis_,
the latter two references use the term "positive integer" and not
natural number. I think you could have gone to the math section of a
library, picked at random and found that the usual convention is
that the natural numbers are 0, 1, 2, ... (and that the term positive
integer is used otherwise).
[snip]
>if one assumes his axioms can those who say 0 is a nn tell me it's
>successor.
I *really* hope that this is rhetorical ...
cheers,
Bill
Daniel Grubb
>different set of Peano axioms.
>1'. 0 is a nn
>2'. 0 is not the succesor of a nn
>3'. every nn has a successor
>4'. the induction principle.
I've actually been partial to including 0 as a natural number because
of the way the natural numbers are *defined*, i.e. as finite ordinals.
Thus, an inductive set A, is one such that
a) \emptyset is in A
b) if x is in A, then (x union {x}) is in A.
The set of naturals is defined to be the smallest inductive set.
Then 0 is identified with the empty set and the successor function
is the map x to (x union {x}). I.e. 0=empty set , 1={emptyset}={0},
2={emptyset,{emptyset}}={0,1}, etc.
The amended Peano axioms then hold. See a good book on set theory.
Chris Thompson
writes:
|> In article <CssIo...@crash.cts.com> ma...@crash.cts.com (Marc Whitaker) writes:
|> #Not so silly. Definitions are important in math (and elsewhere, they
|> #just don't realize it.)
|> #
|> #Natural or Counting numbers -> {1, 2, 3, 4, . . . }
|> #Whole numbers -> {0, 1, 2, 3, 4, . . . }
|> #Integers -> { . . . -3, -2, -1, 0, 1, 2, 3, . . . }
|> #Rational numbers -> I expect you know all this.
|> #
|>
|> This is typical Anglo-Saxon:
I object strongly to that! I am sure that we do _not_ use the terms "whole
numbers" and "integers" differently in the way suggested, and some of us,
at least, would go into etymological hysteria if it was suggested that
we should.
Dare I suggest that it might be an Americanism?
|> "Whole number" is just the same
|> as "Integers", except the latter is Latin-derived. I guess if
|> you try to translate it into any other language, you wouldn't be
|> able to find two corresponding words (example: Ganze Zahlen in German,
|> Entiers in French, en Gehele Getallen in Dutch).
Quite.
Chris Thompson
Cambridge [England] University Computing Service
Internet: ce...@phx.cam.ac.uk
JANET: ce...@uk.ac.cam.phx
Gerald Edgar
> I think you could have gone to the math section of a
>library, picked at random and found that the usual convention is
>that the natural numbers are 0, 1, 2, ...
Only if you pick from the algebra (& logic) end of the shelf.
From the analysis end, most books would not include 0. For example:
Royden, REAL ANALYSIS, page 6: "Natural numbers (positive integers) play..."
Ross, ELEMENTARY ANALYSIS, p 6: "the set {1,2,3,...} of natural numbers"
Courant & John, INTRODUCTION TO CALCULUS AND ANALYSIS, p 1: "The positive
integers or natural numbers 1, 2, 3, ..."
[and, of course, Gillman & Jerison as noted by H. Rabinowitz.]
P. Fritz Cronheim
Since only a handful of students are ever going to become graduate math majors,
a lot of these distinctions are unimportant. (N includes zero! no it
doesn't!). Frankly, it depends, as does all math, on the initial assumptions
and what you are setting out to do. If educationists would just agree to
call {1, 2, 3, ...} the "counting numbers," the problem might go away (but
I doubt it). The artificial distinction between "natural" and "whole" numbers
gets confusing since "Integer" means "whole." Why isn't -1 whole? Frankly,
it's a heck of a lot more important to get kids to understand how to work
with numbers, and logical concepts, IMHO.
re this: Milo.G...@ubik.wmeonlin.sacbbx.com (Milo Gardner) writes:
<<Children should deserve the
best --- show Babylonian, Egyptian, Olmec and Mayan zero early
on in our schools --- and reduce the European stress on Greeks --
and plane geometry --- right?>>
It all depends. They've gotta learn one number notational system first.
And they get zero by grade one, don't they? (I dunno...)
And it's a blast to explore all the other notational systems, too.
("see why we don't want to do long division in Roman Numerals?" "How
many ways can you express 2/3 as the sum of unit fractions, as the
Egyptions did? Besides 1/3 + 1/3, I mean?")
But reducing the stress on "Greeks and plane geometry" is throwing the
baby out with the bath. From what I hear, almost no geometry at all
is taught before grade seven anymore in the US. That's a travesty.
Kids reach grade ten geometry without knowing the formula for the area
of a rectangle, or having ever measured angles with a protractor.
I speak from teaching experience on this. By all means, let's include
units on history of math: Chinese methods for predicting eclipses by
modular arithmetic, Egyptian surveying, Mayan calendars, Arabic foundations
of Algebra (which was *built* on Greek math, which the European governments
(churches) of the time had seriously "reduced the stress on!), and Hindu
mathematics are all valuable and should not be ignored. But don't reduce
the stress on geometry (and algebra!) more than it already is. We've got
third semester calculus students here at UF who moan when asked to add
1/2 and 1/3 by taking a common denominator!
my 2 (1/2 +1/3 +1/6 +1/4 +1/5 +1/10 +1/20 +...) cents
P. Fritz Cronheim
P. Fritz Cronheim
<< I must be getting old because i have heard many responses to the above
question.. when i asked for a reference to the ans. 0 is a n.n. i got a reply
many books state that 0 is a n.n. (new type of reference...many)
however when checking the foundations of the real number system starting
with Peano's axioms i seemes to have read
1. 1 is a nn.
2. 1 is not the succesor of a nn
3. every nn has a successor
4. the induction principle.>> (sound of Ludwig Plutonium spinning dishes!)
Don't you kinda miss Ludwig?
pfc
H. Jurjus
(stuff deleted)
> I must be getting old because i have heard many responses to the above
> question.. when i asked for a reference to the ans. 0 is a n.n. i got a reply
> many books state that 0 is a n.n. (new type of reference...many)
> however when checking the foundations of the real number system starting
> with Peano's axioms i seemes to have read
> 1. 1 is a nn.
> 2. 1 is not the succesor of a nn
> 3. every nn has a successor
> 4. the induction principle.
>
> if one assumes his axioms can those who say 0 is a nn tell me it's
> successor.
>
> herb@ dorsai.dorsai.org
1. 0 is a nn.
2. 0 is not the successor of any nn
3. every nn has a unique successor
4. two different nn's cannot have the same successor
5. induction
But ofcourse I'm not half as old as you ... :-)
H.Jurjus
H. Jurjus
> --
> Herb
>
there's a reaction of mine with title something like N:\mail\.....\nn.txt
I don't know what went wrong, I'm sorry. The title should have been
Re: Natural numbers?
H.Jurjus
Dara Gallagher
> If n.n. are defined by the Peano axioms, we have to ask first why the
> Peano axioms are sufficient to define n.n. They are just the axioms
> of an algebraic theory with an 0-ary function (i.e. constant) "unique
> constant that is no successor" and an unary function "successor".
> Such axioms are obviously not sufficient as a definition (e.g. one
> could not define any particular set by the group axioms). The Peano
> axioms do, however, define somethings because, fortunately, all
> structures satisfying the axioms are isomorphic. In particlar, the two
> structures "n.n. starting with 0" and "n.n. starting with 1" are also
> isomorphic. As the Peamo axioms do not define more than an "isomorphy
> class", the two definitions are actually the same; their difference
> being beyond the scope of the Peano axioms.
0+0=0 but 1+1!=1 ?
Dara.
Sebastian Goette
: 0+0=0 but 1+1!=1 ?
: Dara.
The Peano axioms do not define addition, you have to do that yourself.
Actually, define i o j := i + j - 1, then 1 o 1 = 1, and any isomorphism
H: N u {0} -> N \ {0} will sent '+' to 'o', i.e.: H(i+j) = H(i) o H(j).
Sebastian
Dave Joyce
>And they get zero by grade one, don't they? (I dunno...)
It's interesting that Sesame Street, a PBS TV show for preschoolers, counts up
starting with one, but counts down ending with zero.
--
David E. Joyce Dept. Math. & Comp. Sci.
Internet: djo...@black.clarku.edu Clark University
BITnet: djoyce@clarku Worcester, MA 01610-1477
H. Jurjus
> Surely, at its heart, this entire thread is a matter of semantics
> (albeit fun (:-)). The Greeks, were unable to recognize the
> 'natural' significance of 0. (etc etc)
I was taught that the Greeks started counting with ...2, and 1 was
suspect, as 0 is in this thread (it is ridiculous to count a set with only
1 element). Does anyone have more information on that ?
H.Jurjus
David Kierstead
(albeit fun (:-)). The Greeks, were unable to recognize the
have modern logicians. Like the Supreme Court, we should strive to
abide by the precedent, but should be prepared to recognize the
mistakes of the past. Zero is as natural as One.
But, before the 'old' natural numbers disappear entirely, let me
quote from the preeminent historian of Mathematics (E. T. Bell,
_Development of Mathematics_), referring to the most recent crisis
in Mathematics:
The focus of the last serious trouble was found most
unexpectedly in the speciously innocuous natural numbers
1, 2, 3, ... that, since the days of Pythagoras, had been
eagerly accepted by mathematics as manna from heaven.
Indeed L. Kronecker (1823-1891, German), himself a
confirmed Pythagorean and one of the leading algebraists
and arithmeticians of the nineteenth century, confidently
asserted that "God made the integers; all the rest is the
work of man." By 1910, some of the more wary
mathematicians were inclined to regard the natural
numbers as the most effective net ever invented by the
devil to snare unsuspecting men. Others, of a yet more
mystical sect, maintained that the natural numbers have
nothing supernatural of either kind about them, asserting
that the 'unending sequence' 1, 2, 3, ... is the one
trustworthy 'intuition' vouchsafed to Rouseau's natural
man. The tribes of the Amazon Basin were not consulted.
The tribes of the Amazon Basin would certainly have insisted that Zero
be included [:-).
--David Kierstead (whose opinions aren't anyone's)
Herbert Xu ~{PmV>HI~}
: 1. 1 is a nn.
: 2. 1 is not the succesor of a nn
: 3. every nn has a successor
: 4. the induction principle.
Number systems satisfy this axiom system is not necessarily the natural
number system, e.g., "1, 2, 2, 2, 2, ...".
--
A. B <=> True B. A <=> False
Email: <her...@greathan.apana.org.au>
PGP Key: finger her...@sleeper.apana.org.au
Milo Gardner
level students should not complain about finding the LCM of 1/2 and 1/3.
That is a sad situations.
Fritz's other points -- such as I've thrown the baby out with the
bath water -- when mentioning the possibilty that Greek counting from
1, 2, 3 ... and thus omitting zero --- could be considered to be
replaced ASAP --- and its plane geometry equivalents is strange ---
given that by the 4th grade 75% of USA students fail international
math standards -- when Asian countries suffer only a 10% failure rate.
What price must our children pay to be introduced into European
forms of mathematics -- such as fingers and toes numeracy -- as
proposed by Piaget, a non-mathematician? Children should be exposed
to numeracy issues early on --- and even geometry, algebra and
higher forms of arithmetic --- equally!
It is sad when I see analysis and calculus instructors choosing only
the pythagorean versions of geometry rather than equally valid and
interesting algebra and arithmetic representations. Yes, early
childhood eductors should be exposed to to the equivalencies of
arithmetic, algebra and geometry -- based on number being an IDEA --
and not the mnemonic of 1, 2, 3 , ..., of counting numbers.
Mnemonics and other manipulative like abaci should be shown as
examples of the idea being taught --- and not the idea itself ---
is that one of the central issues that elementary mathematical
educators must face --- to attempt to achieve America 2000?
Milo Gardner
"the history of mathematics does speak -- and it is greatly no
not European in its origins"
Milo Gardner
hybrid -- but I agree with the earlier post that mentioned Pythagoras --
those that do not begin with zero are following mystical Greeks -- and
introducing things that are difficult to unlearn later --- a sad
conidition 75% of our kids at the 4th grade seem not to achieve.
herb rabinowitz
seem to be from analysis and not foundations of arithmetic. it seems that
if one defines the rationals as an extension of the nn by the following
definitions
1. (a,b)+(c,d)=(ad+bc,db)
2. (a,b)*(c,d)=(ac,bd)
the developement of arithmetic could very well be extenmded to
the rationals.
there is no doubt that if one wants completeness, to discuss the cardinality
of the null set etc..we must include 0.
did not mathematics develop from the concrete to the abstract and then
back to the concrete, looking for new models and interpretations.
never had a need for a 0 when counting, adding,multiplying .
one must again ask not from the concept of analysis but from the concept
in the developement of math what was the original concept of a nn.
Gerald Edgar (ed...@math.ohio-state.edu) wrote:
: > I think you could have gone to the math section of a
--
Herb
IMRE BOKOR
: > I think you could have gone to the math section of a
Wolfgang Walter, Analysis I, p17 includes 0 in the natural numbers.
Christian Blatter, Analysis I p.32 includes 0 in the natural numbers.
d.A.
Jan Stevens
: In article <CssIo...@crash.cts.com> ma...@crash.cts.com (Marc Whitaker) writes:
: #May T. Young (myo...@farad.elee.calpoly.edu) wrote:
: #
: #: Silly undergraduate question:
: #: Some authors include zero in the set of natural
: #: numbers, and some do not. Does this usage vary over time, by
: #: geography, or between disciplines of math? What are the
: #: differences between including and excluding zero?
: #
: N.G. de Bruijn has suggested to use the self-explanatory Zeropositives
: for 0, 1, 2, etc. His suggestion was not taken up. Too many
: associations with something else, I guess.
: JWN
According to DIN 1302 and ISO 31-11 the natural numbers (N) include
zero.
Jan.
--
E-mail: ste...@math.uni-hamburg.de
Mathematisches Seminar, Universitaet Hamburg
Bundesstrasse 55
D 20146 HAMBURG, Germany
May T. Young
Jan Stevens <ms3...@math.uni-hamburg.de> wrote:
>
>According to DIN 1302 and ISO 31-11 the natural numbers (N) include
>zero.
The return of the silly, undergraduate question:
What are DIN 1302 and ISO 31-11? Are they available on the Internet?