Why can't 1/0 be defined???
sch...@primenet.com
A silly question really ... someone please refresh my memory as to why 1/0
has not been defined and given properties.
Let me present my case for this strange thing to do. Many years ago, when
mathematicians didn't have a defination for negative numbers they left
equations of the form: 1 + x = 0, unsolvable. They said that there is no
number that we know of that will solve that equation. It almost seems
silly today.
Later on, when mathematicians got an incling of what negative numbers were
and became comfortable with them they could solve all sorts of equations
that wern't solvable before. Then came the dredded equation:
1 + x^2 = 0
and everyone flayled their arms and said: there is no number that can
solve this equation! Then a smart guy named Euler came along and said let
us start working with numbers of the form:
__
/-1 and we shall call it "i" and every one rejoyced at the fact that they
could solve all sorts of new equations, (indeed all polynomial equations).
Still it seems silly that at one time in history there wasn't a concept of
complex numbers. What would we do without complex numbers today?
Now what about the equation:
0 x = 1 (that 0 x is 0 times the variable x)
and define its solution as:
u = 1/0 (u for undefined)
Now we can cite all sorts of new solutions to equations that we threw our
hands up in the air about before, namely:
0 x = 1 Solution: x = u
0 x = -1 Solution: x = -u
0 x = a Solution: x = au (a is a real constant<>0)
1 / x = 0 Solution: x = u
tan x = +/- pi/2, +/- 3pi/2, +/- 5pi/2, etc ... Solution: x = u
_
0 x^2 = 1 Solution: x = /u
0 x^p = 1 Solution: x = u^(1/p) (p is a real constant<>0)
0 x^p = a Solution: x = (au)^(1/p)(p,a are real <> 0)
etc...
The only thing that I am lacking is some concrete properties of u, namely
any linear properties:
a(u) = au = ua (a real)
au + bu = u(a + b) = (a + b)u (a, b real)
and you could relate u to the real number system by the identity:
0u = u0 = 1
obviously if you add:
a + bu you won't get a simpler answer (just like complex numbers).
could you do things like:
(a + bu) * (c + du) = (a + c) + (ad + bc)u + (bd)u^2
what do we know about u? We know that:
lim (x->0+) 1/x = oo
lim (x->0-) 1/x = -oo
which describe te values of many functions, also known as asymptotes, or
poles. Why wouldn't it be convienient to define a number with these
properties? I don't know what it could be used for, but I am sure it
would have some kind of use.
Comments anyone? Flaws in logic anyone???? No this is not a joke, and
mose definately NOT a homework assignment.
Any ideas would be appreciated!
Thanks,
Chris
Andreas Neubacher
In article <587rr9$7...@nnrp1.news.primenet.com>,
sch...@primenet.com writes:
SPC> A silly question really ... someone please refresh my memory as to why 1/0
SPC> has not been defined and given properties.
[...]
SPC> Any ideas would be appreciated!
You may want to look into extensions of the real numbers by infinity. This
may give you the properties you need.
The question of "why has ... not been defined" is also related to "what do
we need ... for". One might introduce new axioms definign the meaning of
1/0. But these axioms should in general be somehow related to real life
and/or be motivated by (fit well together with) other axioms and theorems.
So the answer to your question concerning 1/0 may be that there it doesn't
fit into existing theories and/or that it has no real meaning.
As I said, extensions by infinity come to my mind as being the closest to
what you are thinking of.
Andreas.
PS: Followup-to set to "sci.math", since I believe "sci.math.symbolic"
isn't exactly dedicated to discussing this kind of matters.
-----------------------------------------------------------------------
Andreas NEUBACHER <aneu...@risc.uni-linz.ac.at> | `finger' for PGP
http://www.risc.uni-linz.ac.at/people/aneubach.html | and other info
-----------------------------------------------------------------------
Random selection from my quote file:
When people have to make decisions under conditions which overload
human thinking capabilities, they don't give up, saying the problem
is NP-complete. They use strategies and tactics of *optimal-least-
computation search* and not those of *optimal-shortest-path search*.
-----------------------------------------------------------------------
Erland Gadde
> Now what about the equation:
>
> 0 x = 1 (that 0 x is 0 times the variable x)
>
> and define its solution as:
>
> u = 1/0 (u for undefined)
>
>
> Now we can cite all sorts of new solutions to equations that we threw our
> hands up in the air about before, namely:
>
> 0 x = 1 Solution: x = u
>
> 0 x = -1 Solution: x = -u
>
> 0 x = a Solution: x = au (a is a real constant<>0)
>
> 1 / x = 0 Solution: x = u
This will lead to some severe algebraic difficulties. Usually, when we extend
number systems (from N to Z to Q to R to C), we demand that the extended
system obeys the same algebraic laws as the old one.
For example, if the distributive law: (x+y)z=xz+yz, is to hold in your
system, then
2=1+1=0u+0u=(0+0)u=0u=1, that is: 2=1!
In order to avoid this contradiction, you must give up the distributive law.
Most mathematicians feel that that's a too high price to pay.
Regards,
Erland Gadde
Le Compte de Beaudrap
On 5 Dec 1996 sch...@primenet.com wrote:
<snip>
> Now what about the equation:
>
> 0 x = 1 (that 0 x is 0 times the variable x)
>
> and define its solution as:
>
> u = 1/0 (u for undefined)
>
<snip, snip>
> which describe te values of many functions, also known as asymptotes, or
> poles. Why wouldn't it be convienient to define a number with these
> properties? I don't know what it could be used for, but I am sure it
> would have some kind of use.
>
> Comments anyone? Flaws in logic anyone???? No this is not a joke, and
> mose definately NOT a homework assignment.
>
> Any ideas would be appreciated!
<snipitty doo dah>
I'm not entirely sure why 1/0 hasn't been defined. I think it
might be due to the following property...
Let us examine a vertical line on a cartesian co-ordinate set, say
[x = 0]. Let m be the slope of this line.
Now, as the line "travels" up one unit in zero units forward, is
has a slope m = 1/0. Okay. However, it can also be said to travel two
units up in zero units forward, pi units up in zero units forward,
seventeen units DOWN in zero units forward... you get the drift. In fact,
if m = 1/0, it is also -1/0, 2/0, or any n/0 where n is a non-zero real.
(n=0 would, of course, open up a whole other can of worms.) In fact, by
looking at compex numbers, we could say that n = 1/0 = z/0, where z is
a non-zero complex. Not knowing about quaternions, I postulate this extends
to them too (but as I said, this is a wild postulate).
My position on this is... well, so what? So what if we have a number
1/0 which, multiplied by any nonzero complex n, yields itself. We already
know of another such number (ie, 0): this is not unprecedented. It merely
happens to be the case that this other is infinitely large.
So why hasn't somebody defined 1/0? Beats me. It almost seems
natural to do so in some ways. So, I will take this occasion to "define"
1/0, and represent it by the capital letter G, short for the french word
/grand/, which translated to english is the ultimate understatement: "big".
If it turns out that making such a definition causes inconsistancy, well,
we simply do the same thing as was done for the physical theory of
spherical matter: have a good laugh, say "Oh Well" and pull G out of our
mathematical dictionaries except as a bad turn in the history of mathematics.
As you might have guessed by now, I've done some work on this in
my spare time. Nothing much. Just enough to work out how "G-theory"
affects the number-plane (again, only complexes, no quaternions), how it
helps with cases such as f(x)=1/x, where it approaches -oo on one side,
and +oo on the other side, of zero: a lot about the difinition of G helps
make a lot of "inelgant" case look astonishingly elegant. Probably, as
"i" helped turn x^2 + 1 = 0 from a monster to an ordinary equation.
As for the validity of this kind of definition: beats me, Chris.
I see nothing different between what I just did and what Euler did
(I've seen this kind of thing happen here before, people, please: I'm
comparing technique, not myself and Euler personally. What I just did,
however, is probably more of an abridged version). Perhaps someone
else reading this board has walked this path before, too. As I said, it's
surprising that nobody defined it before -- perhaps, it's for good reasons.
Niel de Beaudrap
----------------------
j...@cpsc.ucalgary.ca
Le Compte de Beaudrap
On 6 Dec 1996, Andreas Neubacher wrote:
> In article <587rr9$7...@nnrp1.news.primenet.com>,
> sch...@primenet.com writes:
> SPC> A silly question really ... someone please refresh my memory as to why 1/0
> SPC> has not been defined and given properties.
> [...]
> SPC> Any ideas would be appreciated!
>
> You may want to look into extensions of the real numbers by infinity. This
> may give you the properties you need.
>
> The question of "why has ... not been defined" is also related to "what do
> we need ... for". One might introduce new axioms definign the meaning of
> 1/0. But these axioms should in general be somehow related to real life
> and/or be motivated by (fit well together with) other axioms and theorems.
> So the answer to your question concerning 1/0 may be that there it doesn't
> fit into existing theories and/or that it has no real meaning.
Here, I disagree. Do you know of any practical purpose
imaginary/complex numbers would have had during Euler's time? I can't.
In fact, I've heard on this board many times (and I personally agree)
that a mathematical concept will usually precede the need for it. That's
what's nice about mathematics: when you need it for something, it usually
already has what you need it for.
The value 1/0 is not necessarily useful for anything at all. But let
us use the pretense, "What if..." After all, that's what math is all about,
when you get down to it: mathematics sutudies not reality, but all possible
realities.
Le Compte de Beaudrap
On 6 Dec 1996, Erland Gadde wrote:
> In article <587rr9$7...@nnrp1.news.primenet.com>, sch...@primenet.com wrote:
>
> > Now what about the equation:
> >
> > 0 x = 1 (that 0 x is 0 times the variable x)
> >
> > and define its solution as:
> >
> > u = 1/0 (u for undefined)
> >
> >
> > Now we can cite all sorts of new solutions to equations that we threw our
> > hands up in the air about before, namely:
> >
> > 0 x = 1 Solution: x = u
> >
> > 0 x = -1 Solution: x = -u
> >
> > 0 x = a Solution: x = au (a is a real constant<>0)
> >
> > 1 / x = 0 Solution: x = u
>
> This will lead to some severe algebraic difficulties. Usually, when we extend
> number systems (from N to Z to Q to R to C), we demand that the extended
> system obeys the same algebraic laws as the old one.
> For example, if the distributive law: (x+y)z=xz+yz, is to hold in your
> system, then
>
> 2=1+1=0u+0u=(0+0)u=0u=1, that is: 2=1!
>
> In order to avoid this contradiction, you must give up the distributive law.
> Most mathematicians feel that that's a too high price to pay.
>
But this uses the definition that "0u" (or, as I wrote it in a
different post, "0G") is equal to 1. Because 1/0 = n/0 for any nonzero
complex "n" (refer to my earlier post), one cannot say that 0G = 1. In
fact, OG can come out to any number, because 0G = 0n / 0 = 0/0.
So, the only reason why the distributive law fails in this case
is because in multiplying G by 0, we get something equivalent to 0/0.
While 1/0 is defined (in this context), 0/0 IS NOT, and probably cannot
be. If anyone can prove me wrong, though, I'm listening.
Joe V
try doing it using hexadecimals, Have fun!
Tom Potter
In <Pine.SUN.3.91.961206131859.12031E-100000@fsf> Le Compte de Beaudrap
<j...@cpsc.ucalgary.ca> writes:
I suggest that there is only one reality,
per person, per unit time.
reality * now
------------- = 1
person
I have a few articles discussing the relationsip
of zero and physical reality at my Web site.
I suggest that a new number system, similar to
surreal numbers would best describe reality.
Please take a look and give me some feedback.
Tom Potter http://pobox.com/~tdp
Miguel Lerma
Le Compte de Beaudrap (j...@cpsc.ucalgary.ca) wrote:
[...]
> Here, I disagree. Do you know of any practical purpose
> imaginary/complex numbers would have had during Euler's time? I can't.
> In fact, I've heard on this board many times (and I personally agree)
> that a mathematical concept will usually precede the need for it. That's
> what's nice about mathematics: when you need it for something, it usually
> already has what you need it for.
Their first use was to solve third degree algebraic equations
with real roots. The amusing thing was that the _real_ roots
came out by using a general formula that yielded square roots
of negative numbers in the intermediate computations, but at
the end they canceled each other and the final result was real.
That was a good reason to try to make sanse of such "imaginary"
or "impossible" numbers: they proved to be useful for computing
"real" solucions.
Miguel A. Lerma
Miguel Lerma
sch...@primenet.com wrote:
> A silly question really ... someone please refresh my memory as to why 1/0
You can "define" 1/0 = infinity and give it some properties compatible
with your needs, such as computing limits, I have seen this done
before. It is rather trivial and does not deserve writing and
entire treatise on it. However you must be aware that the set
of real numbers extended with infinity it is not a "field" any
more, you need to give up some of its algebraic properties.
Also, in complex analysis a meromorphic function is the quotient
of two analytic functions f(z) = p(z)/q(z). Wherever the denominator
vanishes (with a zero of higher order than the numerator) the
function is said to have a "pole". At those points the quotient
has the form "a/0". In the theory of meromorphic functions it
is usual to asign "infinity" as the value of the function at
its poles. So, a meromorphic function takes values in the set
of complex number extended with infinity - which is called the
Riemann sphere. Of course, the Riemann sphere lacks of the
interesting algebraic properties of C (it is not a field),
but has other remarkable mathematical properties that make
of it another truly interesting object.
Miguel A. Lerma
Richard J. Fateman
Below, is the title and abstract of a paper that discusses this issue
at some length, and in a computational context. The paper discusses
the difference between affine and projective infinities, and how a
(common lisp) package implementing it works. It discusses the utility
of such a package for computing with intervals.
{{The people on the program committee of the 1994 ISSAC conference
decided either that it was not interesting enough to publish,
or that I already had enough papers accepted for that conference.
In any case it was rejected.
Perhaps it is not that 1/0 is unknown. Quite the contrary, it is
apparently too well known.}}
You are welcome to read the full text via
http://http.cs.berkeley.edu/~fateman/papers/extrat.ps
............
Computation with the Extended Rational Numbers
\author{ Richard J. Fateman \\
Tak W. Yan\\
University of California, Berkeley}
There are several reasons to change even so basic a component
of a computer algebra system as its underlying number system.
We explain why and how changes to the rational number system may
be useful, especially with respect to support of intervals.
--
Richard J. Fateman
fat...@cs.berkeley.edu http://http.cs.berkeley.edu/~fateman/
James Foster
Isn't the easy answer that if 1/0 = a, then a*0 = 1, and this can't be
true for any a>0. So 1/0 = 0. But then b/0 = 0, too, by the same
argument...regardless of what b is. So a/0 = b/0, and hence a=b
regardless of what a and b are, which is patently silly.
--
James A. Foster email: fos...@cs.uidaho.edu
Laboratory for Applied Logic Dept. of Computer Science
University of Idaho www: http://www.cs.uidaho.edu/~foster
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Carlos A. Felippa
In article <Pine.SUN.3.91.961206125322.12031D-100000@fsf> Le Compte de Beaudrap <j...@cpsc.ucalgary.ca> writes:
>On 5 Dec 1996 sch...@primenet.com wrote:
>
><snip>
>>
>> 0 x = 1 (that 0 x is 0 times the variable x)
>>
>> and define its solution as:
>>
>> u = 1/0 (u for undefined)
>>
>> which describe te values of many functions, also known as asymptotes, or
>> poles. Why wouldn't it be convienient to define a number with these
>> properties? I don't know what it could be used for, but I am sure it
>> would have some kind of use.
>>
>> Comments anyone? Flaws in logic anyone???? No this is not a joke, and
>> mose definately NOT a homework assignment.
>>
>
> I'm not entirely sure why 1/0 hasn't been defined. I think it
>might be due to the following property...
>
Simple. If you define 1/0 and attempt to impose associativity, every
finite number becomes equal to all other numbers. A politically correct
property, for why should 2 > 1? Shouldnt 1 and 2 have equal rights?
But it's not very convenient for mathematics.
Andreas Neubacher
In article <58ndvv$o...@newshound.csrv.uidaho.edu>,
fos...@cs.uidaho.edu (James Foster) writes:
JF> Isn't the easy answer that if 1/0 = a, then a*0 = 1, and this can't be
JF> true for any a>0. So 1/0 = 0. But then b/0 = 0, too, by the same
Yes, 1/0 = a cannot hold for any real number `a'.
The point is that one could define a *new* constant `a' that is different
from any real or complex number and has the property that 1/0 = a. Then
one can start defining additional properties, axioms, etc. and see what
kind of structure this yields.
Andreas.
-----------------------------------------------------------------------
Andreas NEUBACHER http://www.risc.uni-linz.ac.at/people/aneubach/
job: aneu...@risc.uni-linz.ac.at private: andreas....@acm.org
-----------------------------------------------------------------------
Random selection from my quote file:
The ideal scientific mind should be capable of thinking out a point
of abstract knowledge in the interval between its owner falling from
a balloon and reaching the earth.
-----------------------------------------------------------------------
Travis Peery
What about the fact (perhaps I just missed this comment from someone
else...) that if you give definition to a*0 = 1, aka 1/0 = a then you've
just redefined the number zero, like c*0 = 0 for any number c. Once you
do that all of mathematics will be changed! Much of mathematics depends
upon the properties of zero, change that and you must percolate those
changes through the rest of mathematics: I think a mess would result.
There are some functions, or functionals which are defined seemingly ad
hoc, but there definition and use must not conflict with established
mathematics. Mathematics is not purely conventional, nor is it trivial.
We must also not forget that infinity is NOT a number, but more of an
abstraction.
Anyway, that's my two cents for what it's worth.
Travis---
Travis Kidd
Ahh, but infinity *is* a number. And for that matter, all numbers are
"more of an abstraction." I don't see what is so upsetting about
definining 1/0 as infinity. I think things work out great. The
properties of 0 do not have to change that much. c*0=0 for any
number c!=+/-oo. See? It's that simple! And infinity *is* used
in some areas of mathematics. Think of the Dirac delta function.
It is 0 everywhere except at 0, where it is oo. And any integration
that occurs across 0 gives you 1. So 0*oo=1! Nothing wrong with that.
So yes, in order to include infinity you will have to make adjustments
to the "usual" rules of mathematics. But these "adjustments" are nothing
more than exceptions that arise in special, uncommon cases. Acknowledge
them for the special-case exceptions that they are, and get back to your
normal mathematical business.
>Anyway, that's my two cents for what it's worth.
And my 0*oo cents worth. (0*oo is indeterminate, so you have
to decide that it equals 2 from context :-)
>Travis---
-Travis
David Kastrup
tk...@hubcap.clemson.edu (Travis Kidd) writes:
> Ahh, but infinity *is* a number. And for that matter, all numbers are
> "more of an abstraction." I don't see what is so upsetting about
> definining 1/0 as infinity. I think things work out great. The
> properties of 0 do not have to change that much. c*0=3D0 for any
> number c!=3D+/-oo. See? It's that simple!
The problem is not as much working with constants, but working with
variables. If almost all laws break under certain variable values, it
is better to keep these values out.
When you want to use algebra, you want to have the distributive law
hold. Period. Not even having a ring at ones disposal makes
rearranging things a nightmare.
> in some areas of mathematics. Think of the Dirac delta function.
> It is 0 everywhere except at 0, where it is oo. And any integration
> that occurs across 0 gives you 1. So 0*oo=3D1! Nothing wrong with
> that.
A lot is wrong with that. I claim that you have misunderstood the
delta "function", which is really a distribution. What would be,
according to your theory, the integral of the delta function from -1
to 0? and from 0 to 1? What would be the "value" of its derivative at
0?
> So yes, in order to include infinity you will have to make adjustments
> to the "usual" rules of mathematics. But these "adjustments" are nothing
> more than exceptions that arise in special, uncommon cases.
Exceptions are a pain when variables are concerned. Many pupils
cannot even grasp the problems when multiplying an equation with
various non-units and/or zero-dividers. And you expect them to deal
with "numbers" which have much much worse properties and obey much
fewer laws?
--
David Kastrup Phone: +49-234-700-5570
Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
Mark
In article <58oge5$j...@alijku04.edvz.uni-linz.ac.at>,
aneu...@risc.uni-linz.ac.at (Andreas Neubacher) wrote:
>
> In article <58ndvv$o...@newshound.csrv.uidaho.edu>,
> fos...@cs.uidaho.edu (James Foster) writes:
> JF> Isn't the easy answer that if 1/0 = a, then a*0 = 1, and this can't be
> JF> true for any a>0. So 1/0 = 0. But then b/0 = 0, too, by the same
>
> Yes, 1/0 = a cannot hold for any real number `a'.
>
> The point is that one could define a *new* constant `a' that is different
> from any real or complex number and has the property that 1/0 = a. Then
> one can start defining additional properties, axioms, etc. and see what
> kind of structure this yields.
>
Hi
The following is for those people who think that 0u should equal 1.
Remember 1/0 is NOT a real number.
Surely if we define u = 1/0 we are in effect creating a totally new kind of
number.
The point is that given i = sqrt(-1) we are able to construct numbers of the
form z=ai that are linearly independant of real numbers.
This means that there is no real number which when multiplied by or added to
another real number produces the value z above. If this was possible we
would not need i to make sense of x^2+1=0.
In fact there is no real number which can be squared to equal -1.
So with u = 1/0 we are not creating a new kind of real number, but a new set
of numbers of the form x=au which like z above are linearly independant of
real numbers, in that there is no real value defined for 1/0 (try writing it
as a decimal number to any precision you want) that can be achieved by a
linear combination of real numbers.
Therefore 1*u = 1u
2*u = 2u
and 0*u = 0u
why?
Well imagine a geometric representation of real numbers as a line, the unit
of identity on this line is 1(one).
Extend this to include complex numbers by adding another line with an
identity of i=sqrt(-1) at right angles to the first.
Now numbers of the form z=(a+bi) are represented by a point on the familiar
complex plane.
In this system, any number of the form x=a is actually the complex number
x=(a+0i) ie in our 2-dimensional representation, real numbers are complex
numbers with a 0 imaginary component.
Extend this system to include numbers of the form y=cu by (you guessed it)
another line at right angles to the other two, identity is u=1/0.
Now all numbers are points in a volume defined by the three axes above.
They all have the form W=(a+bi+cu) (you can by extension of the polar form
for complex numbers use a spherical geometric form as well).
Now having said all that, it is obvious that all manner of Linear properties
can be assigned to these numbers by extension of complex number properties.
0u has a clear meaning in this context in that it is the third
component of a complex number (a+bi) = (a+bi+0u)
Some things to ponder though are:
x=(a+bi+cu) y=(d+ei+fu)
x*y = {(ad-be)+(bd+ae)i+(cd+af)u+(ce+bf)iu+(cf)u^2}
Can we agree that u^2 = u ?
{ u=1/0, let p=0 then u=1/p, u^2 = 1^2/p^2 = 1/0 = u}
x*y = {(ad-be)+(bd+ae)i+(cd+af+cf)u+(ce+bf)iu}
Also that sqrt(au) = u*sqrt(a)
What does iu mean? { u*sqrt(-1) = sqrt(-u^2) = sqrt(-u) }?
What would you do with numbers like this?
Just a few thoughts.
Regards
Mark.
--
___________________________________________________________________________
M A R K B O T F I E L D...
INTEL outside
_________________________________________________________ m...@argonet.co.uk
electronic monk
Travis Kidd wrote:
> Ahh, but infinity *is* a number. And for that matter, all numbers are
> "more of an abstraction." I don't see what is so upsetting about
> definining 1/0 as infinity. I think things work out great. The
> in some areas of mathematics. Think of the Dirac delta function.
> It is 0 everywhere except at 0, where it is oo. And any integration
0 * oo = any number, not just 1. it is an indeterminante form because
it can be converted into oo/oo. oo/oo = indeterminate. if it was in a
limit (which is the only place one can use oo so freely) like:
lim (2x/x) = 2
x->oo
then oo/oo = 2. or:
lim (x^2)/x = oo
x->oo
in this case, oo/oo = oo. and one more...
lim x/(x^2) = 0
x->oo
so you can see, that oo/oo is equal to any number including 0 and oo,
not just 1.
>
> So yes, in order to include infinity you will have to make adjustments
> to the "usual" rules of mathematics. But these "adjustments" are nothing
> more than exceptions that arise in special, uncommon cases. Acknowledge
> them for the special-case exceptions that they are, and get back to your
> normal mathematical business.
there are mathmaticians (I'm pretty sure anyway) that believe that +oo
and -oo are the same thing, and exist as another pole on the opposite
side of a "sphere" type of universe. this is thought because then one
can think of any parabola, of any conic, as an ellipse with one focal
point at oo (parabola) both on the real plane (ellipse) or both on the
real plane but the figure is "wrapped" around the sphere (hyperbola).
this, i thought, was interesting. but it probably was already discussed
here. these i also think that this is the "adjustment" that Travis was
thinking about. this explains, and adds, some mathamatical properties.
I also think it adds some useless and irritating things to try and
prove.
keep in mind, i'm not directly sure if these properties are true or not.
>
> -Travis
electronic monk
Paul Schlyter
In article <tkidd.850831626@hubcap>,
Travis Kidd <tk...@hubcap.clemson.edu> wrote:
> Travis Peery <pee...@ucs.orst.edu> writes:
>>What about the fact (perhaps I just missed this comment from someone
>>else...) that if you give definition to a*0 = 1, aka 1/0 = a then you've
>>just redefined the number zero, like c*0 = 0 for any number c. Once you
>>do that all of mathematics will be changed! Much of mathematics depends
>>upon the properties of zero, change that and you must percolate those
>>changes through the rest of mathematics: I think a mess would result.
>>There are some functions, or functionals which are defined seemingly ad
>>hoc, but there definition and use must not conflict with established
>>mathematics. Mathematics is not purely conventional, nor is it trivial.
>>We must also not forget that infinity is NOT a number, but more of an
>>abstraction.
>
infinity + 5 = infinity, OK?
infinity + 8 = infinity, OK?
Thus:
infinity + 5 = infinity + 8
If we subtract infinity from both sides of this equation, we get:
5 = 8
We could generalize this into:
<some number> = <any other number>
or even
<any finite number> = <infinity>
since
infinity = infinity + infinity = infinity + any_finite_number
> And for that matter, all numbers are "more of an abstraction."
> I don't see what is so upsetting about definining 1/0 as infinity.
to be a number, since that implies that any number will equal any
other number.
> I think things work out great. The properties of 0 do not have
> to change that much. c*0=0 for any number c!=+/-oo. See? It's
> that simple!
oo = oo + x = oo + y = oo = oo + oo
Subtract oo:
0 = x = y = 0 = oo
for any x and y. Thus any number equals any other number which also
equals infinity. And zero too will then equal infinity.
> And infinity *is* used in some areas of mathematics.
> Think of the Dirac delta function. It is 0 everywhere except at 0,
> where it is oo. And any integration that occurs across 0 gives you 1.
> So 0*oo=1! Nothing wrong with that.
> So yes, in order to include infinity you will have to make adjustments
> to the "usual" rules of mathematics. But these "adjustments" are nothing
> more than exceptions that arise in special, uncommon cases. Acknowledge
> them for the special-case exceptions that they are, and get back to your
> normal mathematical business.
for everyday cases such as addition and subtraction --- see above. If
infinity is a number, then any number equals any other number.
--
----------------------------------------------------------------
Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
Grev Turegatan 40, S-114 38 Stockholm, SWEDEN
e-mail: pau...@saaf.se p...@net.ausys.se pa...@inorbit.com
WWW: http://www.raditex.se/~pausch/ http://spitfire.ausys.se:8003/psr/
Travis Kidd
electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
>> It is 0 everywhere except at 0, where it is oo. And any integration
>> that occurs across 0 gives you 1. So 0*oo=1! Nothing wrong with that.
But it *is* 1.
>there are mathmaticians (I'm pretty sure anyway) that believe that +oo
>and -oo are the same thing, and exist as another pole on the opposite
>side of a "sphere" type of universe.
Back in 9th grade I thought that the number line was really a number
circle, with all finite numbers living at a single point, and the rest
of the circle being infinity. This notion was shattered by the observation
that 3^oo is not the same as 3^(-oo).
>electronic monk
-Travis
Travis Kidd
I've already replied to this through e-mail,
but briefly:
pau...@electra.saaf.se (Paul Schlyter) writes:
>infinity + 5 = infinity, OK?
>infinity + 8 = infinity, OK?
>Thus:
> infinity + 5 = infinity + 8
>If we subtract infinity from both sides of this equation, we get:
> 5 = 8
But you can't simply subtract infinity from infinity and expect
to get zero. This is where your argument is flawed. Infinity
minus infinity is indeterminate, and could be anything.
1=1, so 1^(.5)=1^(.5), so -1=1.
Similar argument. But that doesn't mean square roots don't exist.
> oo = oo + x = oo + y = oo = oo + oo
>Subtract oo:
> 0 = x = y = 0 = oo
Same flawed reasoning.
>> So 0*oo=1! Nothing wrong with that.
>0*oo = any number, not only 1....
But it does equal 1.
>These adjustments would have to be not only for special cases but also
>for everyday cases such as addition and subtraction --- see above. If
>infinity is a number, then any number equals any other number.
Now: a-a=0 for all a.
Adjusted: a-a=0 for all finite a.
Not too hard.
>Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
-Travis
David Kastrup
tk...@hubcap.clemson.edu (Travis Kidd) writes:
> I've already replied to this through e-mail,
> but briefly:
>
> pau...@electra.saaf.se (Paul Schlyter) writes:
> >infinity + 5 =3D infinity, OK?
> >infinity + 8 =3D infinity, OK?
> >Thus:
> > infinity + 5 =3D infinity + 8
> >If we subtract infinity from both sides of this equation, we get:
> > 5 =3D 8
> But you can't simply subtract infinity from infinity and expect
> to get zero. This is where your argument is flawed. Infinity
> minus infinity is indeterminate, and could be anything.
Which is an excellent reason for not making infinity a number. You
are tackling your mathematics too physically: the question is not
whether there *is* a number named infinity, but whether one would like
numbers to be defined in a way which makes infinity a number. For
reasons like the above, one usually wants infinity not to be a
number. Otherwise to many things that generally hold about numbers
even with regard to the simplest algebraic structures get broken.
electronic monk
Travis Kidd wrote:
> >there are mathmaticians (I'm pretty sure anyway) that believe that +oo
> >and -oo are the same thing, and exist as another pole on the opposite
> >side of a "sphere" type of universe.
> Back in 9th grade I thought that the number line was really a number
> circle, with all finite numbers living at a single point, and the rest
> of the circle being infinity. This notion was shattered by the observation
> that 3^oo is not the same as 3^(-oo).
>
> >electronic monk
> -Travis
I hadn't thought of that. you'll have to give me time to think about
that one. but maybe instead of it _being_ -oo, you could be aproaching
oo from the bottom or left.
lim 3^x = oo
x->oo+
lim 3^x = 0
x->oo-
oo isn't a set number and doesn't equal itself, as has been proven in
previous messages, so 3^oo does not nessasarily have to equal 3^oo.
this is very interesting. in some cases infinity could be sensitive to
which direction that you approach it from. in this case, if you
approach it from the negative side, you'll begin to slant downward,
because you'll take the reciprical with the negative exponent. it's
tough to try and prove all this though.
I don't really know if this is true, but it makes a little sense to me.
i'll see if i can find any literature on this because i think it is a
pretty cool idea.
electronic monk
Travis Kidd
electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
>> Back in 9th grade I thought that the number line was really a number
>> circle, with all finite numbers living at a single point, and the rest
>> of the circle being infinity. This notion was shattered by the observation
>> that 3^oo is not the same as 3^(-oo).
>that one. but maybe instead of it _being_ -oo, you could be aproaching
>oo from the bottom or left.
positive number, you cannot approach it from the right.
>lim 3^x = oo
>x->oo+
This makes no sense.
>lim 3^x = 0
>x->oo-
No, this limit is oo.
>oo isn't a set number and doesn't equal itself, as has been proven in
>previous messages, so 3^oo does not nessasarily have to equal 3^oo.
oo indeed is a set number that indeed equals itself. Every number
equals itself. You are correct that 3^oo does not necessarily have
to equal 3^oo. After all, 1^(1/2) does not necessarily have to
equal 1^(1/2). But if you take an infinite power as infinite mul-
tiplication and a negative infinite power as infinite division,
then certainly there is a clear difference between oo and -oo.
>this is very interesting. in some cases infinity could be sensitive to
>which direction that you approach it from. in this case, if you
>approach it from the negative side, you'll begin to slant downward,
>because you'll take the reciprical with the negative exponent. it's
>tough to try and prove all this though.
I don't think it is a matter of proof. Rather, it is axiomatic.
I suppose a proof that oo exists can be given by answering the
question "How many numbers are there?".
>I don't really know if this is true, but it makes a little sense to me.
>i'll see if i can find any literature on this because i think it is a
>pretty cool idea.
Yeah. I do too.
>electronic monk
-Travis
Carl Renneberg
Travis Kidd wrote:
>
> ...
> So yes, in order to include infinity you will have to make adjustments
> to the "usual" rules of mathematics.
Or else look up a treatment of Cantor's transfinite arithmetic in a text
book.
Carl.
Jon Haugsand
pau...@electra.saaf.se (Paul Schlyter) writes:
>
> > 1=1, so 1^(.5)=1^(.5), so -1=1.
> >
> > Similar argument. But that doesn't mean square roots don't exist.
>
> Square roots are different. We all know that taking the square root
> of a number produces two possible results. But subtracting one
> number from some other number should always produce only ONE result.
> Infinity - infinity does not meet this requirement -- therefore
> infinity is not a number.
This may be a matter of definition, but the most common definition of
taking square root of a positive real number x is to take the
*positive* number a such that a*a = x.
--
Jon Haugsand
Dept. of Informatics, Univ. of Oslo, Norway, mailto:jon...@ifi.uio.no
http://www.ifi.uio.no/~jonhaug/, Pho/fax: +47-22852441/+47-22852401
Paul Schlyter
In article <tkidd.850968931@hubcap>,
Travis Kidd <tk...@hubcap.clemson.edu> wrote:
> I've already replied to this through e-mail,
> but briefly:
>
> pau...@electra.saaf.se (Paul Schlyter) writes:
>>infinity + 8 = infinity, OK?
>>Thus:
>> infinity + 5 = infinity + 8
>>If we subtract infinity from both sides of this equation, we get:
> But you can't simply subtract infinity from infinity and expect
> to get zero. This is where your argument is flawed. Infinity
> minus infinity is indeterminate, and could be anything.
the same as zero is that infinity is not a number.
> 1=1, so 1^(.5)=1^(.5), so -1=1.
>
> Similar argument. But that doesn't mean square roots don't exist.
Square roots are different. We all know that taking the square root
of a number produces two possible results. But subtracting one
number from some other number should always produce only ONE result.
Infinity - infinity does not meet this requirement -- therefore
infinity is not a number.
>>Subtract oo:
>> 0 = x = y = 0 = oo
>
> Same flawed reasoning.
>>> So 0*oo=1! Nothing wrong with that.
>>0*oo = any number, not only 1....
>
> But it does equal 1.
>>These adjustments would have to be not only for special cases but also
>>for everyday cases such as addition and subtraction --- see above. If
>>infinity is a number, then any number equals any other number.
>
> Now: a-a=0 for all a.
> Adjusted: a-a=0 for all finite a.
>
> Not too hard.
even for subtraction (and addition).
Why do you want to define "infinity" as a number? Probably because
you don't want to have to treat division by zero as a special case.
But by doing this you've opened a can of worms -- OK, you can now
divide by any number without exception -- but you can no longer add
or subtract any numbers without accounting for special cases.
And even if you define 1/0 = +infinity, you'll have to treat division
with caution. An example:
1/(1/A) = A for all A OK?
Well not for infinities! Consider this:
1/(1/(-infinity)) = 1/0 = +infinity ......
and -infinity is different from +infinity, isn't it?
So you still have to treat division by zero as a special case. So what
did you win? Nothing. What did you lose? A lot -- you now also need
special cases in addition, subtraction, and probably multiplication
as well.
--
----------------------------------------------------------------
Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
electronic monk
Travis Kidd wrote:
> >lim 3^x = oo
> >x->oo+
> This makes no sense.
>
> >lim 3^x = 0
> >x->oo-
> No, this limit is oo.
i wasn't trying to confuse. I meant
lim 3^x = oo
x->+oo
and
lim 3^x = 0
x->-oo
except that i was trying to keep with the idea that +oo and -oo are the
same thing. that was why i used oo+ instead of +oo, so that instead of
"approaching positive infinity" is said "approach infinity from the
right." i just didn't want to contradict myself by saying first that
+oo is the same as -oo and then separating them again.
> >oo isn't a set number and doesn't equal itself, as has been proven in
> >previous messages, so 3^oo does not nessasarily have to equal 3^oo.
> oo indeed is a set number that indeed equals itself. Every number
> equals itself.
oo != oo because oo-oo != 0 and oo/oo != 1. numbers that are the same
(excluding 0) would make all these true. zero is an exception because
it is related to oo very closely. on a polar grid, r = 0 is one pole
and r = oo can be thought of as the "other" pole, for just like on a
globe, if your on the north pole, no matter which direction you walk in,
your heading tward the south pole, of course, on a globe, you can reach
the other pole.
> You are correct that 3^oo does not necessarily have
> to equal 3^oo. After all, 1^(1/2) does not necessarily have to
> equal 1^(1/2).
if you think of 1^(1/2) as one number your right, but it is two numbers
and you can pick from the two which ever you like. ex:
1^(1/2) = x
1 = x^2
0 = x^2 - 1
0 = (x-1)(x+1)
so one picks 1, or -1, for which ever use he finds.
1^(1/2) is not just one number, it is two, and those numbers are always
the same. so 1^(1/2) = 1^(1/2) if you take into account both roots at
the same time.
> But if you take an infinite power as infinite mul-
> tiplication and a negative infinite power as infinite division,
> then certainly there is a clear difference between oo and -oo.
yes, that is because the path you take tward oo leads you tward 0 if you
go around the left side, and if you go right, the graph tends upward,
leading you toward oo. i'll have to think of another example of this
probably.
>
> >this is very interesting. in some cases infinity could be sensitive to
> >which direction that you approach it from. in this case, if you
> >approach it from the negative side, you'll begin to slant downward,
> >because you'll take the reciprical with the negative exponent. it's
> >tough to try and prove all this though.
> I don't think it is a matter of proof. Rather, it is axiomatic.
> I suppose a proof that oo exists can be given by answering the
> question "How many numbers are there?".
that's very true.
> >I don't really know if this is true, but it makes a little sense to me.
> >i'll see if i can find any literature on this because i think it is a
> >pretty cool idea.
> Yeah. I do too.
>
> -Travis
electronic monk
Scott Brown
tk...@hubcap.clemson.edu (Travis Kidd) writes:
>I've already replied to this through e-mail,
>but briefly:
>pau...@electra.saaf.se (Paul Schlyter) writes:
>>infinity + 5 = infinity, OK?
>>infinity + 8 = infinity, OK?
>>Thus:
>> infinity + 5 = infinity + 8
>>If we subtract infinity from both sides of this equation, we get:
>> 5 = 8
>But you can't simply subtract infinity from infinity and expect
>to get zero. This is where your argument is flawed. Infinity
>minus infinity is indeterminate, and could be anything.
But I thought you said infinity is a number? So, what kind
of number did you have in mind? The well-definedness of
subtraction is for most numbers a most useful characteristic.
One of us is confused.
Scott
------------------------------------------------------------------------
| Scott Brown | "I may be speaking from Wolfram Research, Inc., |
| sco...@wolfram.com | but that doesn't mean I'm speaking for them." |
------------------------------------------------------------------------
Travis Kidd
Jon Haugsand <jon...@ejkthyrnir.ifi.uio.no> writes:
>This may be a matter of definition, but the most common definition of
>taking square root of a positive real number x is to take the
>*positive* number a such that a*a = x.
radical sign denotes.
>Jon Haugsand
-Travis
Travis Kidd
sco...@wolfram.com (Scott Brown) writes:
>>But you can't simply subtract infinity from infinity and expect
>>to get zero. This is where your argument is flawed. Infinity
>>minus infinity is indeterminate, and could be anything.
>But I thought you said infinity is a number? So, what kind
>of number did you have in mind?
>The well-definedness of
>subtraction is for most numbers a most useful characteristic.
Agreed. Too bad that oo - oo doesn't have it.
>One of us is confused.
About what?
>Scott
-Travis
Alexander Abian
In article <m2916u7...@mailhost.neuroinformatik.ruhr-uni-bochum.de>,
David Kastrup <d...@mailhost.neuroinformatik.ruhr-uni-bochum.de> wrote:
Abian answers:
To check that, say, 12/4 = 3 is true, it must pass THE ACID TEST,
i.e.,
whether or not 4 x 3 = 12
So, 6/3 = 2 because it passes the acid test 3 X 2 = 6
7/3 = 2 is false because it fails to pass the acid test 3 x 2 =/= 7
NOW,
1/0 is meaningless since for no real number 1/0 = r could possibly
pass the acid test for any real number r. Indeed 0 x r = 0 and is never
equal to 1. So 1/ 0 CANNOT POSSIBLY BE A REAL NUMBER.
And, please, please, please,please INFINITY is NOT a real number. A real
number must be expressible by a decimal representation.
In connection with division by 0 the only possibility is 0/0 because
it passes the acid test. Thus, 0/0 = 4 since 0 x 4 = 0 but so are
true any of the equalities 0/0 = -4, 0/0 = 6.76 0/0 = 8.93. Thus,
0/0 is not uniquely determined and for that reason 0/0 is also should
be forbidden to use. UNLESS there is a decree to define 0/0, say
1 or say 0 (or for that matter even 0/0 = 10. However, the natural
two candidates are 1 or 0 because a/a = 1 for every nonzero a
and 0/a = 0 also for every nonzero a).
And, please, please , please DO NOT talk about where the limit of f(x)/g(x)
involves considerations involving 0/0. In the above we are talking
strictly concerning 0/0 where no functions or limiting processes are
involved.
--
--------------------------------------------------------------------------
ABIAN MASS-TIME EQUIVALENCE FORMULA m = Mo(1-exp(T/(kT-Mo))) Abian units.
ALTER EARTH'S ORBIT AND TILT - STOP GLOBAL DISASTERS AND EPIDEMICS
ALTER THE SOLAR SYSTEM. REORBIT VENUS INTO A NEAR EARTH-LIKE ORBIT
TO CREATE A BORN AGAIN EARTH (1990)
electronic monk
Dik T. Winter wrote:
...
> > For reasons like the above, one usually wants infinity not to be a
> > number.
infinity is a never ending limit. it is the x value that 2^(-x) equals
zero. it is not a "number" in the normal sense of the word since it
does not exist.
...
> And the axioms still say that given
> some value "oo" (which would be the result of applying "/" to "1" and
> "0") there must be some value "-oo" such that "oo" + "-oo" = 0.
there is. ex:
lim (x+(-x)) = 0
x->oo
just the same as:
lim (x+(-x)) = 0
x-> (any number)
> So, pray Travis, how would you modify the axioms and definitions of
> "-" and "/" such that you could introduce "oo".
no need to modify... they all work. infinity is a CONCEPT it does not
exist. that is why we must use limits instead of strictly doing math
with oo. we can only APPROACH oo, we can never get there.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
electronic monk
Dik T. Winter
In article <m2916u7...@mailhost.neuroinformatik.ruhr-uni-bochum.de> David Kastrup <d...@mailhost.neuroinformatik.ruhr-uni-bochum.de> writes:
> tk...@hubcap.clemson.edu (Travis Kidd) writes:
> > But you can't simply subtract infinity from infinity and expect
> > to get zero. This is where your argument is flawed. Infinity
> > minus infinity is indeterminate, and could be anything.
>
> are tackling your mathematics too physically: the question is not
> whether there *is* a number named infinity, but whether one would like
> reasons like the above, one usually wants infinity not to be a
> number.
It is even much simpler. Mathematics is based on a set of axioms.
Numbers form a field according to these axioms. Basically the axioms
are as follows (oh, this is loooong ago):
1. Addition of all numbers and multiplication of all numbers form a
halfgroup. That is given numbers a, b and c:
1 a: a + b and a * b are defined.
1 b: a + b = b + a and a * b = b * a (commutative law).
1 c: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)
(associative law).
2. They additive halfgroup has a unit (i.e. it is groups):
2 a: there is a unit (0) for addition such that a + 0 = a.
3. Addition and multiplication combine to form a ring:
3 a: the distributive law holds:
a * (b + c) = (a * b) + (a * c).
3 b: the additive group has inverses, so for each e there is a (-e)
with e + (-e) = 0.
4. They form a field:
4 a: there is a unit (1) for multiplication such that a * 1 = a.
4 b: the multiplicative group with the exception of 0 has inverses,
so for each e != 0 there is a (e^(-1)) with e * (e^(-1)) = 1.
As a shorthand we write "a - b" for "a + (-b)" and "a / b" for
"a * (b^(-1))". There are many number systems that obey these axioms
(take for instance integer arithmetic mod p, with p a prime).
What apparently many people find offensive is the exception in (4).
However, the axioms as given tell enough about the structure to make
the exception necessary when you want to talk about interesting
structures. You can prove (based on these axioms) that a*0 = 0
(and many more things). If you allow a multiplicative inverse of
0 there is only one structure left: the single element set {0}
with trivial addition and multiplication.
I saw comparisons between 0/0 and ln(1). Based on the axioms above
the first is impossible (0/0 is just a shorthand for 0*(0^(-1)), and
0 does not have an inverse). The second is a function (and not analytic
to boot). Even Abian got it wrong, 0/0 is *not* multivalued; the
definitions and axioms just rule it out as making any sense. It
becomes different when you start to define "/" as some function, rather
than a shorthand (but in that case you have to define your function
completely). So we can define "/" as a function such that "1/0" has
some value, but now we are burdened with a complete definition.
Similarly we could define "-" as a function, but also in that case
we have to define it completely. And the axioms still say that given
some value "oo" (which would be the result of applying "/" to "1" and
"0") there must be some value "-oo" such that "oo" + "-oo" = 0.
So, pray Travis, how would you modify the axioms and definitions of
"-" and "/" such that you could introduce "oo".
David
pau...@electra.saaf.se (Paul Schlyter) wrote:
Sorry. No time to read all articles right now. However, to jump in;
was "Arithmetic" not defined as "*Finite* Arithmetic" in order to
avoid such complications and speculations? That is, the concept of
infinity implies that it is NOT a finite number, and therefore not
part of the arithmetic of finite numbers.
Travis Kidd
d...@cwi.nl (Dik T. Winter) writes:
>It is even much simpler. Mathematics is based on a set of axioms.
>Numbers form a field according to these axioms.
>Basically the axioms
>are as follows (oh, this is loooong ago):
>1. Addition of all numbers and multiplication of all numbers form a
> halfgroup. That is given numbers a, b and c:
> 1 a: a + b and a * b are defined.
> 1 b: a + b = b + a and a * b = b * a (commutative law).
> 1 c: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c)
> (associative law).
>2. They additive halfgroup has a unit (i.e. it is groups):
> 2 a: there is a unit (0) for addition such that a + 0 = a.
>3. Addition and multiplication combine to form a ring:
> 3 a: the distributive law holds:
> a * (b + c) = (a * b) + (a * c).
> 3 b: the additive group has inverses, so for each e there is a (-e)
> with e + (-e) = 0.
>4. They form a field:
> 4 a: there is a unit (1) for multiplication such that a * 1 = a.
> 4 b: the multiplicative group with the exception of 0 has inverses,
> so for each e != 0 there is a (e^(-1)) with e * (e^(-1)) = 1.
For infinities, there is a case where each of these properties holds.
However, there are some where some don't.
>(and many more things). If you allow a multiplicative inverse of
>0 there is only one structure left: the single element set {0}
>with trivial addition and multiplication.
Not true.
>I saw comparisons between 0/0 and ln(1). Based on the axioms above
>the first is impossible (0/0 is just a shorthand for 0*(0^(-1)), and
>0 does not have an inverse). The second is a function (and not analytic
>to boot). Even Abian got it wrong, 0/0 is *not* multivalued; the
>definitions and axioms just rule it out as making any sense. It
>becomes different when you start to define "/" as some function, rather
>than a shorthand (but in that case you have to define your function
>completely). So we can define "/" as a function such that "1/0" has
>some value, but now we are burdened with a complete definition.
Why not define / as a relation that is a function on the finite domain?
>Similarly we could define "-" as a function, but also in that case
>we have to define it completely. And the axioms still say that given
>some value "oo" (which would be the result of applying "/" to "1" and
>"0") there must be some value "-oo" such that "oo" + "-oo" = 0.
Yes. And there is. All is well.
>So, pray Travis, how would you modify the axioms and definitions of
>"-" and "/" such that you could introduce "oo".
Well, first of all, the axioms would remain the same when applied
to finite numbers.
Here are the additional things I would add:
oo - oo has as a solution every real number, oo, and -oo.
1/0 has as a solution +oo or -oo.
0/0 has the same solution set as oo - oo.
Otherwise the axioms remain the same.
This is pretty quick, and while I don't see a mistake, I may have left
alot out. Feel free to ask for details.
>dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
-Travis
Paul Schlyter
In article <tkidd.851306477@hubcap>,
Please give one single example of a set, except {0}, which follows
all the axioms above AND allows a multiplicative inverse of zero...
If you cannot do this, your claim "not true" loses its credibility.
>>Similarly we could define "-" as a function, but also in that case
>>we have to define it completely. And the axioms still say that given
>>some value "oo" (which would be the result of applying "/" to "1" and
>>"0") there must be some value "-oo" such that "oo" + "-oo" = 0.
>
> Yes. And there is. All is well.
is indeterminate rather than zero....
>>So, pray Travis, how would you modify the axioms and definitions of
>>"-" and "/" such that you could introduce "oo".
>
> Well, first of all, the axioms would remain the same when applied
> to finite numbers.
>
> Here are the additional things I would add:
>
> oo - oo has as a solution every real number, oo, and -oo.
> 1/0 has as a solution +oo or -oo.
> 0/0 has the same solution set as oo - oo.
>
> Otherwise the axioms remain the same.
>
> This is pretty quick, and while I don't see a mistake, I may have left
> alot out. Feel free to ask for details.
1. Please give the additive inverse of oo such that oo + (-oo) is
always zero. In your sentence "Yes. And there is. All is well" you
claimed such an inverse does exist.
2. Please give one single example of a set, except {0}, which follows
all the axioms above AND allows a multiplicative inverse of zero. In
your sentence "Not true" you claimed such a set exists, yet you didn't
give one single example....
electronic monk
Paul Schlyter wrote:
> Please give one single example of a set, except {0}, which follows
> all the axioms above AND allows a multiplicative inverse of zero...
> If you cannot do this, your claim "not true" loses its credibility.
>
> >>Similarly we could define "-" as a function, but also in that case
> >>we have to define it completely. And the axioms still say that given
> >>some value "oo" (which would be the result of applying "/" to "1" and
> >>"0") there must be some value "-oo" such that "oo" + "-oo" = 0.
> >
> > Yes. And there is. All is well.
>
> Please stop contradicting yourself! Below you claim that oo + (-oo)
> is indeterminate rather than zero....
what he means is that oo - oo CAN be equal to zero. ex:
lim (x - x) = 0
x->oo
but also oo - oo could be anything, including oo and 0.
lim (2x - x) = oo
x->oo
> Yes:
>
> 1. Please give the additive inverse of oo such that oo + (-oo) is
> always zero. In your sentence "Yes. And there is. All is well" you
> claimed such an inverse does exist.
infinity is NOT a number. it was thought up to explain never ending
systems. there are said that there are an infinite amount of numbers,
this just means that no matter how high you count, you'll never reach a
point where there are no more numbers. but the actual number "infinity"
is not a number and that is why oo can only be used inside limits. you
can only say that you are _approaching_ infinity, you can never actualy
express the idea of being at infinity. one can prove most of the rules
of infinity using common sense. ex:
1/oo = 0
this says, "as the denominator of a fraction gets very very large, and
the top stays the same, the whole thing seems to approach zero."
oo/oo = indeterminite
"as the numerator of a fraction and the denominator of a fraction get
very large, it is imposible to tell what it converges on."
1/0 = oo
"as the denominator of a fraction gets very very small and the top stays
the same, the whole thing seems to never stop getting larger."
oo - oo = indeterminite
"if one subtracts two numbers that are bothe getting very very large, it
is imposible to determine what number it converges on."
this is the only way i can think of proving them.
i left out -oo only because i figured that was understood.
> --
> ----------------------------------------------------------------
> Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
> Grev Turegatan 40, S-114 38 Stockholm, SWEDEN
> e-mail: pau...@saaf.se p...@net.ausys.se pa...@inorbit.com
> WWW: http://www.raditex.se/~pausch/ http://spitfire.ausys.se:8003/psr/
electronic monk
Alexander Abian
In article <m2916u7...@mailhost.neuroinformatik.ruhr-uni-bochum.de>,
David Kastrup <d...@mailhost.neuroinformatik.ruhr-uni-bochum.de> wrote:
Abian answers:
To check that, say, 12/4 = 3 is true, it must pass THE ACID TEST,
i.e.,
whether or not 4 x 3 = 12
So, 6/3 = 2 because it passes the acid test 3 X 2 = 6
7/3 = 2 is false because it fails to pass the acid test 3 x 2 =/= 7
NOW,
1/0 is meaningless since for no real number 1/0 = r could possibly
pass the acid test for any real number r. Indeed 0 x r = 0 and is never
equal to 1. So 1/ 0 CANNOT POSSIBLY BE A REAL NUMBER.
And, please, please, please,please INFINITY is NOT a real number. A real
number must be expressible by a decimal representation.
In connection with division by 0 the only apparent possibility is 0/0
because it passes the acid test. Thus, 0/0 = 4 since 0 x 4 = 0 but
so are true any of the equalities 0/0 = -4, 0/0 = 6.76 0/0 = 8.93.
Thus, 0/0 is not uniquely determined and for that reason 0/0 is also
should be FORBIDDEN to use. Because the non-uniqueness of 0/0 would
entail a contradiction. Indeed say, 0/0 = 2 and 0/0 = 7 would imply the
contradiction 2 = 7.
Dik T. Winter
In article <32BEDE...@sqruhs.ruhs.uwm.edu> don...@sqruhs.ruhs.uwm.edu writes:
> infinity is NOT a number.
This is correct if you define oo as some limiting thing.
> can only say that you are _approaching_ infinity, you can never actualy
> express the idea of being at infinity. one can prove most of the rules
> of infinity using common sense. ex:
>
> 1/oo = 0
>
> this says, "as the denominator of a fraction gets very very large, and
> the top stays the same, the whole thing seems to approach zero."
But this is the "gotcha". Why should in this expression "1" stay the
same?
>
> oo/oo = indeterminite
>
> "as the numerator of a fraction and the denominator of a fraction get
> very large, it is imposible to tell what it converges on."
And another "gotcha". In general when numerator and denominator of a
fraction get very large it is very possible to tell what it converges to.
Moreover, "indeterminate" is apparently another non-number.
>
> 1/0 = oo
...
> i left out -oo only because i figured that was understood.
Well, here is why I do not understand it. Why is 1/0 = oo and not = - oo?
There is no way to tell *how* 0 is a limiting value (and, yes, I do
understand why IEEE f-p arithmetic has both +oo and -oo).
You can get some form of consistency by introducing two concepts: oo
and I (indeterminate). The rules of the game are as follows (where
finite means not oo nor I) and they are given in order of precedence:
1. Any expression involving I has the value I.
2. oo/oo = I.
3. 0/0 = I.
4. Any expression divided by oo has 0 as value.
5. Any expression divided by 0 has oo as value.
6. 0*oo = I.
7. Multiplying any other value with oo results in oo.
8. Subtracting oo from oo results in I.
9. Adding or subtracting any other value to or from oo results in oo.
10. In all other cases normal arithmetic applies.
(I think these are all.) The standard axioms for numbers do not apply,
but this is close enough to what you want, and actually IEEE f-p
arithmetic is close to this. The only difference is that IEEE allows
+oo and -oo and this creates havoc when you want to define complex
numbers.
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
Dik T. Winter
In article <59nale$8...@news.iastate.edu> ab...@iastate.edu (Alexander Abian) writes:
> NOW,
> 1/0 is meaningless since for no real number 1/0 = r could possibly
> pass the acid test for any real number r. Indeed 0 x r = 0 and is never
> equal to 1. So 1/ 0 CANNOT POSSIBLY BE A REAL NUMBER.
Abian, you are focussing on real numbers. There are a lot more number
system than just real numbers (but I agree, 1/0 cannot possibly be a number).
> In connection with division by 0 the only apparent possibility is 0/0
> because it passes the acid test. Thus, 0/0 = 4 since 0 x 4 = 0 but
> so are true any of the equalities 0/0 = -4, 0/0 = 6.76 0/0 = 8.93.
And here is the problem. You apparently define / as passing some acid
test. That is not the way / is normally defined!
Dik T. Winter
Axioms parafrased from the definitions by me:
1 a: definition ( a+b and a*b are defined )
1 b: commutative law ( a+b = b+a and a*b = b*a )
1 c: associative law ( (a+b)+c = a+(b+c) and (a*b)*c = a*(b*c) )
2 a: additive unit ( a+0 = a )
3 a: distributive law ( a*(b+c) = (a*b)+(a*c) )
3 b: additive inverse ( a+(-a) = 0 )
4 a: multiplicative unit ( a*1 = a )
4 b: multiplicative inverse except for 0 ( a*(a^(-1)) = 1 )
In article <tkidd.851306477@hubcap> tk...@hubcap.clemson.edu (Travis Kidd) writes:
> For infinities, there is a case where each of these properties holds.
> However, there are some where some don't.
Yes, there are fields with infinities, but in those cases the inverses of
the inifinities are not 0, but something different again.
>
> > If you allow a multiplicative inverse of
> >0 there is only one structure left: the single element set {0}
> >with trivial addition and multiplication.
> Not true.
Let's say that x is the multiplicative inverse:
1 = 0.x (definition of multiplicative inverse)
= (0 + 0).x (because 0 = 0 + 0)
= 0.x + 0.x (distributive law)
= 1 + 1 = 2.
So if we accept the set of axioms above and when we introduce a
multiplicative inverse of 0, this immediately leads to 1 = 2, and
also to 0 = 1, so we have a single element set.
> Why not define / as a relation that is a function on the finite domain?
Ok, in that case feel free to define it as a function on the set of
numbers. Note however that the set of numbers has to fit the axioms
given above, so there is still no place for the multiplicative inverse
of 0.
> >So, pray Travis, how would you modify the axioms and definitions of
> >"-" and "/" such that you could introduce "oo".
> Well, first of all, the axioms would remain the same when applied
> to finite numbers.
Ok, so oo is not a number. Good so far.
>
> Here are the additional things I would add:
> oo - oo has as a solution every real number, oo, and -oo.
> 1/0 has as a solution +oo or -oo.
> 0/0 has the same solution set as oo - oo.
(I do not understand the concept of "has as a solution", I think you mean
in the second case that / when applied to 1 and 0 has two values, oo and
-oo.)
So apparently - and / are 2-parameter multi-valued functions that can
be applied to the set consisting of the numbers and in addition to that
oo and -oo. Still a lot of things are missing. What is 1*oo?
What is 2*oo? What is (-1)*oo? What is 0*oo? What is oo*oo?
What is oo+oo? (Note that the concept of defined in 1a implies that
there is a unique definition.) What axioms are valid for oo and -oo?
> Otherwise the axioms remain the same.
Let's see. Apparently the concepts of additive and multiplicative inverses
no longer hold for oo, so those axioms do not hold for oo. So that is two
of the axioms we lose for infinities. Also oo+(-oo) is not defined, so
that is also part of an axiom that does not hold.
What if we are doing calculations in the complex field?
Dik T. Winter
> >It is even much simpler. Mathematics is based on a set of axioms.
> >Numbers form a field according to these axioms.
> You mean the complex numbers form a field according to these axioms.
One I did not even see on first reading. No, I did not mean that.
The axioms where those for a "field". Rational numbers form a field,
algebraic numbers form a field, real numbers form a field, complex
numbers form a field, integers mod p (p prime) form a field, John
Conways numbers form a field, he has defined the algebraically complete
field On2 of characteristic 2, we have the quadratic fields, and so on.
In general, when a mathematician is speaking about numbers he is
speaking about objects within a field. (And quaternions or hamiltonians
do not form a field but a "skew field", i.e. commutativity does not hold
for multiplication. But by extensions those are also called numbers.)
Some of those fields contain infinities, but none has a multiplicative
inverse of 0.
electronic monk
Dik T. Winter wrote:
>
> In article <32BEDE...@sqruhs.ruhs.uwm.edu> don...@sqruhs.ruhs.uwm.edu writes:
> > infinity is NOT a number.
>
> This is correct if you define oo as some limiting thing.
that is what i define oo to be. the limit of an always increasing
number.
> > 1/oo = 0
> >
> > this says, "as the denominator of a fraction gets very very large, and
> > the top stays the same, the whole thing seems to approach zero."
>
> But this is the "gotcha". Why should in this expression "1" stay the
> same?
i am confused. what do you mean "Why should in this expression '1' stay
the same?" do you mean the first expression?
> > oo/oo = indeterminite
> >
> > "as the numerator of a fraction and the denominator of a fraction get
> > very large, it is imposible to tell what it converges on."
>
> And another "gotcha". In general when numerator and denominator of a
> fraction get very large it is very possible to tell what it converges to.
> Moreover, "indeterminate" is apparently another non-number.
this is only true if you have functions in the form f(x)/g(x) in which
f(x) -> oo and g(x) -> oo. one cannot just look at oo/oo and tell what
number it represents. for instance:
lim (4x +1)/(x + 2) = oo/oo = 4
x->oo
both top and bottom approach oo, but you can't just look at oo/oo and
say, "that's four!" it is not possible unless you know more about how
fast both top and bottom are getting very very large. that's why
indeterminante was invented, to give a name to limits that cannot be
determined from the given information.
> > 1/0 = oo
> ...
> > i left out -oo only because i figured that was understood.
>
> Well, here is why I do not understand it. Why is 1/0 = oo and not = - oo?
> There is no way to tell *how* 0 is a limiting value (and, yes, I do
> understand why IEEE f-p arithmetic has both +oo and -oo).
this is true, what i meant was that i was always dealing with positive
values and was therefor approaching 0 from the right, making it +oo. i
just didn't want to go back and change it. plus there are a lot of
people that believe that positive and negative oo are the same thing.
but you are right, i should have included -oo in my answers.
> You can get some form of consistency by introducing two concepts: oo
> and I (indeterminate). The rules of the game are as follows (where
> finite means not oo nor I) and they are given in order of precedence:
> 1. Any expression involving I has the value I.
> 2. oo/oo = I.
> 3. 0/0 = I.
> 4. Any expression divided by oo has 0 as value.
> 5. Any expression divided by 0 has oo as value.
> 6. 0*oo = I.
> 7. Multiplying any other value with oo results in oo.
> 8. Subtracting oo from oo results in I.
> 9. Adding or subtracting any other value to or from oo results in oo.
> 10. In all other cases normal arithmetic applies.
these are all true, as far as i can tell anyway, yet i stated most of
these before. i just wanted to state them using words as to make them
easier to understand. i did use oo as a limiting value. as i stated
before, that was what infinity was created for, a limit for something
that is always increasing.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
electronic monk
electronic monk
Dik T. Winter wrote:
> Let's say that x is the multiplicative inverse:
> 1 = 0.x (definition of multiplicative inverse)
> = (0 + 0).x (because 0 = 0 + 0)
> = 0.x + 0.x (distributive law)
> = 1 + 1 = 2.
> So if we accept the set of axioms above and when we introduce a
> multiplicative inverse of 0, this immediately leads to 1 = 2, and
> also to 0 = 1, so we have a single element set.
this is a common argument, the fact that 0 times its inverse does not
seem to equal 1, but instead, any number, including 0 and its inverse.
this leads to the point that zero does not have an inverse. and in
reality, it doesn't. yes, most people woould jump and say that oo is
it's inverse, but oo does not exist as just one number. plus, the
definition of multiplicative inverses comes from this:
x and y are inverses if and only if xy = 1. so, x = 1/y and y = 1/x.
so when you try to solve for the inverse of zero, you get x = 1/0. but
what happens when you multiply both sides times a number, n. nx = n/0.
divide both sides by n and you get x = n/(n*0) or x = n/0. so this
means that to get the inverse of 0 you can just say 0*x = n where n =
any finite number except 0. this means that x equals anything that can
make 0*x != 0. and since any finite number times 0 equals 0, it must be
infinite. and it turns out to be any infinite number, or, oo. you can
also achieve the same results this way:
0*x = 1
n*(0*x) = n
0*x = n
> Let's see. Apparently the concepts of additive and multiplicative inverses
> no longer hold for oo, so those axioms do not hold for oo. So that is two
> of the axioms we lose for infinities. Also oo+(-oo) is not defined, so
> that is also part of an axiom that does not hold.
of course not, these axioms were defined before oo was introduced.
infinity adds a whole new dimension to mathimatics.
Travis Kidd
electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
>what he means is that oo - oo CAN be equal to zero.
>infinity is NOT a number.
Disagreed.
> it was thought up to explain never ending
>systems. there are said that there are an infinite amount of numbers,
>this just means that no matter how high you count, you'll never reach a
>point where there are no more numbers.
You can't count to infinity in a finite amount of time. But that doens't
mean infinity is not a number.
>electronic monk
-Travis
Travis Kidd
electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
>of course not, these axioms were defined before oo was introduced.
>infinity adds a whole new dimension to mathimatics.
>electronic monk
-Travis
electronic monk
Travis Kidd wrote:
>
> electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
> >infinity is NOT a number.
> Disagreed.
infinity is not a number in the normal sense of the word "number." oo
has no numerical value and cannot be represented using normal numbers
and digits.
> >electronic monk
> -Travis
electronic monk
Travis Kidd
electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
>> >infinity is NOT a number.
>> Disagreed.
>infinity is not a number in the normal sense of the word "number." oo
>has no numerical value and cannot be represented using normal numbers
>and digits.
say so? I'll agree to that.
>electronic monk
-Travis
Plamen Stefanov
This question does not make much sense. Who said that 1/0 cannot
be defined? It can, for example 1/0=13. This definition is stupid,
you will probably say. Well, it is, but no more than any other
definition. Why on Earth we need a definition of 1/0, or even 0/0?
Carl Renneberg
You might like to look up a textbook with covers Cantor's transfinite
numbers, and infinite sets.
The smallest transfinite ("beyond finite") number is called "aleph
null" (a Hebrew symbol, can't draw it here).
The set of all integers is infinite. Aleph null equals the size
of that set. Aleph null is infinite, but you perform perform
arithmetic with it.
Often you'll come across the phrase "infinite but countable" in
Comp. Sci. texts - that means that the set under discussion can
be placed in a one-to-one correspondence with the integers. The
set is infinite, but has the same "size" (cardinality) as the
integers, namely aleph null.
If you take the power set (the set of all subsets) of the integers,
you get another infinite set, but it's larger. The power set
is always larger - this applies to finite and infinite sets.
The power set of the integers has a size equal to another
transfinite number called aleph one (another Hebrew symbol).
You *cannot* form a one-to-one correspondence between the integers
and the power set of the integers (first proved by Cantor with his
famous "diagonal" argument); no matter how you try to construct
the correspondence, there will still be members of the power set
left over (infinitely many in this case). Aleph one is larger than
aleph null.
Performing operations with transfinite numbers may seem unsettling,
but no more unsettling than irrational numbers must have seemed to
the ancient Greeks.
Carl.
electronic monk
undefined means that it never can be defined, or it is infintity (or
-oo), this is what 1/0 is. indeterminante means that the value must be
determined by the context it is used in, this is 0/0 or oo/oo. one
cannot just simply give a value to a arithmetic statement. for instance
i cannot just say that 2 + 2 = 3. now, 1/0 and 0/0 must be given a
value in order to evaluate limits. the basic derivitive limit is always
equal to 0/0, but it can equal anything. we couldn't do any calculus
without these indeterminante and undefined forms. math people decided
that 1/0 is oo or -oo and 0/0 could be any one value including oo and 0,
and since oo/oo is the same as 0/0, it could also attain any value.
they came to these conclusions with limits.
electronic monk
Anonymous
The limit of 1/x as x --> 0 is infinity.
When you talk about 1/0 directly there is at times a difficulty
connecting it with the general numerical system. If you wish to be
sloppy, you can say it is infinity without noting limits. However, it
is usually defined in terms of limits.
a suggestion,
-X
Jim Carr
In a forged article <32CA9D...@efgh.net> ab...@efgh.net (-X) writes:
>
>The limit of 1/x as x --> 0 is infinity.
positive or negative?
--
James A. Carr <j...@scri.fsu.edu> | "The half of knowledge is knowing
http://www.scri.fsu.edu/~jac/ | where to find knowledge" - Anon.
Supercomputer Computations Res. Inst. | Motto over the entrance to Dodd
Florida State, Tallahassee FL 32306 | Hall, former library at FSCW.
Plamen Stefanov
John Briggs, VAX system manager, x4411
In article <32CA9D...@efgh.net>, Anonymous <ab...@efgh.net> writes:
> The limit of 1/x as x --> 0 is infinity.
>
> When you talk about 1/0 directly there is at times a difficulty
> connecting it with the general numerical system. If you wish to be
> sloppy, you can say it is infinity without noting limits. However, it
> is usually defined in terms of limits.
The limit of 1/x as x --> 0 does not exist. At least not in the reals.
Definition:
The limit of a function f at point x is a number l such that for
every epsilon there is a delta such that for all y != x where
abs(x-y) < delta, abs(f(y)-l) < epsilon.
(real numbers assumed)
(clauses relating to the domain of f omitted for brevity)
One could, I suppose, mess with this definition, using a distance
metric other than abs(x-y) and adding oo to the domain of the metric.
That would allow you to define the limit of 1/x at x=0 as oo.
And it would even allow you to define the limit of 1/x at x=oo as 0.
That wouldn't be the definition of limit I learned in school.
The range of such a limit function wouldn't be the reals.
And, as you point out, it still wouldn't define 1/0.
John Briggs vax...@alpha.vitro.com
ibokor
Anonymous (ab...@efgh.net) wrote:
: The limit of 1/x as x --> 0 is infinity.
:
If you're talking about the real numbers, then
what you have claimed is not true.
Even if you were to be sloppy about the use
of the word "limit", there would still not be
a limit, since if x is negative, so is 1/x
and if x is positive, so is 1/x. Hence the only
possible limit would be 0, the limit would
have to be non-positive and non-negative.
d.A.
electronic monk
John Briggs, VAX system manager, x4411 wrote:
> The limit of 1/x as x --> 0 does not exist. At least not in the reals.
that's right, it is an infinite number.
> One could, I suppose, mess with this definition, using a distance
> metric other than abs(x-y) and adding oo to the domain of the metric.
> That would allow you to define the limit of 1/x at x=0 as oo.
> And it would even allow you to define the limit of 1/x at x=oo as 0.
one doesn't need to redifine anything. lim 1/x as x->oo is 0.
> That wouldn't be the definition of limit I learned in school.
> The range of such a limit function wouldn't be the reals.
a limit does not have to exist in the reals. limits were made in order
to define functions at infinite areas, this is why we _approach_
things. you can approach oo all you want, you just can't get there. a
limit just says what it looks like the function would be if it COULD be
at the point your approaching.
> And, as you point out, it still wouldn't define 1/0.
that is true. lim 1/x as x->0 does not exist and is considered to be
either -oo or +oo. that's why it is more appropriate to say that lim
1/x as x->0 is _infinite_. this also avoids all the confusion with
levels of infinity.
>
> John Briggs vax...@alpha.vitro.com
electronic monk
lim 1/x = oo
x->0+
lim 1/x = -oo
x->0-
hence
lim 1/x = DNE
x->0
it does not exist because it is different from the left than from the
right. but both limits ARE infinite, meaning that 1/0 is an infinite
number, either +oo of -oo. limits were made to evaluate functions at
infinite and indeterminante values, that is their point in existance.
electronic monk
Darrell Ryan
electronic monk wrote:
>
> lim 1/x = oo
> x->0+
>
> lim 1/x = -oo
> x->0-
>
> hence
>
> lim 1/x = DNE
> x->0
>
> it does not exist because it is different from the left than from the
> right. but both limits ARE infinite, meaning that 1/0 is an infinite
> number, either +oo of -oo. limits were made to evaluate functions at
> infinite and indeterminante values, that is their point in existance.
>
> electronic monk
Actually, the limit does not exist because you get oo (or -oo). When we
write oo or -oo as an answer to a limit problem, we are at the same time
saying that the limit does not exist. The oo part tells us a little
about *why* the limit does not exist. The left-hand limit having to
equal the right-hand limit in order for the limit to exist doesn't
really apply here since oo and -oo are not real numbers.
Anyway, to answer the original question:
1/0 is undefined simply because there is no number that you can multiply
by 0 and get 1 as an answer. Sure we can talk about limits and the
like, but that does not define division by zero. In other words:
lim x-->0 f(x) and f(0) are often two very different things.
____________________________________________________________
Darrell Ryan
e-mail dr...@edge.net
personal website http://edge.edge.net/~dryan
company website http://www.edge.net/stmc
Dik T. Winter
In article <32CC49...@sqruhs.ruhs.uwm.edu> don...@sqruhs.ruhs.uwm.edu writes:
> that is true. lim 1/x as x->0 does not exist and is considered to be
> either -oo or +oo.
Or simply not distinguish -oo and +oo. To distinguish them may lead to
severe problems when you are switching to complex numbers. The fact that
at one stage IEEE switched in their definition of floating-point numbers
from a projective infinity to a dual infinity leads to severe problems
stating basic properties about complex f-p math. That is, unless you
switch to a polar representation of complex quantities, but polar complex
math is numerically much less stable than an euclidian equivalent.
electronic monk
>
i think he meant that lim 1/x as x->0 is infinitE. meaning that it is
an infinite number, either - or + oo.
>
> --
> James A. Carr <j...@scri.fsu.edu> | "The half of knowledge is
knowing
> http://www.scri.fsu.edu/~jac/ | where to find knowledge" -
Anon.
> Supercomputer Computations Res. Inst. | Motto over the entrance to
Dodd
> Florida State, Tallahassee FL 32306 | Hall, former library at
FSCW.
electronic monk
electronic monk
Dik T. Winter wrote:
>
> In article <32CC49...@sqruhs.ruhs.uwm.edu>
don...@sqruhs.ruhs.uwm.edu writes:
> > that is true. lim 1/x as x->0 does not exist and is considered to
be
> > either -oo or +oo.
>
> Or simply not distinguish -oo and +oo. To distinguish them may lead
to
> severe problems when you are switching to complex numbers. The fact
that
> at one stage IEEE switched in their definition of floating-point
numbers
> from a projective infinity to a dual infinity leads to severe
problems
> stating basic properties about complex f-p math. That is, unless you
> switch to a polar representation of complex quantities, but polar
complex
> math is numerically much less stable than an euclidian equivalent.
i do believe that -oo and +oo are the same thing. it's just that not
many others do, so i must explain other things without mentioning that
-oo and +oo could be the same thing. the basic proof that people used
against me for -oo != +oo is that
lim 2^x as x-->oo != lim 2^x as x-->-oo
this situation is explained because negative exponents were invented to
aid the rules of math along.
because it was already discovered that:
(2^x)/(2^y) = 2^(x-y) when x and y were both positive numbers and x > y,
people asked what would happen if
x <= y. so, they said that negative exponents just flipped the number
and zero exponent always equaling one. this is because:
(x^n)/(x^n) = x^(n-n) = x^0 and (x^n)/(x^n) = 1 so x^0 = 1.
also...
(x^n)/(x^(n+1)) = x^(-1) and (x^n)/(x^(n+1)) = 1/x so x^(-1) = 1/x.
so
2^oo --> oo and
2^(-oo) --> 1/(2^oo) --> 1/oo --> 0
they are completly different types of problems.
-oo is just the sybol we use to say that we are approaching oo from the
right instead of the left. and when you go through all the negative
numbers to get to oo, your number gets smaller and smaller.
another misconception due to rules of exponents is 0^0
0^0 = (0^n)/(0^n) = 0/0 which is indeterminite.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland,
+31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
electronic monk
Jim Carr
Jim Carr wrote:
|
| In a forged article <32CA9D...@efgh.net> ab...@efgh.net (-X) writes:
| >
| >The limit of 1/x as x --> 0 is infinity.
|
| positive or negative?
don...@sqruhs.ruhs.uwm.edu writes:
>
>i think he meant that lim 1/x as x->0 is infinitE. meaning that it is
>an infinite number, either - or + oo.
And I meant to find out if he has ever plotted 1/x, and whether
he meant that this number is both positive and negative, and in
which formal sense he used the term "the limit exists".
Owen Holden
electronic monk wrote:
>
> ibokor wrote:
> >
> > Anonymous (ab...@efgh.net) wrote:
> > : The limit of 1/x as x --> 0 is infinity.
> > :
> >
> > If you're talking about the real numbers, then
> > what you have claimed is not true.
> >
> > Even if you were to be sloppy about the use
> > of the word "limit", there would still not be
> > a limit, since if x is negative, so is 1/x
> > and if x is positive, so is 1/x. Hence the only
> > possible limit would be 0, the limit would
> > have to be non-positive and non-negative.
> >
> > d.A.
>
> x->0+
>
> lim 1/x = -oo
> x->0-
>
> hence
>
> lim 1/x = DNE
> x->0
>
> it does not exist because it is different from the left than from the
> right. but both limits ARE infinite, meaning that 1/0 is an infinite
> number, either +oo of -oo. limits were made to evaluate functions at
> infinite and indeterminante values, that is their point in existance.
>
> electronic monk
Existence is indeed the issue for 1/0. This expression has nothig to do
with limits. 1/0 cannot be defined because contradictions would result
in our reasoning. 0^0, ln(0), also do not exist and therefore can not
equal any object finite or infinite. 0/0 cannot equal anything
(certainly not 1) without admitting inconsistency into our logic.
Owen Holden (My first try at news groups...I could not resist your
topic)
ibokor
electronic monk (don...@sqruhs.ruhs.uwm.edu) wrote:
: John Briggs, VAX system manager, x4411 wrote:
:
: > The limit of 1/x as x --> 0 does not exist. At least not in the reals.
:
: that's right, it is an infinite number.
Which infinite number?
d.A.
electronic monk
it could be any one of them depending on the level of zero. in this
case lim 1/x as x-->0 = the basic infinite number. i am not familiar
with the names of them. lim 1/(x^2) as x-->0 > lim 1/x as x-->0. also
lim 2/x as x-->0 > lim 1/x as x--> 0. and so on. it depends on the
level of zero in each case. the smaller the zero, the bigger the oo.
PS in all limits, i mean from the right, the positive limit.
electronic monk
Erik Max Francis
Anonymous wrote:
> The limit of 1/x as x --> 0 is infinity.
No, it isn't. The limit does not exist.
For a limit to exist, both handed limits must exist and be the same.
lim{x -> 0-} (1/x) = -oo, but lim{x -> 0+} (1/x) = +oo.
Therefore lim{x -> 0} (1/x) does not exist.
--
Erik Max Francis | m...@alcyone.com
Alcyone Systems | http://www.alcyone.com/max/
San Jose, California | 37 20 07 N 121 53 38 W
&tSftDotIotE | R^4: the 4th R is respect
"You must surely know if man made heaven | Then man made hell"
Tim Hollebeek
In article <5ar10s$321$4...@gruvel.une.edu.au>, ibo...@metz.une.edu.au writes:
> electronic monk (don...@sqruhs.ruhs.uwm.edu) wrote:
> : John Briggs, VAX system manager, x4411 wrote:
> :
> : > The limit of 1/x as x --> 0 does not exist. At least not in the reals.
> :
> : that's right, it is an infinite number.
>
> Which infinite number?
Go to the real number line. It's the last on the right. You can't miss it.
[sorry, couldn't resist]
---------------------------------------------------------------------------
Tim Hollebeek | Disclaimer :=> Everything above is a true statement,
Electron Psychologist | for sufficiently false values of true.
Princeton University | email: t...@wfn-shop.princeton.edu
----------------------| http://wfn-shop.princeton.edu/~tim (NEW! IMPROVED!)
Norbert Kolvenbach
> Agreed.
>
> Disagreed.
>
> >systems. there are said that there are an infinite amount of numbers,
> >this just means that no matter how high you count, you'll never reach a
> >point where there are no more numbers.
> You can't count to infinity in a finite amount of time. But that doens't
> mean infinity is not a number.
>
> >electronic monk
define the inferse of infinity, concerning multiplication.
NoKo
"Careful with this VAX, Eugene!"
ibokor
electronic monk (don...@sqruhs.ruhs.uwm.edu) wrote:
: ibokor wrote:
: >
: > Which infinite number?
: >
: > d.A.
:
: it could be any one of them depending on the level of zero.
What does "the level of zero" mean?
: in this
: case lim 1/x as x-->0 = the basic infinite number.
What is "the basic infinite number"?
From my recollection of set theory etc, the smallest
infinite cardinal is \aleph_0, the cardinality of the
set of natural numbers.
The smallest infinite ordinal is \omega, the ordinality
of the same set.
How do you incorporate this into your set of real numbers,
or how do you extend the reals to incorporate them?
: i am not familiar
: with the names of them. lim 1/(x^2) as x-->0 > lim 1/x as x-->0. also
: lim 2/x as x-->0 > lim 1/x as x--> 0. and so on. it depends on the
: level of zero in each case. the smaller the zero, the bigger the oo.
:
Please explain this notin of "level of zero" How can there
be different sized zeroes?
: PS in all limits, i mean from the right, the positive limit.
:
What does that mean? How does the positive real number x tend
to infinity from the right?
d.A.
Erik Max Francis
electronic monk wrote:
> i think he meant that lim 1/x as x->0 is infinitE. meaning that it is
> an infinite number, either - or + oo.
If it's either-or, that means the limit does not exist. For a limit to
exist the handed limits must agree, and they do not (one is -oo, one is
+oo).
Erik Max Francis
electronic monk wrote:
> i do believe that -oo and +oo are the same thing.
Mathematically, they are not; there are even separate definitions for the
two. A limit of +oo means that the function increases without bound, and
-oo means that it decreases without bound. They both mean that there is no
finite limit, but similarly they are not identical.
U Lange
Erik Max Francis (m...@alcyone.com) wrote:
: electronic monk wrote:
:
: > i do believe that -oo and +oo are the same thing.
:
: Mathematically, they are not;
Depends. For the Alexandroff-compactification of the real line, there is
only one infinite point and
lim 1/x = oo (In the topology of that compact space, of course)
x->0
(Thus one might say -oo = oo in that space). There are other
compactifications with two distinct infinite points -oo and oo, and
in these topological spaces, the above limit does not exist.
: there are even separate definitions for the
: two. A limit of +oo means that the function increases without bound, and
: -oo means that it decreases without bound. They both mean that there is no
: finite limit, but similarly they are not identical.
--
Ulrich Lange Dept. of Chemical Engineering
University of Alberta
la...@gpu.srv.ualberta.ca Edmonton, Alberta, T6G 2G6, Canada
Michael Weiss
Anonymous wrote:
The limit of 1/x as x --> 0 is infinity.
Erik Max Francis <m...@alcyone.com> writes:
No, it isn't. The limit does not exist.
For a limit to exist, both handed limits must exist and be the same.
The English mathematician G.H.Hardy once wrote that the 19th century
British mathematicians didn't accomplish much compared with their
European contemporaries, because "they had not the habit of
definition".
Or to trot out a more famous quote:
"The question is who is to be master, that's all"
--Humpty Dumpty, quoted by Lewis Carroll
There are *two* common definitions of this sort of limit, both
mathematically important.
One involves completing the real line by adding *distinct* points +oo
and -oo; the structure you get is topologically equivalent to a closed
unit interval. This is concept that Erik has in mind (perhaps only
implicitly). With this structure, the function 1/x indeed has no
limit as x-->0.
(More precisely put: let R be the real line, let R+ be the extended
real line, as defined above, and let R* be the real line with the
point 0 omitted. So f(x)=1/x is a continuous function from R* to R.
Then f(x) has no extension to a continuous function from R to R+.
Don't you just love precision?)
But you can *also* complete the real line by adding a *single* point
oo. You get a structure topologically equivalent to a circle. In
this context, the limit of 1/x as x-->0 is indeed oo.
(I.e., with *this* definition of R+, f(x) *can* be continuously
extended to a function from R to R+, and even to a function from R+ to
itself. Just set f(oo)=0 and f(0)=oo.)
The second definition looks even more natural if you extend it to the
complex plane. Adding a single point "at infinity" gives you,
topologically speaking, a sphere. This is known as the Riemann
sphere, and plays a major role in complex analysis. For one thing,
any rational function gives a well-defined mapping from the Riemann
sphere into itself. This is pretty damn convenient.
Or if you want a physical example: one sometime hears of negative
temperatures, which are (we are told) *hotter* than any positive
temperature. What's going on?
Well, in statistical thermodynamics, it turns out that the important
parameter is not T, but 1/T. For some systems, you can raise their
temperature (suitably defined) to infinity with a finite influx of
energy. So 1/T goes to 0 through positive values. Add a little
*more* energy, and 1/T becomes negative, as does T.
This all looks very natural if you set up your definitions so that the
limit of 1/T as T-->0 is oo.
Moral: left as an exercise to the reader.
electronic monk
Erik Max Francis wrote:
>
> electronic monk wrote:
>
> > i do believe that -oo and +oo are the same thing.
>
the
> two. A limit of +oo means that the function increases without bound,
and
> -oo means that it decreases without bound. They both mean that there
is no
> finite limit, but similarly they are not identical.
think of the xy plane as a sphere instead. and make (0,0) one pole and
(oo,oo) the other pole. now, in fact, the point (oo,oo) is actualy
infinitely big, but we are going to think of it as one point. if a line
were to start at the origin and angle out in any direction, it would
approach infinity, but, if it were to approach it from the bottom or
around the left of the sphere, we would say that it was approaching -oo,
but instead, we can say that it is actualy approaching oo- (from the
left). just think about it. all conics can be thought of as ellipses.
a parabola is an ellipse with one focal point at infinity. a hyperbola
can be thought of as an ellipse wraped aroung the spherical plane
backwards. this is a true idea, don't get me wrong. i'm not nearly the
first to mention it. but that is what i mean when i say i think that
they are the same thing.
>
> --
> Erik Max Francis | m...@alcyone.com
> Alcyone Systems |
http://www.alcyone.com/max/
> San Jose, California | 37 20 07 N 121 53 38
W
> &tSftDotIotE | R^4: the 4th R is
respect
> "You must surely know if man made heaven | Then man made hell"
electronic monk
Darrell Ryan
electronic monk wrote:
>
>
> like i said before, we can think of zero having levels in the same way
> that we can think of infinity as having levels. if we think of zero
> only as a limit, then it will make sense. x^2 > x for all x>0,
Huh? Since when is (1/2)^2 > 1/2 ?????
> and then we could say 1/(x^2) > 1/x for all x>0.
Huh? Since when is 1/(2^2) > 1/2 ?????
>so, lim 1/(x^2) as x-->0+ > lim 1/x as x-->0+.
Not so. lim x-->0+ [1/(x^2)] = +infinity (the limit does not exist.
*Even if* lim x-->0- = +infinity, which it does)
and lim x-->0+ (1/x) = +infinity (the limit does not exist)
Remember, when we write +infinity or -infinity as an answer to a limit
problem, we are really saying, "The limit does not exist." The infinity
part tells us the reason *why* the limit does not exist.
Of course, none of this really has anything to do with why 1/0 is
undefined. What number can you multiply by 0 and get an answer of 1?
Infinity? Nada. Infinity is not even a number. 'Nuff said.
electronic monk
ibokor wrote:
>
> electronic monk (don...@sqruhs.ruhs.uwm.edu) wrote:
> : ibokor wrote:
> : >
> : > Which infinite number?
> : >
> : > d.A.
> :
> : it could be any one of them depending on the level of zero.
>
> What does "the level of zero" mean?
i'm using zero as a limit. the line y = x approaches zero at the same
rate that x approaches zero. on the other hand, the line y = 2x
approaches zero twice as fast as x approaches zero. therefor we can
think of
lim 2x as x-->0+ > lim x as x-->0+ because 2x > x for all x>0. your
right in saying that 0=0, because that seems obvious, but if 0 was
always equal to 0 then wouldn't 0/0 = 1?
>
> : in this
> : case lim 1/x as x-->0 = the basic infinite number .
>
> What is "the basic infinite number"?
when i say "basic infinite number" i mean the infinity with a rate of
increase of one to one. for instance the line y = x approaches infinite
at the same rate x approaches infinity. 1/x approches oo and the same
rate that x approaches zero. 2x approaches oo at twice the rate that x
approaches oo. i called this one to one relationship with x the basic
infinite number just cause it sounded nice.
>
> From my recollection of set theory etc, the smallest
> infinite cardinal is \aleph_0, the cardinality of the
> set of natural numbers.
>
> The smallest infinite ordinal is \omega, the ordinality
> of the same set.
how can you define the smallest infinite number? that would be like
trying to find the smallest finite number before zero, or finding the
number "right next to" another number on a number line.
>
> How do you incorporate this into your set of real numbers,
> or how do you extend the reals to incorporate them?
>
> : i am not familiar
> : with the names of them. lim 1/(x^2) as x-->0 > lim 1/x as x-->0.
also
> : lim 2/x as x-->0 > lim 1/x as x--> 0. and so on. it depends on
the
> : level of zero in each case. the smaller the zero, the bigger the
oo.
> :
>
> Please explain this notin of "level of zero" How can there
> be different sized zeroes?
like i said before, we can think of zero having levels in the same way
that we can think of infinity as having levels. if we think of zero
only as a limit, then it will make sense. x^2 > x for all x>0, and then
we could say 1/(x^2) > 1/x for all x>0. so, lim 1/(x^2) as x-->0+ >
lim 1/x as x-->0+. this means simply that x^2 approaches zero much
faster than x does, so when i say level of zero, i just mean how fast a
function approaches zero. but, like i said before, if 0 always equaled
0, then 0/0 = 1, which it doesn't.
>
> : PS in all limits, i mean from the right, the positive limit.
> :
>
> What does that mean? How does the positive real number x tend
> to infinity from the right?
i mean that x approaches zero from the right in all my limits.
>
> d.A.
electronic monk
electronic monk
Norbert Kolvenbach wrote:
> If infinity is a number, please define the Operations (+,-,*),
> define the inferse of infinity, concerning multiplication.
infinity's inverse is zero because lim 1/x as x-->oo = 0
>
> NoKo
> "Careful with this VAX, Eugene!"
electronic monk
electronic monk
Darrell Ryan wrote:
>
> electronic monk wrote:
> >
> >
> > like i said before, we can think of zero having levels in the same
way
> > that we can think of infinity as having levels. if we think of
zero
> > only as a limit, then it will make sense. x^2 > x for all x>0,
>
whoops, haha, meant to say 2x instead of x^2.
> >so, lim 1/(x^2) as x-->0+ > lim 1/x as x-->0+.
>
> Not so. lim x-->0+ [1/(x^2)] = +infinity (the limit does not exist.
> *Even if* lim x-->0- = +infinity, which it does)
>
> and lim x-->0+ (1/x) = +infinity (the limit does not exist)
>
> Remember, when we write +infinity or -infinity as an answer to a
limit
> problem, we are really saying, "The limit does not exist." The
infinity
> part tells us the reason *why* the limit does not exist.
no, a limit does not exist when the right hand limit differs from the
left hand one. oo is a perfectly good answer for a limit. lim 1/(x^2)
as x-->0 IS oo, because lim 1/(x^2) as x-->0+ = lim 1/(x^2) as x-->0-.
we can say that the number _approaches_ infinity. we never actualy
refer to it _as_ infinity. but again, i mean to say 2x instead of
x^2...sorry.
>
> Of course, none of this really has anything to do with why 1/0 is
> undefined. What number can you multiply by 0 and get an answer of 1?
> Infinity? Nada. Infinity is not even a number. 'Nuff said.
yer right though, oo is not a number. it is a concept and can only be
refered to as a limit.
>
> ____________________________________________________________
> Darrell Ryan
> e-mail dr...@edge.net
> personal website http://edge.edge.net/~dryan
> company website http://www.edge.net/stmc
electronic monk
Darrell Ryan
>
> Rubbish.
I second.
>
> "b" is the multiplicative inverse of "a" _if and only if_ a*b=b*a=1.
Now there's a statement I can believe in!
>
> So do you claim that 0*infinity = 1? If not, then infinity is _not_ the
> inverse of zero. I am sorry, but I have to say it: This absolutely
> pointless dicussion is mainly driven by lack of knowledge about basic
> algebraic concepts.
Agreed. I would add that a lack of knowledge is one thing, but making
up "definitions" for the things you don't understand is the real
fallacy. It is good to wonder about concepts like infinity, division by
zero being undefined, and the like, but I have *never* heard or read any
"definitions" like the ones posted in this thread. Sorry, but just
because it sounds good does not make it so.
One final attempt to explain it:
The graph of y = 1/x has a vertical asymptote at x = 0. This means
there is *no point* on the graph when x = 0. We can talk about the
limit as x approaches 0 all we want, but this has *nothing* to do with
the graph when x = 0. "Approaches" and "equals" are two very different
things.
Remember from CAL I class? The existence or non-existence of f(x) at x
= c has *no bearing* on the existence or non-existence of the LIMIT of
f(x) as x APPROACHES c. It appears to me (IMHO) that electronic monk
believes that the limit of f(x) as x approaches c is the same thing as
f(c). As we know, it often is (depending on the function) but is is
generally *not*. What I'm trying to say is this: 1/x at x = 0 is NOT
infinity just because lim x-->0+ is. That is gobbly goo.
As a side note, I reiterate that the statement "lim x-->0+ (1/x) =
+infinity" implies THE LIMIT DOES NOT EXIST. The "+infinity" part tells
us *why* it does not exist. The closer x gets to 0 (from the right),
the BIGGER y gets. The y value does not "approach" anything. It
increases WITHOUT BOUND.
John Briggs, VAX system manager, x4411
In article <32D29551...@alcyone.com>, Erik Max Francis <m...@alcyone.com> writes:
> electronic monk wrote:
>
>> i do believe that -oo and +oo are the same thing.
>
> Mathematically, they are not; there are even separate definitions for the
> two. A limit of +oo means that the function increases without bound, and
> -oo means that it decreases without bound. They both mean that there is no
> finite limit, but similarly they are not identical.
It all depends on the set of definitions that you use.
1. -oo and +oo can be identical.
2. -oo and +oo can be different.
3. -oo and +oo can be non-existent.
Try projecting the real line onto a circle and see 1.
Try projecting the real line onto a closed interval and see 2.
Don't bother projecting the real line and see 3.
Is it too much to ask that people state their definitions before making
unsupported claims about 1/0, limits and infinities?
John Briggs vax...@alpha.vitro.com
U Lange
electronic monk (don...@sqruhs.ruhs.uwm.edu) wrote:
: infinity's inverse is zero because lim 1/x as x-->oo = 0
Rubbish.
"b" is the multiplicative inverse of "a" _if and only if_ a*b=b*a=1.
So do you claim that 0*infinity = 1? If not, then infinity is _not_ the
inverse of zero. I am sorry, but I have to say it: This absolutely
pointless dicussion is mainly driven by lack of knowledge about basic
algebraic concepts.
BlackTopPilot
electronic monk wrote, respondingo to:
> > define the inferse of infinity, concerning multiplication.
>
> infinity's inverse is zero because lim 1/x as x-->oo = 0
Great! This is just what I've been looking for... a number wherein I can
multiply it by it's inverse and get something other than the set's unique
identity.
eg: (the limit of one over x, as x gets as large as you wish)*zero = er,
uh, z e r o .....
let's try another definition that suits our needs . . . .
<Hd,+> BlackTopPilot
David Kastrup
electronic monk <don...@sqruhs.ruhs.uwm.edu> writes:
> Norbert Kolvenbach wrote:
>
> > If infinity is a number, please define the Operations (+,-,*),
> > define the inferse of infinity, concerning multiplication.
>
> infinity's inverse is zero because lim 1/x as x-->oo =3D 0
Ah, the old truth: every function is forced to be equal to its limits
at all of its points, so every reasoning about limits is a reasoning
about values.
I encounter this reasoning so often that I could just throw up... my
hands in disgust.
--
David Kastrup Phone: +49-234-700-5570
Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
Institut f=FCr Neuroinformatik, Universit=E4tsstr. 150, 44780 Bochum, Germa=
ny
John
John Briggs, VAX system manager, x4411 wrote:
>
0*PI = 0*1997
and
PI = 1997
Quod Erat Unhandy!!!
electronic monk
BlackTopPilot wrote:
> eg: (the limit of one over x, as x gets as large as you wish)*zero =
er,
> uh, z e r o .....
actualy,
lim (x * 1/x) = oo * 0 = 1
x-->oo
>
> <Hd,+> BlackTopPilot
electronic monk
electronic monk
David Kastrup wrote:
> > infinity's inverse is zero because lim 1/x as x-->oo = 0
>
> Ah, the old truth: every function is forced to be equal to its limits
> at all of its points, so every reasoning about limits is a reasoning
> about values.
i cannot use oo anywhere outside of the limit sign, i must use limits.
and you do know that 1/x approaches zero, and if it were to get to
infinity, it would equal zero. just the same as .99999[repeat] equals
one if you were to calculate all it's digits to infinity.
> I encounter this reasoning so often that I could just throw up... my
> hands in disgust.
i want you to prove to me that lim 1/x as x-->oo does NOT equal zero.
only then can you "throw your hands up in disgust."
> --
> David Kastrup Phone: +49-234-700-5570
> Email: d...@neuroinformatik.ruhr-uni-bochum.de Fax: +49-234-709-4209
> Institut für Neuroinformatik, Universitätsstr. 150, 44780 Bochum, Germany
electronic monk
Darrell Ryan
electronic monk wrote:
>
> i want you to prove to me that lim 1/x as x-->oo does NOT equal zero.
> only then can you "throw your hands up in disgust."
Yes. lim x-->oo (1/x) is indeed 0. This is not a rigid "proof," but
one can easily see this by realizing that the graph of 1/x has a
horizontal asymptote at y = 0. You can graph it and evaluate it for
large values of x and, sure enough, y gets closer to zero. Guess how
you find horizontal asymptotes (analytically)? You guessed it. Take
the limit as x approaches +infinity and -infinity.
But, at the same time, I have not read anything that suggests that
anyone disagrees with that. IMHO, the disagreements are with your
suggestions that "infinity's inverse is zero because lim 1/x as x-->oo =
0" and statements to the effect that 1/0 is defined as oo.
Norbert Kolvenbach
electronic monk wrote:
>
> Norbert Kolvenbach wrote:
>
> > If infinity is a number, please define the Operations (+,-,*),
> > define the inferse of infinity, concerning multiplication.
>
>
> >
> > "Careful with this VAX, Eugene!"
>
> electronic monk
WWWOOUUUU!!! - So by now we have 0^(-1) = oo and 0 * oo = 1 !
That´s good. Enuf for today... (hahahah...)
NoKo
"Careful with that VAX, Eugene!"
Dik T. Winter
In article <32D587...@sqruhs.ruhs.uwm.edu> don...@sqruhs.ruhs.uwm.edu writes:
> > eg: (the limit of one over x, as x gets as large as you wish)*zero =
> er,
> > uh, z e r o .....
>
> actualy,
>
> lim (x * 1/x) = oo * 0 = 1
> x-->oo
Better read correctly, he was writing about:
lim (1/x * 0)
x-->oo
--
dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
Paul Schlyter
In article <32D587...@sqruhs.ruhs.uwm.edu>,
electronic monk <don...@sqruhs.ruhs.uwm.edu> wrote:
>BlackTopPilot wrote:
>
> > eg: (the limit of one over x, as x gets as large as you wish)*zero =
>er,
> > uh, z e r o .....
>
>actualy,
>
>lim (x * 1/x) = oo * 0 = 1
>x-->oo
1 = oo * 0 = oo*(0 + 0) = oo*0 + oo*0 = 1 + 1 = 2
Q.E.D.
--
----------------------------------------------------------------
Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
Grev Turegatan 40, S-114 38 Stockholm, SWEDEN
e-mail: pau...@saaf.se p...@net.ausys.se pa...@inorbit.com
WWW: http://www.raditex.se/~pausch/ http://spitfire.ausys.se:8003/psr/
Mike McCarty
In article <E3IHG...@cwi.nl>, Dik T. Winter <d...@cwi.nl> wrote:
)In article <32CC49...@sqruhs.ruhs.uwm.edu> don...@sqruhs.ruhs.uwm.edu writes:
) > that is true. lim 1/x as x->0 does not exist and is considered to be
) > either -oo or +oo.
)
)Or simply not distinguish -oo and +oo. To distinguish them may lead to
)severe problems when you are switching to complex numbers. The fact that
)at one stage IEEE switched in their definition of floating-point numbers
)from a projective infinity to a dual infinity leads to severe problems
)stating basic properties about complex f-p math. That is, unless you
)switch to a polar representation of complex quantities, but polar complex
)math is numerically much less stable than an euclidian equivalent.
Numerically less stable for what operations? It depends on application,
you know.
Mike
--
----
char *p="char *p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}
This message made from 100% recycled bits.
I don't speak for DSC. <- They make me say that.
Dominique Bernardi
In article (Dans l'article) <32C6530D...@sci-log.apana.org.au>, Carl
Renneberg <renn...@sci-log.apana.org.au> wrote (écrivait) :
>The power set of the integers has a size equal to another
>transfinite number called aleph one (another Hebrew symbol).
Well, not quite. Aleph one is defined to be the smallest cardinality larger
than aleph zero. The continuum hypothesis says that it is in fact the
cardinality of the power set of the integers, but it has been shown that
the continuum hypothesis is undecidable in ZFC.
--
Dominique Bernardi, Théorie des Nombres
Université Pierre et Marie Curie
4 place Jussieu - F75005 Paris Tel (33-1) 44275441
bern...@math.jussieu.fr
Dave Seaman
In article <32D03E...@idirect.com>,
Owen Holden <oho...@idirect.com> wrote:
>Existence is indeed the issue for 1/0. This expression has nothig to do
>with limits. 1/0 cannot be defined because contradictions would result
>in our reasoning. 0^0, ln(0), also do not exist and therefore can not
>equal any object finite or infinite. 0/0 cannot equal anything
>(certainly not 1) without admitting inconsistency into our logic.
One nit. The expression 0^0 does indeed have a value, which has
nothing to do with limits. This is discusses in the sci.math FAQ,
which gives several reasons why 0^0 = 1, at least in the case where the
exponent is considered to be an integer.
My favorite reason: 0^0 is the cardinality of the class of functions
mapping the empty set to itself, which is one.
Dave Seaman
--
Dave Seaman dse...@purdue.edu
++++ stop the execution of Mumia Abu-Jamal ++++
++++ if you agree copy these lines to your sig ++++
++++ see http://www.xs4all.nl/~tank/spg-l/sigaction.htm ++++
electronic monk
Paul Schlyter wrote:
>
> In article <32D587...@sqruhs.ruhs.uwm.edu>,
> electronic monk <don...@sqruhs.ruhs.uwm.edu> wrote:
> >BlackTopPilot wrote:
> >
> > > eg: (the limit of one over x, as x gets as large as you
wish)*zero =
> >er,
> > > uh, z e r o .....
> >
> >actualy,
> >
> >lim (x * 1/x) = oo * 0 = 1
> >x-->oo
>
> 1 = oo * 0 = oo*(0 + 0) = oo*0 + oo*0 = 1 + 1 = 2
>
> Q.E.D.
what i mean is that:
lim (x * 1/x) = 1
x->oo
lim (x * 1/x) = 0 * oo
x->oo
so 0*oo can equal one.
>
> --
> ----------------------------------------------------------------
> Paul Schlyter, Swedish Amateur Astronomer's Society (SAAF)
> Grev Turegatan 40, S-114 38 Stockholm, SWEDEN
> e-mail: pau...@saaf.se p...@net.ausys.se pa...@inorbit.com
> WWW: http://www.raditex.se/~pausch/
http://spitfire.ausys.se:8003/psr/
electronic monk
Darrell Ryan
electronic monk wrote:
>
> Paul Schlyter wrote:
> >
> > In article <32D587...@sqruhs.ruhs.uwm.edu>,
> > electronic monk <don...@sqruhs.ruhs.uwm.edu> wrote:
> > >BlackTopPilot wrote:
> > >
> > > > eg: (the limit of one over x, as x gets as large as you
> wish)*zero =
> > >er,
> > > > uh, z e r o .....
> > >
> > >actualy,
> > >
> > >lim (x * 1/x) = oo * 0 = 1
> > >x-->oo
> >
> > 1 = oo * 0 = oo*(0 + 0) = oo*0 + oo*0 = 1 + 1 = 2
1 = 2? That's a contradiction. 0*oo is an indeterminate form.
> >
> > Q.E.D.
>
> what i mean is that:
>
> lim (x * 1/x) = 1
> x->oo
>
> lim (x * 1/x) = 0 * oo
> x->oo
>
> so 0*oo can equal one.
>
No it can't. At least not based on that reasoning. 0*oo is an
*indeterminate* form, therefore we cannot make any conclusions about
what the limit is by looking at that form.
If I were evaluating this limit, I would simply cancel the x's and get:
lim x-->oo (1)
which is 1.
But if you insist on taking the limits of the two factors individualy,
it is done like this:
lim x-->oo [ x * (1/x) ]
Direct substitution yields oo*0. This is an indeterminate form, so use
L'Hopital's Rule.
But in order to apply L'Hopitals Rule, we need the indeterminate form
0/0 or oo/oo. So we end up multiplying the two factors anyway to get
x/x. Direct substitution now yields the indeterminate form oo/oo so now
we can apply L'Hopital's Rule. Taking the derivative of the numerator
and denominator (of x/x) produces:
lim x-->oo (1/1)
which is 1.
So, yes it is true that the limit is 1. But *not* because 0*oo = 1.
Try evaluating:
lim x-->oo [ (e^-x)*(sqrt x) ]
You will also get 0*oo, but after re-writing the expression where it
fits the indeterminate form oo/oo and applying L'Hopital's Rule, you
will see that this limit is 0. Does that mean that 0*oo is now 0
instead of 1? I think not. This illustrates why 0*oo is an
indeterminate form.
Jim Hunter
electronic monk wrote:
>
> ...
> that was what he was saying, and that equals zero. the problmem with
> that is that he should of expressed 0 as a limit.
> ...
The problem with this whole concept is that it has _nothing_ to do with
the _algebraic_ reasons why 1/0 is not definable consistently.
What sort of logical monstrosity is being constructed here? Doesn't this
seem to be a little "unbalanced", combining the few algebraic axioms
needed for a field with the many topological axioms and theorems needed
to
derive the idea of convergence. Don't you think we would be a little
cruel to our children to make them learn the theory of Banach spaces
before they can divide.
Anyway, there's a simpler way do this. What you're going to
end up with this derivation is logically equivalent to the real numbers
on the closed interval [-1,1]. 1 will play the role of what you're
trying to define as oo, and -1 corresponds to -oo. Make it easier on
yourself and work with this interval. That is develop an _algebra_
on this interval.
---
Jim
electronic monk
Dik T. Winter wrote:
>
> In article <32D587...@sqruhs.ruhs.uwm.edu>
don...@sqruhs.ruhs.uwm.edu writes:
wish)*zero =
> > er,
> > > uh, z e r o .....
> >
> > actualy,
> >
> > lim (x * 1/x) = oo * 0 = 1
> > x-->oo
>
> lim (1/x * 0)
> x-->oo
you mean as x-->0 or you mean lim [x-->oo] (x * 0). but true enough,
that was what he was saying, and that equals zero. the problmem with
that is that he should of expressed 0 as a limit. this is because since
we can never get to infinity, we must also never be able to reach zero.
i expressed the limit this way.
> --
> dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland,
+31205924131
> home: bovenover 215, 1025 jn amsterdam, nederland;
http://www.cwi.nl/~dik/
electronic monk