[25354] in Perl-Users-Digest
Perl-Users Digest, Issue: 7597 Volume: 10
daemon@ATHENA.MIT.EDU (Perl-Users Digest)
Mon Jan 3 21:53:37 2005
Date: Mon, 3 Jan 2005 15:21:49 -0800 (PST)
From: Perl-Users Digest <Perl-Users-Request@ruby.OCE.ORST.EDU>
To: Perl-Users@ruby.OCE.ORST.EDU (Perl-Users Digest)
Perl-Users Digest Mon, 3 Jan 2005 Volume: 10 Number: 7597
Today's topics:
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <george@briar.demon.co.uk>
Re: Is zero even or odd? <george@briar.demon.co.uk>
Re: Is zero even or odd? <dak@gnu.org>
Re: Is zero even or odd? <jmw@jmwa.demon.contraspam.yuk>
Re: Is zero even or odd? <jmw@jmwa.demon.contraspam.yuk>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <jmw@jmwa.demon.contraspam.yuk>
Re: Is zero even or odd? (Matthew Russotto)
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <dak@gnu.org>
Re: Is zero even or odd? <george_coxanti@spambtinternet.com.invalid>
Re: Is zero even or odd? <george_coxanti@spambtinternet.com.invalid>
Re: Is zero even or odd? <dak@gnu.org>
Re: Is zero even or odd? <jmw@jmwa.demon.contraspam.yuk>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? <jfields@austininstruments.com>
Re: Is zero even or odd? mmeron@cars3.uchicago.edu
Re: Is zero even or odd? mmeron@cars3.uchicago.edu
Re: Is zero even or odd? <dseaman@no.such.host>
Re: Is zero even or odd? <george@briar.demon.co.uk>
Re: Is zero even or odd? <george@briar.demon.co.uk>
Re: Is zero even or odd? <george_coxanti@spambtinternet.com.invalid>
Re: Is zero even or odd? <see@sig.com>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <see@sig.com>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <hadrainc@earthlink.net>
Re: Is zero even or odd? <see@sig.com>
Re: Is zero even or odd? <tilford@ugcs.caltech.edu>
Re: Is zero even or odd? <a.newmane.remove@eastcoastcz.com>
Re: Is zero even or odd? <a.newmane.remove@eastcoastcz.com>
Re: Is zero even or odd? <a.newmane.remove@eastcoastcz.com>
Re: Is zero even or odd? <a.newmane.remove@eastcoastcz.com>
Digest Administrivia (Last modified: 6 Apr 01) (Perl-Users-Digest Admin)
----------------------------------------------------------------------
Date: Wed, 29 Dec 2004 11:48:52 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <nsl5t0lsbrmrrqptc4opo7ifk2vh77v35s@4ax.com>
On Wed, 29 Dec 2004 13:38:59 +0100, Michael Mendelsohn
<invalid@msgid.michael.mendelsohn.de> wrote:
>John Fields schrieb:
>> On Wed, 29 Dec 2004 03:03:12 +0000 (UTC), George Cox
>> >John Fields wrote:
>> >>
>> >> x
>> >> y = lim x->0 --- = 1
>> >> x
>> >
>> >This is true, but it is true because
>> >
>> > x
>> > y = lim x->0 --- = lim x->0 1 = 1.
>> > x
>> >
>> >The "1" on the left of the equals sign is there because the two x's
>> >cancelled.
>>
>> ---
>> And if those x's were zeroes when they were cancelled, that still
>> results in a quotient of 1, so 0/0 = 1
>
>The point is that you can't cancel them when x may be 0, because
>division by 0 is undefined.
---
And there's the rub! My point is that it _should_ be defined, and
defined like this:
0 1
--- = 1, --- = oo,
0 0
---
>However, in the lim x->0 case x approaches
>0, but never really equals zero, so the cancellation may be performed.
>
>> >does not allow you to conclude that 0/0 = 1.
---
I disagree.
If we say
x
y = lim --- = 1
x +oo -> -oo x
Then y will be equal to 1 for every instance of x except when x
crosses over from + to -?
In my view that's preposterous, and the mere parroting of "division by
zero is disallowed because it's undefined" a convenient dodge. No
insult intended.
---
>> ---
>> It does if the case where X = 0 can result in a cancellation because
>> the numerator and denominator were equal and a quotient of 1 resulted
>> because of that cancellation.
>
>It is desirable for mathematics not to allow two contradictory
>statements such as 0/0=1 and 0/0=2 to be true at the same time; and your
>resoning allows these statements to be true in some cases (we did this
>in the Ohm's law episode, where you never said my reasoning was wrong,
>it was merely not the thing you wanted to prove).
---
Actually, what I didn't want to do was to get into a long, off-topic
harangue about Ohm's law (Ohm's formula, actually. Ohm's law is an
entirely different thing and is used to determine whether conductors
are 'Ohmic'. That is, if their resistance remains constant when
current through them is made to vary), which requires that for
resistance to be _measured_ a known voltage _must_ be placed across it
and the resulting charge flowing through the resistance determined.
Your example eschewed the _practical_ requirements in order to
contrive a desired outcome and, as such, was irrelevant.
But, proceeding along that tack for a while, maybe we can use our
"exquisite" technology to advantage here by assuming that we have
managed to craft two identical zero ohm resistors (superconducting, if
you like) and that we can force equal numbers of electrons through
each of them in equal amounts of time. Then, since Q = It, the
current flowing in each resistor will be I = Q/t, and the ratio of the
currents will be I1/I2.
Since Q/t will be identical for both resistors, I1/I2 will be 1 for
any current. Now, let's say that we inject fewer and fewer pairs of
electrons per unit time into the 'rig' and that eventually we inject
none. Since we have agreed that as long as I1=I2 then I1/I2 will be
equal to 1, does 0/0 not satisfy that requirement?
---
>0/0=1 and 0/0=2 leads to 1=2, which is undesirable.
>
>lim x->0 x/x = 1 and lim x->0 2x/x = 2 doesn't have this problem.
>It is downright fine and dandy, because
>2 lim x->0 x/x = lim x->0 2x/x
>
>It is more inconvenient to write, so maybe you could agree to rephrasing
>that 0/0=1 where 0/0 is shorthand for lim x->0 x/x ?
>(You would decidedly be in a minority if you used that shorthand).
---
I don't have a problem with either, but it seems to be skirting that
taboo point where x = 0.
Interesting thread...
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 17:50:50 -0000
From: "George Dishman" <george@briar.demon.co.uk>
Subject: Re: Is zero even or odd?
Message-Id: <cquqnq$mq9$1@news.freedom2surf.net>
"vonroach" <hadrainc@earthlink.net> wrote in message
news:d9g5t0dcbv3f9g7u3hn142no3i8mrkkpve@4ax.com...
> On Tue, 28 Dec 2004 22:16:07 -0000, "George Dishman"
> <george@briar.demon.co.uk> wrote:
>
>>The task at hand is to show whether the rule
>>is valid or otherwise. Can you do that?
>
> Otherwise? Accept your opinion? Crazy George is really crazy!
I'll take that as a "No" then.
George
------------------------------
Date: Wed, 29 Dec 2004 18:14:03 -0000
From: "George Dishman" <george@briar.demon.co.uk>
Subject: Re: Is zero even or odd?
Message-Id: <cqus3b$n5r$1@news.freedom2surf.net>
"John Fields" <jfields@austininstruments.com> wrote in message
news:uhn3t0h1kar5l2kbiu5dj5id3bgk5rt54k@4ax.com...
> On Tue, 28 Dec 2004 22:13:54 -0000, "George Dishman"
> <george@briar.demon.co.uk> wrote:
<snip>
>>> For the normal order of precedence, my argument is that in order for
>>> x/x = 1 to be true, the numerator and the denominator must both be
>>> equal at the time of the division.
>>
>>So far you are assuming the result you are
>>trying to prove, that the ratio of numerator
>>and denominator is to be 1.
>>
>>> That is, for any set of x's where
>>> x = x, x/x = 1. If that's true, and 0 = 0, then 0/0 must be equal to
>>> 1. The proof is the proof of exhaustion.
>>
>>I like that however while the statement 0=0
>>is obviously true, it isn't exhaustive. We
>>also have
>>
>> 0 = k * 0
>>
>>hence by your method
>>
>> 0/0 = limit as x->0 of (k*x)/x
>>
>>hence
>>
>> 0/0 = k for all constant k
>>
>>You still fail to prove the claim of
>>uniqueness for the value of 1.
>
> ---
> Ah, but :-) the normal order of precedence dictates that the
> multiplication be performed first,
That is incorrect, multiplication and division
are of equal precedence.
> so my method first reduces k*0 to 0
> by virtue of the multiplication, then the division by zero is
> performed to yield the ratio of 1.
>
>
> (k*0) -> 0
> ------ --- = 1
> 0 -> 0
>
> Interestingly, your method also requires the quotient of 0/0 to be 1,
> otherwise the multiplication by k wouldn't yield k as the product!
Not at all. I have avoided writing this as a limit
for the reason Mati gives but will follow your lead
since you wrote:
> .. That, I believe, proves my point, my point being:
>
> x
> y = lim x->0 --- = 1
> x
Let
k*x
y(x) = ---
x
Clearly y(x) = k for all non-zero finite values of x and
limit y(x)
x->0 ---- = k
x
By your argument that the limit can be used to
define the value at zero, we can infer:
y(0)
---- = k
0
but of course y(0) = 0 hence
0
--- = k
0
for all finite k.
George
------------------------------
Date: Wed, 29 Dec 2004 19:22:22 +0100
From: David Kastrup <dak@gnu.org>
Subject: Re: Is zero even or odd?
Message-Id: <x5is6lrmld.fsf@lola.goethe.zz>
John Fields <jfields@austininstruments.com> writes:
> On Wed, 29 Dec 2004 13:38:59 +0100, Michael Mendelsohn
> <invalid@msgid.michael.mendelsohn.de> wrote:
>
>>The point is that you can't cancel them when x may be 0, because
>>division by 0 is undefined.
>
> ---
> And there's the rub! My point is that it _should_ be defined, and
> defined like this:
>
>
> 0 1
> --- = 1, --- = oo,
> 0 0
And that's nonsensical.
It gives us, for example,
1 = 0/0 = (0+0)/0 = (0/0) + (0/0) = 2
>>However, in the lim x->0 case x approaches 0, but never really
>>equals zero, so the cancellation may be performed.
>>
>>> >does not allow you to conclude that 0/0 = 1.
>
> ---
> I disagree.
Well, so you are talking inconsistent nonsense. You must not expect
that it will impress anybody much.
> If we say
>
> x
> y = lim --- = 1
> x +oo -> -oo x
This notation is complete and utter meaningless hogwash. What you
_can_ say is that
lim (x->v) (x/x) = 1
for _all_ v in R (and actually also if v is +oo or -oo, which is not a
number, but a neighborhood in some sense of the word).
But you can equally well say that
lim (x->v) (2*x/x) = 2
for _all_ v in R. For v=0, both limits are of the _form_ 0/0, and
that means that any such limiting _form_ is undefined without further
qualification. But you don't need limiting arguments to show that 0/0
can't be defined: mere algebra is sufficient.
> Then y will be equal to 1 for every instance of x except when x
> crosses over from + to -?
Your notation above is rubbish to start with, so assigning any meaning
to it is likely to be rubbish too.
> In my view that's preposterous, and the mere parroting of "division
> by zero is disallowed because it's undefined" a convenient dodge.
Well, not being able to choose a consistent value is _the_ perfect
reason for leaving it undefined. You call it a dodge, other's call it
a necessity.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
------------------------------
Date: Wed, 29 Dec 2004 18:24:27 +0000
From: John Woodgate <jmw@jmwa.demon.contraspam.yuk>
Subject: Re: Is zero even or odd?
Message-Id: <1NnIFqCbZv0BFwJ+@jmwa.demon.co.uk>
I read in sci.electronics.design that John Fields <jfields@austininstrum
ents.com> wrote (in <nsl5t0lsbrmrrqptc4opo7ifk2vh77v35s@4ax.com>) about
'Is zero even or odd?', on Wed, 29 Dec 2004:
>I don't have a problem with either, but it seems to be skirting that
>taboo point where x = 0.
I think this thread has shown that there are two sorts of people, those
that associate taboos with 0 and those that don't.
My view is that logically-inferred values of expressions involving zero
should be accepted unless they result in contradictions. I do not
support a priori restrictions on interpretation, and I do not accept
that 'undefined' is a valid show-stopper.
--
Regards, John Woodgate, OOO - Own Opinions Only.
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk
------------------------------
Date: Wed, 29 Dec 2004 18:26:13 +0000
From: John Woodgate <jmw@jmwa.demon.contraspam.yuk>
Subject: Re: Is zero even or odd?
Message-Id: <wtwJ93CFbv0BFwLl@jmwa.demon.co.uk>
I read in sci.electronics.design that George Dishman
<george@briar.demon.co.uk> wrote (in <cqus3b$n5r$1@news.freedom2surf.net
>) about 'Is zero even or odd?', on Wed, 29 Dec 2004:
> 0
> --- = k
> 0
>
>for all finite k.
Yes. John Fields' value of 1 is just ONE valid solution. Not wrong, but
not the whole solution.
--
Regards, John Woodgate, OOO - Own Opinions Only.
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk
------------------------------
Date: Wed, 29 Dec 2004 12:48:27 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <ljr5t0116eo1l2s73egemhtl5ohg2mrgf0@4ax.com>
On Wed, 29 Dec 2004 13:58:54 +0100, Michael Mendelsohn
<invalid@msgid.michael.mendelsohn.de> wrote:
>John Fields schrieb:
>> On Mon, 27 Dec 2004 20:58:44 +0100, Michael Mendelsohn
>> <invalid@msgid.michael.mendelsohn.de> wrote:
>>
>> >Below, you remove the short from my diagram.
>> >However, you also remove the power supply, which achieves the same
>> >thing.
>>
>> ---
>> I don't know what you mean, since the + and - terminals are there and
>> I refer to the voltage across the resistance as being 1V.
>
>You eventually lower the voltage to 0V.
>That's what I achieved with the short.
>
>> >> The proper circuit:
>> >>
>> >> +---(V)---+
>> >> | |
>> >> (-)---o---[R]---o---(A)---o---(+)
>> >>
>> >> Will yield the proper results if examined using Ohm's law.
>> >>
>> >> Assuming that the voltage across the resistance is 1V and the current
>> >> through it is 1A, then the resistance will be:
>> >>
>> >> E 1V
>> >> R = --- = ---- = 1 ohm (1)
>> >> I 1A
>> >>
>> >
>> >Assuming that the voltage across the resistance is 2V and the current
>> >through it is 1A, then the resistance will be: 2 ohm.
>>
>> ---
>> Why would I want to do that? I'm specifically setting up a set of
>> conditions to illustrate _my_ point, not yours.
>
>I am trying to illustrate that I can make a point that 0/0=2.
>You cannot discard my point without adding extra information about your
>set of conditions.
---
I thought that by interspersing your remarks among mine you were being
arbitrary. I see now that wasn't the case; thanks.
---
>This extra information is not present in the 0/0 term, but it _is_
>explicitly written in lim x->0 x/x and lim x->0 2x/x , respectively.
>
>> >> Now, if we go to the more general case of:
>> >>
>> >> x
>> >> y = ---
>> >> x
>> >>
>> >> we can see that for any value of x, as x goes to zero, y will remain
>> >> constant, and exactly equal to 1. Therefore,
>> >>
>> >>
>> >> 0
>> >> --- = 1
>> >> 0
>> >
>> >This is only true because you assumed a resistor of 1 ohm. If you assume
>> >a resistor of 2 ohm, then 0/0 = 2.
>>
>> ---
>> Yes, of course. But I didn't "assume" a resistance of one ohm, I
>> selected the voltage and current to force the resistance to one ohm.
>> ---
>>
>> >Again, you can only state with concidence that 0/0 = 1 in this case
>> >because you already *know* that the resistance is 1; you have not
>> >computed it from 0/0, because the 0/0 quotient doesn't help you to know
>> >that the resistance is 1 ohm.
>>
>> ---
>> The game being to prove that 0/0 = 1, I'm not looking so much for a
>> resistance of 1 ohm as I am a set of values which when divided by
>> themselves will result in a quotient of 1.
>
>
>You want 1 ohm, that's what you bring into the computation. You're
>setting everything up so that 1 ohm results, which means it's circular
>reasoning.
---
I disagree. All I was trying to do was to set up the initial
conditions to give me an equal numerator and denominator as a starting
point for a proof that, eventually, 0/0 = 1.
---
>If you hadn't set everything up that way, the 0V/0A measurement would
>leave you stumped as to the value of the resistor, and 0/0=??? then.
---
0/0 would still be = 1 but yes, of course, to the rest of it. A
_measurement_ would be impossible without a known, non-zero voltage
forcing charge through the resistance.
---
>Again, the set of values that are divided by itself is x/x for all x in
>R\{0}, and lim x->0 x/x = 1 as well.
>
>
>> Let's make them each equal to 1E-40:
>>
>> 1E-40
>> x = ------- = 1
>> 1E-40
>>
>> Damn! That x is still equal to 1!
>>
>> It seems that no matter what we do, as long as the numerator and
>> denominator are equal, the quotient will always be 1. So, if the
>> smallest number we can come up with is 0, and if 0 = 0, then it seems
>> we can say:
>>
>> 0
>> x = --- = 1
>> 0
>
>You are doing the limit argument nicely.
>Now you arrived at x=1 not by dividing by 0 outright, but by taking
>larger values and going closer to 0. I would explicitly write what you
>did, so your last formula would change to
> r
>x = lim --- = 1
> r->0 r
>
>This is mathematical shorthand for your reasoning that any nonzero value
>plugged into r/r is 1, so it makes sense to take r/r=1 for r=0 as well.
---
Yes, thanks.
---
>> >> The value of x is unimportant. What does matter is that the numerator
>> >> and denominator be numerically equal.
>> >
>> >No, it matters that they are algebraically equal.
>>
>> ---
>> Yeah, good point. they have to have the same sign in order for the
>> quotient to come out positive.
>
>You misunderstood my point. My point is that you have to arrive at x/x
>or r/r before you plug in the zero. Conmsider: if you had x=2r/r, put in
>r=0 to get x=2*0/0 and use 2*0=0 to simplify to x=0/0, then the
>numerator is algebraically 0=2r and the denominator is algebraically
>0=r, and then
> 2r
>x = lim ---- = 2
> r->0 r
>which would lead you to conclude that 0/0=2, in this case. You set your
>case up so that x=r/r, i.e. numerator and denominator are algebraically
>the same *before* you plug in the zero.
---
OK, but it seems to me that if you subscribe to
2r
x = lim ---- = 2
r->0 r
when r = 0, then 0/0 _must_ be equal to 1, otherwise x could not have
been equal to 2. That is, if r/r = y and 2y = 2, then y = 1.
---
>This is not true for R=E/I, which is why you can't make a measurement in
>the E=0, I=0 case (unless you have extra information).
>
>> >If I give you a resistor to measure, but no power supply, you will
>> >measure 0A and 0V, but must the resistor always be 1 ohm, then?
>>
>> ---
>> Since R = E/I, I won't be able to make a measurement, so the cat will
>> be both dead and alive.
>
>You seem to grasp that so well. You can certainly measure 0V and 0A, but
>you can't determine R. For what reason do not grasp the analogy that 0/0
>represents the same condition in mathematics?
---
Because it doesn't?
In the latter case we're trying to determine whether, when equal
quantities are divided into each other, the result will always be
equal to 1, while in the former we're dealing with with the interplay
between physically different entities. I.e., 1 apple/1 orange = ???
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 18:41:59 +0000
From: John Woodgate <jmw@jmwa.demon.contraspam.yuk>
Subject: Re: Is zero even or odd?
Message-Id: <WNcA1kD3pv0BFwMQ@jmwa.demon.co.uk>
I read in sci.electronics.design that David Kastrup <dak@gnu.org> wrote
(in <x5is6lrmld.fsf@lola.goethe.zz>) about 'Is zero even or odd?', on
Wed, 29 Dec 2004:
>John Fields <jfields@austininstruments.com> writes:
>
>> On Wed, 29 Dec 2004 13:38:59 +0100, Michael Mendelsohn
>> <invalid@msgid.michael.mendelsohn.de> wrote:
>>
>>>The point is that you can't cancel them when x may be 0, because
>>>division by 0 is undefined.
'Undefined' is a human artefact. Anything can be defined; if the
definition results in contradictions, it's a bad definition.
>>
>> ---
>> And there's the rub! My point is that it _should_ be defined, and
>> defined like this:
>>
>>
>> 0 1
>> --- = 1, --- = oo,
>> 0 0
>
>And that's nonsensical.
>
>It gives us, for example,
>1 = 0/0 = (0+0)/0 = (0/0) + (0/0) = 2
0/0 can't be *defined* as 1 because, as has been repeatedly
demonstrated, that definition results in contradictions, such as the one
illustrated above.
The only definition that does not result in contradictions is that 0/0
is 'any number'. 1 is just one solution.
--
Regards, John Woodgate, OOO - Own Opinions Only.
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk
------------------------------
Date: Wed, 29 Dec 2004 12:54:40 -0600
From: russotto@grace.speakeasy.net (Matthew Russotto)
Subject: Re: Is zero even or odd?
Message-Id: <mLCdnQBp1bftYE_cRVn-uQ@speakeasy.net>
In article <1NnIFqCbZv0BFwJ+@jmwa.demon.co.uk>,
John Woodgate <noone@yuk.yuk> wrote:
>I read in sci.electronics.design that John Fields <jfields@austininstrum
>ents.com> wrote (in <nsl5t0lsbrmrrqptc4opo7ifk2vh77v35s@4ax.com>) about
>'Is zero even or odd?', on Wed, 29 Dec 2004:
>
>>I don't have a problem with either, but it seems to be skirting that
>>taboo point where x = 0.
>
>I think this thread has shown that there are two sorts of people, those
>that associate taboos with 0 and those that don't.
>
>My view is that logically-inferred values of expressions involving zero
>should be accepted unless they result in contradictions. I do not
>support a priori restrictions on interpretation, and I do not accept
>that 'undefined' is a valid show-stopper.
So what do you do when there are multiple logically-inferred values,
as with 0^0? Or when the logically-inferred value breaks a lot of
other things, as with 0/0.
BTW, what's log 0? arctan 0?
------------------------------
Date: Wed, 29 Dec 2004 13:05:39 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <4iv5t05qmkv65t15hgg107nbrpekr8ncm0@4ax.com>
On Wed, 29 Dec 2004 08:11:36 GMT, mmeron@cars3.uchicago.edu wrote:
>In article <r984t09csj6s1ksek81lcij9g5695uv6pd@4ax.com>, John Fields <jfields@austininstruments.com> writes:
>>
>>OK, look at it this way:
>>
>>These two number lines are behind a screen, so the numbers can't be
>>seen, but arranged in such a way that when a number is picked from
>>one, the same number will be picked from the other and those two
>>numbers will be divided and the quotient presented for viewing.
>>
>>-1 0 1
>>-|-------------------------|-------------------------|-
>>
>>
>>-1 0 1
>>-|-------------------------|-------------------------|-
>>
>>I fail to see why, at one infinitesimal spot on the line(s), the
>>division becomes intractable and the machine refuses to output a 1.
>>
>Sigh. Forget machines. Is the definition of division clear to you?
>the result of the division a/b is such a number c that c*b = a.
>That's *all* there is to it. So, the result of 0/0 is a number c such
>that c*0 = 0. Can you tell me what c is?
>
>OK, here is a little exercise for you. At x = 0, sin(x) is also zero.
>And, so is any positive power of either x or sin(x). So, using the
>Taylor expansion for sin(x), evaluate the following limits:
>
>1) lim_x->0 sin(x)/x
>2) lim_x->0 sin(x)/x^2
>3) lim_x->0 (sin(x))^2/x
---
In the context of the problem at hand, the definition of division
isn't in question. What's being discussed is whether zero, being
equal to itself will yield a quotient of 1 when it's divided into
itself. As for the rest of it, I will follow the advice you proffer
in your dotsig and forego arguing with you.
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 13:10:09 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <7606t0tbv7p90fh1vuj4dam88m10hipgtu@4ax.com>
On Wed, 29 Dec 2004 14:41:42 GMT, vonroach <hadrainc@earthlink.net>
wrote:
>On Tue, 28 Dec 2004 22:16:07 -0000, "George Dishman"
><george@briar.demon.co.uk> wrote:
>
>>The task at hand is to show whether the rule
>>is valid or otherwise. Can you do that?
>
>Otherwise? Accept your opinion? Crazy George is really crazy!
---
Do you find it impossible to answer a simple question with a 'yes' or
a 'no'?
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 13:28:34 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <p616t0lmvsths2q28o91unu83kf23hmb1g@4ax.com>
On Wed, 29 Dec 2004 15:12:41 GMT, vonroach <hadrainc@earthlink.net>
wrote:
>On Tue, 28 Dec 2004 22:12:09 -0600, John Fields
><jfields@austininstruments.com> wrote:
>
>>And if those x's were zeroes when they were cancelled, that still
>>results in a quotient of 1, so 0/0 = 1
>
>Chuckle....0/0 is meaningless and you are wrong and bordering on
>lunacy.
---
Just a few more steps, then, and I'll be on your side of the border.
;)
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 20:31:32 +0100
From: David Kastrup <dak@gnu.org>
Subject: Re: Is zero even or odd?
Message-Id: <x5acrwsxyj.fsf@lola.goethe.zz>
John Woodgate <jmw@jmwa.demon.contraspam.yuk> writes:
> David Kastrup <dak@gnu.org> wrote:
>>
>>It gives us, for example,
>>1 = 0/0 = (0+0)/0 = (0/0) + (0/0) = 2
>
> 0/0 can't be *defined* as 1 because, as has been repeatedly
> demonstrated, that definition results in contradictions, such as the
> one illustrated above.
>
> The only definition that does not result in contradictions is that
> 0/0 is 'any number'. 1 is just one solution.
It does not result in contradictions? Since when?
any number = 0/0 = (2*0)/0 = 2*(0/0) = 2*any number = any even number
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
------------------------------
Date: Wed, 29 Dec 2004 19:31:13 +0000 (UTC)
From: George Cox <george_coxanti@spambtinternet.com.invalid>
Subject: Re: Is zero even or odd?
Message-Id: <41D30608.EF4CAD96@spambtinternet.com.invalid>
George Dishman wrote:
>
> <snip> hence
>
> 0
> --- = k
> 0
>
> for all finite k.
No 0/0 is undefined, it has _no_ value.
------------------------------
Date: Wed, 29 Dec 2004 19:34:47 +0000 (UTC)
From: George Cox <george_coxanti@spambtinternet.com.invalid>
Subject: Re: Is zero even or odd?
Message-Id: <41D306DB.AF065B9D@spambtinternet.com.invalid>
David Kastrup wrote:
>
> .... What you
> _can_ say is that
>
> lim (x->v) (x/x) = 1
>
> for _all_ v in R (and actually also if v is +oo or -oo, which is not a
> number, but a neighborhood in some sense of the word).
>
> But you can equally well say that
>
> lim (x->v) (2*x/x) = 2
>
> for _all_ v in R. For v=0, both limits are of the _form_ 0/0, and
> that means that any such limiting _form_ is undefined without further
> qualification.
May I disagree? lim_{x->0} x/x is lim_{x->0} 1 is 1.
------------------------------
Date: Wed, 29 Dec 2004 20:37:28 +0100
From: David Kastrup <dak@gnu.org>
Subject: Re: Is zero even or odd?
Message-Id: <x51xd8sxon.fsf@lola.goethe.zz>
George Cox <george_coxanti@spambtinternet.com.invalid> writes:
> David Kastrup wrote:
>>
>> .... What you
>> _can_ say is that
>>
>> lim (x->v) (x/x) = 1
>>
>> for _all_ v in R (and actually also if v is +oo or -oo, which is not a
>> number, but a neighborhood in some sense of the word).
>>
>> But you can equally well say that
>>
>> lim (x->v) (2*x/x) = 2
>>
>> for _all_ v in R. For v=0, both limits are of the _form_ 0/0, and
>> that means that any such limiting _form_ is undefined without further
>> qualification.
>
> May I disagree?
Only if you understand what I am talking about.
> lim_{x->0} x/x is lim_{x->0} 1 is 1.
That _is_ further qualification of the limiting form 0/0.
--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
------------------------------
Date: Wed, 29 Dec 2004 19:39:25 +0000
From: John Woodgate <jmw@jmwa.demon.contraspam.yuk>
Subject: Re: Is zero even or odd?
Message-Id: <lyq8axDtfw0BFw46@jmwa.demon.co.uk>
I read in sci.electronics.design that Matthew Russotto
<russotto@grace.speakeasy.net> wrote (in <mLCdnQBp1bftYE_cRVn-
uQ@speakeasy.net>) about 'Is zero even or odd?', on Wed, 29 Dec 2004:
>In article <1NnIFqCbZv0BFwJ+@jmwa.demon.co.uk>,
>John Woodgate <noone@yuk.yuk> wrote:
>>I read in sci.electronics.design that John Fields <jfields@austininstrum
>>ents.com> wrote (in <nsl5t0lsbrmrrqptc4opo7ifk2vh77v35s@4ax.com>) about
>>'Is zero even or odd?', on Wed, 29 Dec 2004:
>>
>>>I don't have a problem with either, but it seems to be skirting that
>>>taboo point where x = 0.
>>
>>I think this thread has shown that there are two sorts of people, those
>>that associate taboos with 0 and those that don't.
>>
>>My view is that logically-inferred values of expressions involving zero
>>should be accepted unless they result in contradictions. I do not
>>support a priori restrictions on interpretation, and I do not accept
>>that 'undefined' is a valid show-stopper.
>
>So what do you do when there are multiple logically-inferred values,
>as with 0^0?
What multiple values do you mean? As far as I know, 0^0 = 1 results in
no contradictions. So it is ONE solution. Maybe you know of others;
that doesn't bug me.
>Or when the logically-inferred value breaks a lot of
>other things, as with 0/0.
Which 'logically-inferred value'? This is an equation whose solution is
'any number'. That doesn't bug me, either.
>
>BTW, what's log 0?
The limit Lim(x->0+)(log x) exists, and the value '-oo' creates no
contradictions that I am aware of. Your view may differ.
>arctan 0?
Ooh! That's a REALLY tough one! Maybe you didn't mean it, or you posed a
trick question.
--
Regards, John Woodgate, OOO - Own Opinions Only.
The good news is that nothing is compulsory.
The bad news is that everything is prohibited.
http://www.jmwa.demon.co.uk Also see http://www.isce.org.uk
------------------------------
Date: Wed, 29 Dec 2004 14:06:04 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <jg36t05j4urih20p91kn1ujj2edhljf0d5@4ax.com>
On Wed, 29 Dec 2004 15:28:27 GMT, vonroach <hadrainc@earthlink.net>
wrote:
>On Tue, 28 Dec 2004 17:32:57 -0600, John Fields
><jfields@austininstruments.com> wrote:
>
>>1/0 is a small number?
>
>No it is meaningless gibberish.
---
Beauty is in the eye of the beholder.
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 14:11:48 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <hk36t098e1l7qh76nd0isp88i648m1dirq@4ax.com>
On Wed, 29 Dec 2004 15:39:44 GMT, vonroach <hadrainc@earthlink.net>
wrote:
>On Wed, 29 Dec 2004 01:37:39 +0000 (UTC), George Cox
><george_coxanti@spambtinternet.com.invalid> wrote:
>
>>> Then it seems to me like y gets pretty big when x gets pretty small!
>>
>>You're confused. If y = 1/x then y gets pretty big when x gets pretty
>>small.
>>
...
>x=0 yields a meaningless result.
---
'Red' is meaningless to a blind man.
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 14:14:01 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <mu36t01qmvufravdcsi8rr0uh9i2ph7t5b@4ax.com>
On Wed, 29 Dec 2004 15:43:11 GMT, vonroach <hadrainc@earthlink.net>
wrote:
>On Tue, 28 Dec 2004 19:53:07 -0600, John Fields
><jfields@austininstruments.com> wrote:
>
>>Then what would be the proper way to write it, please?
>>
>There isn't one. Division by 0 is meaningless.
If you have no money, how large is the set of that which you can't
afford?
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 14:33:47 -0600
From: John Fields <jfields@austininstruments.com>
Subject: Re: Is zero even or odd?
Message-Id: <1246t0pqj08eobqcv3fbvu6aetlfc540ub@4ax.com>
On Wed, 29 Dec 2004 15:46:12 GMT, vonroach <hadrainc@earthlink.net>
wrote:
>On Wed, 29 Dec 2004 02:58:46 +0000 (UTC), George Cox
><george_coxanti@spambtinternet.com.invalid> wrote:
>
>>John Fields wrote:
>>>
>>> Then what would be the proper way to write it, please?
>>
>>Write what? That 1/x gets big as x gets small?
>>
>> 1/x -> +infinity as x -> +0
>>
>> 1/x -> -infinity as x -> -0.
>Both are meaningless. Just crap piled higher and deeper as in Ph D.
---
I see. In order not to have to deal with what you don't undestand you
label it 'crap' declare it 'meaningless' and wait for the fog to lift.
--
John Fields
------------------------------
Date: Wed, 29 Dec 2004 20:36:50 GMT
From: mmeron@cars3.uchicago.edu
Subject: Re: Is zero even or odd?
Message-Id: <CzEAd.3$45.2768@news.uchicago.edu>
In article <6je3t0h8uvl1rgkch8r42h8cb4ems0ma31@4ax.com>, Michele Dondi <bik.mido@tiscalinet.it> writes:
>On Tue, 28 Dec 2004 18:21:18 +0000 (UTC), Dave Seaman
><dseaman@no.such.host> wrote:
>
>>using about half a dozen or so different C compilers on various platforms, and
>>every single one of them printed 1.000000. Similarly for most other
>>programming languages that I have tried. However, the mathematical definition
>
>Since this was (cross)posted -for reasons that I can not imagine- also
>to clpmisc, to bring it _slightly_ more on topic:
>
> $ perl -le 'print 0**0'
> 1
>
>BTW: my good 'ol faithful HP-28s tells me the same.
>
>
>PS: the abondance of crankery this thread is revealing is tending to
>make me feel homesick of my sci.math days... (has JSH pop out with an
>FLT-based proof about 0^0 being... <whatever>?!?)
>
Shhhh. Don't encourage him.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
------------------------------
Date: Wed, 29 Dec 2004 21:09:25 GMT
From: mmeron@cars3.uchicago.edu
Subject: Re: Is zero even or odd?
Message-Id: <92FAd.8$45.2809@news.uchicago.edu>
In article <4iv5t05qmkv65t15hgg107nbrpekr8ncm0@4ax.com>, John Fields <jfields@austininstruments.com> writes:
>On Wed, 29 Dec 2004 08:11:36 GMT, mmeron@cars3.uchicago.edu wrote:
>
>>In article <r984t09csj6s1ksek81lcij9g5695uv6pd@4ax.com>, John Fields <jfields@austininstruments.com> writes:
>>>
>>>OK, look at it this way:
>>>
>>>These two number lines are behind a screen, so the numbers can't be
>>>seen, but arranged in such a way that when a number is picked from
>>>one, the same number will be picked from the other and those two
>>>numbers will be divided and the quotient presented for viewing.
>>>
>>>-1 0 1
>>>-|-------------------------|-------------------------|-
>>>
>>>
>>>-1 0 1
>>>-|-------------------------|-------------------------|-
>>>
>>>I fail to see why, at one infinitesimal spot on the line(s), the
>>>division becomes intractable and the machine refuses to output a 1.
>>>
>>Sigh. Forget machines. Is the definition of division clear to you?
>>the result of the division a/b is such a number c that c*b = a.
>>That's *all* there is to it. So, the result of 0/0 is a number c such
>>that c*0 = 0. Can you tell me what c is?
>>
>>OK, here is a little exercise for you. At x = 0, sin(x) is also zero.
>>And, so is any positive power of either x or sin(x). So, using the
>>Taylor expansion for sin(x), evaluate the following limits:
>>
>>1) lim_x->0 sin(x)/x
>>2) lim_x->0 sin(x)/x^2
>>3) lim_x->0 (sin(x))^2/x
>
>---
>In the context of the problem at hand, the definition of division
>isn't in question.
I notice that you carefully avoided dealing with my little exercise:-)
> What's being discussed is whether zero, being
>equal to itself will yield a quotient of 1 when it's divided into
>itself.
In order to find out what o divided by itself, or indeed anything
divided by anything, yields, you do need the definition of division.
Mathematical operations do not have an existance independent of their
definitions.
> As for the rest of it, I will follow the advice you proffer
>in your dotsig and forego arguing with you.
>
You're very welcome.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
------------------------------
Date: Wed, 29 Dec 2004 21:19:02 +0000 (UTC)
From: Dave Seaman <dseaman@no.such.host>
Subject: Re: Is zero even or odd?
Message-Id: <cqv706$6v1$2@mailhub227.itcs.purdue.edu>
On Wed, 29 Dec 2004 19:39:25 +0000, John Woodgate wrote:
> I read in sci.electronics.design that Matthew Russotto
><russotto@grace.speakeasy.net> wrote (in <mLCdnQBp1bftYE_cRVn-
>>
>>So what do you do when there are multiple logically-inferred values,
>>as with 0^0?
> What multiple values do you mean? As far as I know, 0^0 = 1 results in
> no contradictions. So it is ONE solution. Maybe you know of others;
> that doesn't bug me.
There can't be very many others. Consider:
Let x = 0^0.
Then x^2 = (0^0)^2
= 0^(0*2)
= 0^0
= x,
from which we find that the only possibilities are x = 0 or x = 1.
Of those possibilities, 0^0 = 1 is the only logical choice, for the various
reasons that have already been discussed.
--
Dave Seaman
Judge Yohn's mistakes revealed in Mumia Abu-Jamal ruling.
<http://www.commoncouragepress.com/index.cfm?action=book&bookid=228>
------------------------------
Date: Wed, 29 Dec 2004 22:19:41 -0000
From: "George Dishman" <george@briar.demon.co.uk>
Subject: Re: Is zero even or odd?
Message-Id: <cqvafs$rf6$1@news.freedom2surf.net>
"George Cox" <george_coxanti@spambtinternet.com.invalid> wrote in message
news:41D30608.EF4CAD96@spambtinternet.com.invalid...
> George Dishman wrote:
>>
>> <snip> hence
>>
>> 0
>> --- = k
>> 0
>>
>> for all finite k.
>
> No 0/0 is undefined, it has _no_ value.
If you go back down the thread a bit, you'll see
that was my point, John's attempted proof can be
used equally well to show it is any other value
therefore it fails as a proof that the value is
uniquely 1.
George
------------------------------
Date: Wed, 29 Dec 2004 22:59:47 -0000
From: "George Dishman" <george@briar.demon.co.uk>
Subject: Re: Is zero even or odd?
Message-Id: <cqvcr2$s5f$1@news.freedom2surf.net>
"John Fields" <jfields@austininstruments.com> wrote in message
news:ljr5t0116eo1l2s73egemhtl5ohg2mrgf0@4ax.com...
>
> .... All I was trying to do was to set up the initial
> conditions to give me an equal numerator and denominator as a starting
> point for a proof that, eventually, 0/0 = 1.
I think that's the key point that Michael and
I are trying to get across John. You have to
select specific initial conditions to get the
answer you want and if you choose different
conditions you get a different answer.
The bottom line is that if you know a and b and
c * b = a
then
c = a/b
lets you find c.
c = 0/0
is a way of asking a question, what number when
multiplied by zero gives the answer zero. Any
finite number satisfies that requirement, not
just one.
George
------------------------------
Date: Wed, 29 Dec 2004 23:26:22 +0000 (UTC)
From: George Cox <george_coxanti@spambtinternet.com.invalid>
Subject: Re: Is zero even or odd?
Message-Id: <41D33D27.66E419BD@spambtinternet.com.invalid>
George Cox wrote:
>
> It is not always the case that
>
> lim x -> A f(x) = f(A).
Indeed lim x -> A f(x) can exist even when f(A) doesn't. Example: f(x)
= x/x.
------------------------------
Date: Thu, 30 Dec 2004 00:32:39 GMT
From: "Nicholas O. Lindan" <see@sig.com>
Subject: Re: Is zero even or odd?
Message-Id: <H0IAd.2285$JC2.1423@newsread2.news.atl.earthlink.net>
"George Cox" <george_coxanti@spambtinternet.com.invalid> wrote
> No 0/0 is undefined, it has _no_ value.
No it has value: 0/0 == 0/0 = 0/0 * 1 etc.
It just doesn't have any other value. (Yet)
--
Nicholas O. Lindan, Cleveland, Ohio
Consulting Engineer: Electronics; Informatics; Photonics.
Remove spaces etc. to reply: n o lindan at net com dot com
psst.. want to buy an f-stop timer? nolindan.com/da/fstop/
------------------------------
Date: Thu, 30 Dec 2004 00:38:32 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <iej6t0l5sh3vm1hj5l2dr5haadh0btpd55@4ax.com>
On Wed, 29 Dec 2004 15:07:12 +0000 (UTC), Dave Seaman
<dseaman@no.such.host> wrote:
>> No 0^0 is meaningless .
>
>Your rants are meaningless, since you never include supporting arguments and
>you never answer contrary arguments, even when documented with authoritative
>references.
Your `arguments' are meaningless.
------------------------------
Date: Thu, 30 Dec 2004 00:39:53 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <hhj6t0d88tq1v24hg0o0of5rdkeghqj75e@4ax.com>
On Wed, 29 Dec 2004 13:28:34 -0600, John Fields
<jfields@austininstruments.com> wrote:
>Just a few more steps, then,
Nope, you are already there.
------------------------------
Date: Thu, 30 Dec 2004 00:41:22 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <mkj6t053sqdadksnh6b2fcqmmnsdsd2bg7@4ax.com>
On Wed, 29 Dec 2004 14:14:01 -0600, John Fields
<jfields@austininstruments.com> wrote:
>If you have no money, how large is the set of that which you can't
>afford?
Meaningless comment.
------------------------------
Date: Thu, 30 Dec 2004 00:42:44 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <6mj6t0da3atbgl04h7mibjut9n4t5gusrq@4ax.com>
On Wed, 29 Dec 2004 14:33:47 -0600, John Fields
<jfields@austininstruments.com> wrote:
>I see. In order not to have to deal with what you don't undestand you
>label it 'crap' declare it 'meaningless' and wait for the fog to lift.
Fog never lifts on meaningless crap as posted by you.
------------------------------
Date: Thu, 30 Dec 2004 00:45:24 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <nqj6t0lvp18kll2nqs9jbs2c887irsfns8@4ax.com>
On Wed, 29 Dec 2004 11:48:52 -0600, John Fields
<jfields@austininstruments.com> wrote:
>John Fields
You bore with this nonspeak garbage. Go troll elsewhere Tonto.
------------------------------
Date: Thu, 30 Dec 2004 00:47:30 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <juj6t01et7qlg3enbhbdg3sfe5mmvc5oni@4ax.com>
On Wed, 29 Dec 2004 18:41:59 +0000, John Woodgate
<jmw@jmwa.demon.contraspam.yuk> wrote:
>'Undefined' is a human artefact
Not in mathematics, you idiot. Run along and troll elsewhere.
------------------------------
Date: Thu, 30 Dec 2004 00:48:54 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <a2k6t05akmiijpjpgpr84gl78aj4d25emi@4ax.com>
On Wed, 29 Dec 2004 19:34:47 +0000 (UTC), George Cox
<george_coxanti@spambtinternet.com.invalid> wrote:
>May I disagree?
Not until you master your subject.
------------------------------
Date: Thu, 30 Dec 2004 00:50:27 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <m4k6t0to5m9e7hqsqe0uuc3d3e5s9h6p33@4ax.com>
On Wed, 29 Dec 2004 20:37:28 +0100, David Kastrup <dak@gnu.org> wrote:
> 0/0.
Another idiot. Probably all go to same elementary school.
------------------------------
Date: Thu, 30 Dec 2004 00:51:33 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <97k6t0h8ngjf50g712fjn85l7jstsbe4ag@4ax.com>
On Wed, 29 Dec 2004 18:24:27 +0000, John Woodgate
<jmw@jmwa.demon.contraspam.yuk> wrote:
>I think this thread has shown that there are two sorts of people, those
>that associate taboos with 0 and those that don't.
That is because you are a dumbass.
------------------------------
Date: Thu, 30 Dec 2004 00:53:01 GMT
From: "Nicholas O. Lindan" <see@sig.com>
Subject: Re: Is zero even or odd?
Message-Id: <NjIAd.2316$JC2.1246@newsread2.news.atl.earthlink.net>
"David Kastrup" <dak@gnu.org> wrote
> > >It gives us, for example,
> > >1 = 0/0 = (0+0)/0 = (0/0) + (0/0) = 2
Well, that's wrong. Obviously wrong. But constructing
a false statement doesn't have much value that I can see.
Try:
0 + 0 = 2 * 0 <> 0
1 = 0/0
Then:
1 + 1 = 0/0 + 0/0 = (0+0)/0 = 2 * 0/0 = 2
> > The only definition that does not result in contradictions is that
> > 0/0 is 'any number'. 1 is just one solution.
I'll buy it. Now is '1' the best solution of all the many that can be
picked?
> ... 0/0 is not defined ...
0/0 is defined by what people define it as, just like anything else.
Thousand years ago, '0' wasn't defined.
To say 'it is not defined' is as to say 'here there be monsters',
and suddenly everybody (most everbody, well, would you believe >0)
wants to go there. It is the undiscovered country.
And: "It really peeeses the Anglos off, man."
- In response to the interviewer
asking a low-rider why he covers
his car with rainbow tape.
--
Nicholas O. Lindan, Cleveland, Ohio
Consulting Engineer: Electronics; Informatics; Photonics.
Remove spaces etc. to reply: n o lindan at net com dot com
psst.. want to buy an f-stop timer? nolindan.com/da/fstop/
------------------------------
Date: Thu, 30 Dec 2004 00:53:17 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <6ak6t0ps035n41ftgds600kavam9tvklo6@4ax.com>
On Wed, 29 Dec 2004 19:39:25 +0000, John Woodgate
<jmw@jmwa.demon.contraspam.yuk> wrote:
> As far as I know,
Less than a Plank unit.
------------------------------
Date: Thu, 30 Dec 2004 00:55:33 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <7fk6t0l15decdpdb5j3vs1la8dq330e1qg@4ax.com>
On Wed, 29 Dec 2004 18:26:13 +0000, John Woodgate
<jmw@jmwa.demon.contraspam.yuk> wrote:
>Yes. John Fields'
is an idiot.
------------------------------
Date: Thu, 30 Dec 2004 00:58:21 GMT
From: vonroach <hadrainc@earthlink.net>
Subject: Re: Is zero even or odd?
Message-Id: <vik6t0dvb37vjkmnovkastv5vgfp4vh1ip@4ax.com>
On Wed, 29 Dec 2004 21:09:25 GMT, mmeron@cars3.uchicago.edu wrote:
>In order to find out what o divided by itself, or indeed anything
>divided by anything, yields, you do need the definition of division.
Which states that division by 0 is not defined and therefore
meaningless.
------------------------------
Date: Thu, 30 Dec 2004 01:04:25 GMT
From: "Nicholas O. Lindan" <see@sig.com>
Subject: Re: Is zero even or odd?
Message-Id: <tuIAd.6699$qf5.4423@newsread3.news.atl.earthlink.net>
> is a way of asking a question, what number when
> multiplied by zero gives the answer zero.
1, it's an identity
> Any finite number satisfies that requirement, not
> just one.
See, now there you go again, thinking there is only one zero
and there's nothing that it can do.
If, though, you allow 0 to be treated as any other imaginary
number then:
2 * 0 = 0 + 0 <> 0
Two times zero doesn't give the answer 0.
_Only_ one times zero does.
And so on ...
0 * 0 = 0^2 <> 0
Demand Zero Equality!
Free your imagination!
Pervert young minds!
Annoy your math teacher!
--
Nicholas O. Lindan, Cleveland, Ohio
Consulting Engineer: Electronics; Informatics; Photonics.
Remove spaces etc. to reply: n o lindan at net com dot com
psst.. want to buy an f-stop timer? nolindan.com/da/fstop/
------------------------------
Date: Thu, 30 Dec 2004 01:28:32 GMT
From: "Mark J. Tilford" <tilford@ugcs.caltech.edu>
Subject: Re: Is zero even or odd?
Message-Id: <slrnct6jpm.rv2.tilford@ralph.earthlink.net>
On Wed, 29 Dec 2004 20:31:32 +0100, David Kastrup <dak@gnu.org> wrote:
> John Woodgate <jmw@jmwa.demon.contraspam.yuk> writes:
>
>> David Kastrup <dak@gnu.org> wrote:
>>>
>>>It gives us, for example,
>>>1 = 0/0 = (0+0)/0 = (0/0) + (0/0) = 2
>>
>> 0/0 can't be *defined* as 1 because, as has been repeatedly
>> demonstrated, that definition results in contradictions, such as the
>> one illustrated above.
>>
>> The only definition that does not result in contradictions is that
>> 0/0 is 'any number'. 1 is just one solution.
>
> It does not result in contradictions? Since when?
>
> any number = 0/0 = (2*0)/0 = 2*(0/0) = 2*any number = any even number
>
Any number, not any integer.
--
------------------------
Mark Jeffrey Tilford
tilford@ugcs.caltech.edu
------------------------------
Date: Wed, 29 Dec 2004 23:31:10 -0800
From: "Alfred Z. Newmane" <a.newmane.remove@eastcoastcz.com>
Subject: Re: Is zero even or odd?
Message-Id: <33hp81F401lr4U1@individual.net>
Dave Seaman wrote:
> On Tue, 28 Dec 2004 23:08:28 GMT, vonroach wrote:
>> On Tue, 28 Dec 2004 18:21:18 +0000 (UTC), Dave Seaman
>> <dseaman@no.such.host> wrote:
>
>>> The Macintosh calculator returns 1. So do most Hewlett-Packard
>>> calculators that I have tried, and at least one by Texas
>>> Instruments that I can recall. Likewise Maple and MATLAB (but not
>>> Mathematica).
>
>> You rely on a computer? How slipshod.
>
> I rely on the mathematical arguments I have previously posted with
> references to recognized authoritative authors (Suppes, Lang, and the
> sci.math FAQ).
Just a minor nit pick, not meaning to sound anal, but I find it hard to
accept a news group's FAQ as a concret authority of Math. It may be an
authority of the nes group it elf (guidelines, etc), and maybe perhaps,
since you're refering to a math news group, it can hold some weight in
FAQ questions, it cannot possibly take presidence over publichsed and
fully accepted documents.
(Unless there is something very special about this particular FAQ, such
as being fully recognized by the main stream math community as a
concrete authority.)
------------------------------
Date: Wed, 29 Dec 2004 23:42:00 -0800
From: "Alfred Z. Newmane" <a.newmane.remove@eastcoastcz.com>
Subject: Re: Is zero even or odd?
Message-Id: <33hpscF3vn0v6U1@individual.net>
vonroach wrote:
> On Tue, 28 Dec 2004 09:51:12 -0800, "Alfred Z. Newmane"
> <a.newmane.remove@eastcoastcz.com> wrote:
>
>> 2^0 = 1
>> 1^0 = 1
>> 0^0 = ERROR, DOMAIN (hence the limit)
>> (-1)^0 = 1
>> (-2)^0 = 1
>
> (-1^1/2)^0 =?
> or ( i )^0 =?
Both would result in 1 (I've never seen a limit preventing this, at
least when complex numbers are allowed (that is, (real, imag), which the
result above is (1, 0).)
(Also checked on my TI86.)
---------------------
((-1)^(1/2))^0
(1,0)
i
(0,1)
i^0
(1,0)
---------------------
------------------------------
Date: Wed, 29 Dec 2004 23:48:27 -0800
From: "Alfred Z. Newmane" <a.newmane.remove@eastcoastcz.com>
Subject: Re: Is zero even or odd?
Message-Id: <33hq8fF3vrepkU1@individual.net>
Dave Seaman wrote:
> On Tue, 28 Dec 2004 23:01:17 GMT, vonroach wrote:
>> On Tue, 28 Dec 2004 09:51:12 -0800, "Alfred Z. Newmane"
>> <a.newmane.remove@eastcoastcz.com> wrote:
>
>>> 2^0 = 1
>>> 1^0 = 1
>>> 0^0 = ERROR, DOMAIN (hence the limit)
>>> (-1)^0 = 1
>>> (-2)^0 = 1
>
>> (-1^1/2)^0 =?
>> or ( i )^0 =?
>
> i^0 = exp(0*log(i)) = exp(0) = 1,
>
> where log(i) is multivalued (= i*pi/2 + 2*n*pi*i), but it doesn't
> particularly matter which value you choose. And (-i)^0 = 1 by similar
> analysis.
Exactly. My TI86 reflects this quite nicely:
---------------------
0*log (i)
(0,0)
e^(0*log (i))
(1,0)
e^0
1
---------------------
(And just FYI, (1,0) is the same as 1, it's just in complex
(real,imaginary) form.)
------------------------------
Date: Wed, 29 Dec 2004 23:57:22 -0800
From: "Alfred Z. Newmane" <a.newmane.remove@eastcoastcz.com>
Subject: Re: Is zero even or odd?
Message-Id: <33hqp8F40nd3aU1@individual.net>
mmeron@cars3.uchicago.edu wrote:
[...]
> Sigh. Forget machines. Is the definition of division clear to you?
> the result of the division a/b is such a number c that c*b = a.
> That's *all* there is to it. So, the result of 0/0 is a number c such
> that c*0 = 0. Can you tell me what c is?
/Thank/ /you/, that is the best explanation/proof I've seen in this
entire thread about.
c = undefined, since it cannot be one value, since /anything/ * 0 = 0.
Your proof demonstrates this perfectly.
------------------------------
Date: 6 Apr 2001 21:33:47 GMT (Last modified)
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