Re: Is zero even or odd?

From: Michael Mendelsohn (invalid_at_msgid.michael.mendelsohn.de)
Date: 12/29/04 

Date: Wed, 29 Dec 2004 13:38:59 +0100

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. 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.
>
> ---
> 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).

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).

Cheers
Michael 

-- 
Still an attentive ear he lent        Her speech hath caused this pain
But could not fathom what she meant   Easier I count it to explain
She was not deep, nor eloquent.       The jargon of the howling main
                 -- from Lewis Carroll: The Three Usenet Trolls

Relevant Pages

  • Re: Is zero even or odd?
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  • Re: Is zero even or odd?
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