# Re: 2.0.30 - its all a numbers game, I tell ya!

Matthew Kirkwood (weejock@ferret.lmh.ox.ac.uk)
Tue, 11 Mar 1997 14:49:31 +0000 (GMT)

On Tue, 11 Mar 1997, Richard B. Johnson inflicted the following upon me:

> Since this continues....
>
> N/N = 1 For all N

No. N/N = 1 For all N != 0

As I pointed out earlier, the division operator is _not_defined_ when
zero is the second argument.

n / m = whatever, if m != 0

The implementation of whatever is irrelevant; the point is that, when
m == 0, n / m does not equal anything. It does not equal infinity; it
*has*no*value*.

> Therefore:
> 0/0 = 1
> Proof: (actually a conjecture)
> Numbers from -N to +N are LINER, there is no discontinuity through zero.

The numbers themselves are not linear. They are values. A function to,
or from these may, or may not, be linear. The division function is not
linear in its second argument. Surely that is obvious from the graph of
f(x) = 1/x, which IS discontinuous through zero. f(0) is simply not
defined, since 1/0 is not defined.

> Therefore:
> +N/+N = 1
> -N/-N = 1
> And everything in between...
[snip!!]
> All the real numbers between - N and + N, when divided by themselves must
> equal 1. Therefore 0/0 = 1.

What meaningless twaddle. 0/0 is not defined. anything/0 is not defined.
The limit of 0/n as n approaches 0, is defined. And _does_ equal 0. But,
since division is not defined for second arguments equal to zero, the
function g(x) = 0/x is not defined at zero, and thus is not continuous at
zero. By the definition of continuity, since g is not continuous, the
limit of g(x) as x tends to zero does not equal g(0).

> Now, if N was infinity, and it was divided by the same N, it too would equal
> 1. However, one must be sure that it is the same infinity from the same set.
> You can't be too sure about that because a number series converges at
> infinity losing some properties in the process. If N was infinity....

This all assumes that there exists a greatest natural number.
Conjecture: "There is no greatest natural number"
Proof: 1. Assume false, ie there is some N such that for all natural
numbers n, n <= N.
2. n+1 > n, for all natural numbers n
Then 3. N+1 > N
Contradiction. Therefore there is no greatest natural number

> N + N = N
> N * N = N

Er... no. The only number with these properties is 0.

> N / N should be 1, but maybe not.

And since you've just fixed N = 0, N/N is not defined.

> N - N should = 0, but maybe not.

No. What you mean is:

lim (N-N)
n -> inf <- Look up an analysis textbook for the definition of "limit"

= {Since N - N = 0 for all N}

lim 0
n -> inf

= {n bound, but not referenced}

0

> Such a problem does not exist at zero, therefore the proof stands.

If N was infinity?? No, no, no!!! There is no such object as infinity.
Infinity is a useful collection of properties which describe "arbitrarily
large".

If you insist on N being infinity, then which one do you choose? Do you
choose the countable infinity (the one indexed by the natural numbers)
which can list integers, fractions, and plenty other stuff too, or the
uncountable one (the number of REAL numbers)??

You seem to forget that whatever arithmetic (or mathematics) you do with a
computer are essentially finite. Not even countable. You might be able
to make them arbitrarily large, but that _does_not_ make them infinite.

(In a brief nod towards the real purpose of the list) I'm afraid this
means that, no matter _how_ much work is put into Linux, we will never
reach version 1/0 :-(

Matthew.

PS. Please excuse my pedantry, but having had all of this forced into me
in algebra and analysis courses last year, I get upset when people
produce fallacies like these.

```--
Matthew Kirkwood  |  Mail:  weejock@ferret.lmh.ox.ac.uk
LMH JCR,          |  Web:   http://www-jcr.lmh.ox.ac.uk/~weejock/
Oxford OX2 6QA,   |  Microsoft: "Where do you want to go today?"
England.          |  Linux: "Where you want to go _now_?"
```