On 20/09/2011 05:55 PM, Mike Raiford wrote:

> 1/0 is undefined, however the limit of 1/n as n approaches 0 is
> infinity. :)

When approached from above, it's positive infinity. When approached from 
below, it's negative infinity...

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On 9/20/2011 9:55, Mike Raiford wrote:
> 1/0 is undefined, however the limit of 1/n as n approaches 0 is infinity. :)

It depends on your math, really. The only reason people invented "limits" is 
because they thought dividing by zero was somehow impure, like "irrational" 
numbers used to be.

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Darren New, San Diego CA, USA (PST)
   How come I never get only one kudo?

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On 9/20/2011 1:08, Invisible wrote:
> On 20/09/2011 02:06 AM, Darren New wrote:
>> On 9/19/2011 11:20, Orchid XP v8 wrote:
>>> Fact is, 0 * infinity = 0, so it's not a sensible answer.
>>
>> I can't imagine how you can say dividing by zero is undefined, but then
>> go and define multiplying zero by infinity. :-)
>
> If you multiply *anything* by zero, the result is always zero.

No it isn't, if you don't take into account the existence of infinity.

> pre-existing definition of how multiplication by zero works.

The "pre-existing definition" doesn't take into account infinity.

Yes, multiplying any real number by zero gives you zero. That isn't saying 
anything about infinity.

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Darren New, San Diego CA, USA (PST)
   How come I never get only one kudo?

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On 9/20/2011 1:47, Roman Reiner wrote:
> 0/0 = 1

No it isn't. :-)

-- 
Darren New, San Diego CA, USA (PST)
   How come I never get only one kudo?

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On 20-9-2011 22:27, Darren New wrote:
> On 9/20/2011 1:08, Invisible wrote:
>> On 20/09/2011 02:06 AM, Darren New wrote:
>>> On 9/19/2011 11:20, Orchid XP v8 wrote:
>>>> Fact is, 0 * infinity = 0, so it's not a sensible answer.
>>>
>>> I can't imagine how you can say dividing by zero is undefined, but then
>>> go and define multiplying zero by infinity. :-)
>>
>> If you multiply *anything* by zero, the result is always zero.
>
> No it isn't, if you don't take into account the existence of infinity.
>
>> pre-existing definition of how multiplication by zero works.
>
> The "pre-existing definition" doesn't take into account infinity.
>
> Yes, multiplying any real number by zero gives you zero. That isn't
> saying anything about infinity.
>

Put another way: multiplying any number by zero gives zero, multiplying 
any number by infinity gives infinity (give or take a sign)

Hence 0 * infinity is both zero and infinity.
Or actually it isn't, it is undefined.

The actual rules are:
multiplying any *finite* number by zero gives zero,
multiplying any *non-zero, positive* number by infinity gives infinity

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On 20-9-2011 17:14, clipka wrote:
> Am 20.09.2011 13:53, schrieb Le_Forgeron:
>
>> Now for the happy part: you sell for 1c, all taxes are to be round up
>> (you cannot cheat on the state...) so you earn nothing but 1c of VAT,
>> minus the fixed rates of the accountant, the bank and all... in fact,
>> you loose.
>
> I would guess this might depend on what country you live in.

In the netherlands all tax calculations are 'rounded' to a whole number 
of euros in the direction that is most convenient for the tax payer.

Earnings are rounded down, expenses rounded up and tax to be paid down 
again.

So yes, it depends.


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Am 20.09.2011 22:25, schrieb Darren New:
> On 9/20/2011 9:55, Mike Raiford wrote:
>> 1/0 is undefined, however the limit of 1/n as n approaches 0 is
>> infinity. :)
>
> It depends on your math, really. The only reason people invented
> "limits" is because they thought dividing by zero was somehow impure,
> like "irrational" numbers used to be.

Wrong. First of all they were invented for the general case where f(x) 
is an element of the real (or complex) numbers for certain x, or is 
undefined due to ambiguities. While f(x)=1/x is the most prominent 
example, there are others, e.g.:

f(x) = arctan x
f(x) = x^0
f(x) = 0^x
f(x) = g'(x), with g(x) = |x|
f(x) = |g'(x)|, with g(x) = |x|

Also it was not invented because someone thought that some mathematical 
operations were "somehow impure", but to systematically prove some 
assumptions that had simply been taken for granted in earlier 
mathematical work. It is also a great tool for working with functions 
where lim[x->X]f(x) depends on from which direction you approach X, or 
to examine the behaviour of a function as the parameter approaches 
(positive or negative) infinity.

Besides, there is no such thing in (modern day) mathematics as "somehow 
impure"; there is just "defined" and "undefined", or "proven" and 
"unproven". And x=1/0 /is/ undefined on the set of real (or complex) 
numbers, because there is no solution to x*0=1 in that domain.

For any domain that includes an element inf:=1/0 satisfying inf*0=1, 
each and every mathematical property of the body of real (or complex) 
numbers must be re-evaluated with respect to this new element inf, and 
exceptions need to be established for various mathematical operations. 
For instance, if you want inf to have the property that inf+x=inf for 
x!=0, you'll obviously need to break even such simple properties as 
y+x=y <=> x=0.

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On 9/20/2011 19:16, clipka wrote:
> Also it was not invented because someone thought that some mathematical
> operations were "somehow impure", but to systematically prove some
> assumptions that had simply been taken for granted in earlier mathematical
> work.

OK. I had been taught that the whole concept of "limits" was invented 
because while there was a working set of math that involved calculating 
stuff about 0/0 for example, it wasn't really quite right.

> Besides, there is no such thing in (modern day) mathematics as "somehow
> impure";

Of course. But the guy who invented calculus is the same guy who specified 
that things are "defined" and "undefined" and "proven" and "unproven." :-) I 
wasn't talking about modern math, obviously.

> And x=1/0 /is/ undefined on the set of real (or complex)
> numbers, because there is no solution to x*0=1 in that domain.
>

Sure. Never argued against that.

> For any domain that includes an element inf:=1/0 satisfying inf*0=1, each
> and every mathematical property of the body of real (or complex) numbers
> must be re-evaluated with respect to this new element inf, and exceptions
> need to be established for various mathematical operations. For instance, if
> you want inf to have the property that inf+x=inf for x!=0, you'll obviously
> need to break even such simple properties as y+x=y <=> x=0.

Yep.

-- 
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   How come I never get only one kudo?

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On Tue, 20 Sep 2011 10:47:13 +0200, Roman Reiner <lim### [at] gmxde> wrote:

> Invisible <voi### [at] devnull> wrote:
>> On 20/09/2011 02:06 AM, Darren New wrote:
>> > On 9/19/2011 11:20, Orchid XP v8 wrote:
>> >> Fact is, 0 * infinity = 0, so it's not a sensible answer.
>> >
>> > I can't imagine how you can say dividing by zero is undefined, but  
>> then
>> > go and define multiplying zero by infinity. :-)
>>
>> If you multiply *anything* by zero, the result is always zero. This
>> isn't exactly news. We're not "defining" anything, we're just using the
>> pre-existing definition of how multiplication by zero works.
>
> You're wrong there. 0 * inf = 0 is indeed a definition as the result of  
> the
> left-hand side is undefined in the same manner as dividing by zero is  
> (actually,
> the concepts of multiplication and division are mathematically the very  
> same
> thing).
> see, if you define 1/0 = inf, and treat 0 and inf as ordinary numbers,  
> then 0 *
> inf = 0/0 = 1 contrary to your claim. This contradiction is the result  
> of using
> inf as a number when, in fact, it isn't.
>
> Regards Roman
>
>

So there needs to be a rule of order - similar to doing calculations in  
brackets first.
What I mean is in th case of 0/0 - which rule should be applied first?
A) 0/anything =0
B) anything divided by itself =1
C) anything divided by 0 is undefined



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-Nekar Xenos-

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> On Tue, 20 Sep 2011 10:47:13 +0200, Roman Reiner <lim### [at] gmxde> wrote:
>
>> Invisible <voi### [at] devnull> wrote:
>>> On 20/09/2011 02:06 AM, Darren New wrote:
>>> > On 9/19/2011 11:20, Orchid XP v8 wrote:
>>> >> Fact is, 0 * infinity = 0, so it's not a sensible answer.
>>> >
>>> > I can't imagine how you can say dividing by zero is undefined, but
>>> then
>>> > go and define multiplying zero by infinity. :-)
>>>
>>> If you multiply *anything* by zero, the result is always zero. This
>>> isn't exactly news. We're not "defining" anything, we're just using the
>>> pre-existing definition of how multiplication by zero works.
>>
>> You're wrong there. 0 * inf = 0 is indeed a definition as the result
>> of the
>> left-hand side is undefined in the same manner as dividing by zero is
>> (actually,
>> the concepts of multiplication and division are mathematically the
>> very same
>> thing).
>> see, if you define 1/0 = inf, and treat 0 and inf as ordinary numbers,
>> then 0 *
>> inf = 0/0 = 1 contrary to your claim. This contradiction is the result
>> of using
>> inf as a number when, in fact, it isn't.
>>
>> Regards Roman
>>
>>
>
> So there needs to be a rule of order - similar to doing calculations in
> brackets first.
> What I mean is in th case of 0/0 - which rule should be applied first?
> A) 0/anything =0
> B) anything divided by itself =1
> C) anything divided by 0 is undefined
>
>
>
The exact case depends on /how/ you reatch 0/0.

If you follow a x/x, (x^n)/x, x/(x^n)... suite, then you have the case B 
and the result is 1. As the value of x become infinitesimally small, 
your result stay, or converge to 1.
If you have 0/x, "x" been about any expression, then the result should 
be zero.
In other cases, it's undefined, +infinity or -infinity.

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