LudovicoVan
Posts:
2,971
From:
London
Registered:
2/8/08
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Re: 0^0=1
Posted:
May 6, 2012 1:05 PM
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"Jussi Piitulainen" <jpiitula@ling.helsinki.fi> wrote in message
news:qotlil5w8tt.fsf@ruuvi.it.helsinki.fi...
> LudovicoVan writes:
>
>> AFAIK, from the basics of number theory, 0^0 is just like 0/0 and it
>> should be undefined (in standard, point-like arithmetic). Then,
>> sometimes for convenience of calculation, sometimes because it makes
>> sense re the specific mathematical object we are using, one can
>> decide to define 0^0 to have a specific value, which amounts to
>> redefining the power function with the addition of a special case.
>> The drawback with this approach is that the corresponding operations
>> lose some of their properties, like commutativity, or
>> "invertibility", etc. etc. -- that is, I cannot be more precise than
>> that, but sometimes one gets to prove 1=0 with these "extensions",
>> etc...
>
> Sheer FUD. There is no number-theoretic reason to leave 0^0 undefined,
> and there are good reasons to define it. It's not a special case. It's
> a special case when it is left undefined, which is usually done for no
> stated reason and sometimes for bogus reasons such as 0^1/0^1 being
> undefined, or 0^0 being indeterminate as a limiting form.
>
> By all means, present the reasoning. I've been watching this space
> since the times when the topic used to be raised every other week and
> seen only nervous cluelesness raised against the definition.
(I'll talk about 0/0 because 0^0 can be shown to be equivalent.)
On the contrary, there are all reasons: in the case 0/0, that is, by
definition of division, "the number which, when multiplied by 0, gives 0",
and you just do not have ground to chose a specific value among all the
possible values that would fit the bill. But the point is even more
technical than that, as you do get to (easily) prove that 1==0 if you adopt
any such definitions. This is how I would show it:
1
= 1 + 1 - 1
= 0/0 + 0/0 - 1
= (0+0)/(0*0) - 1 -- 0/0 is defined, isn't it?
= 0/0 - 1
= 1 - 1
= 0
Hence please re-read what I said, because one thing is to patch a function,
other thing is the properties of the corresponding operations and the
manipulations you are then entitled to perform. In particular, you are
surely entitled to take 0/0:=1 in order to attain this or that specific
computational goal, but you have to put careful and strict limits on where
that definition can be used, otherwise all you will get is an inconsistent
system. And these limits are what we usually denote as "exceptions", in the
broad sense.
-LV
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