On 24.06.2016 02:14, H. S. Teoh via Digitalmars-d wrote:
On Fri, Jun 24, 2016 at 01:58:01AM +0200, Timon Gehr via Digitalmars-d wrote:
On 24.06.2016 01:18, H. S. Teoh via Digitalmars-d wrote:
On Thu, Jun 23, 2016 at 11:14:08PM +0000, deadalnix via Digitalmars-d wrote:
On Thursday, 23 June 2016 at 22:53:59 UTC, H. S. Teoh wrote:
This argument only works for discrete sets. If n and m are reals,
you'd need a different argument.
For reals, you can use limits/continuation as argument.
The problem with that is that you get two different answers:
lim x^y = 0
x->0
but:
lim x^y = 1
y->0
...
That makes no sense. You want lim[x->0] x^0 and lim[y->0] 0^y.
Sorry, I was attempting to write exactly that but with ASCII art. No
disagreement there.
So it's not clear what ought to happen when both x and y approach 0.
The problem is that the 2-variable function f(x,y)=x^y has a
discontinuity at (0,0). So approaching it from some directions give
1, approaching it from other directions give 0, and it's not clear
why one should choose the value given by one direction above
another. ...
It is /perfectly/ clear. What makes you so invested in the continuity
of the function 0^y? It's just not important.
I'm not. I'm just pointing out that x^y has an *essential*
discontinuity at (0,0),
Which just means that there is no limiting value for that point.
and the choice 0^0 = 1 is a matter of
convention. A widely-adopted convention, but a convention nonetheless.
It does not change the fact that (0,0) is an essential discontinuity of
x^y.
...
No disagreement here. Nothing about this is 'arbitrary' though. All
notation is convention, but not all aspects of notations are arbitrary.
[...]
not something that the mathematics itself suggest.
...
What kind of standard is that? 'The mathematics itself' does not
suggest that we do not define 2+2=5 while keeping all other function
values intact either, and it is still obvious to everyone that it
would be a bad idea to give such succinct notation to such an
unimportant function.
Nobody said anything about defining 2+2=5. What function are you
talking about that would require 2+2=5?
...
There exists a function that agrees with + on all values except (2,2),
where it is 5. If we call that function '+', we can still do algebra on
real numbers by special casing the point (2,2) in most theorems, but we
don't want to.
It's clear that 0^0=1 is a choice made by convenience, no doubt made to
simplify the statement of certain theorems, but the fact remains that
(0,0) is a discontinous point of x^y.
Yup.
At best it is undefined, since it's an essential discontinuity,
Nope. x=0 is an essential discontinuity of sgn(x) too, yet sgn(0)=0.
just like x=0 is an essential discontinuity of 1/x.
That is not why 1/0 is left undefined on the real numbers. It's a
convention too, and it is not arbitrary.
What *ought* to be the value of 0^0 is far from
clear; it was a controversy that raged throughout the 19th century and
only in recent decades consensus began to build around 0^0=1.
...
This is the 21st century and it has become clear what 0^0 should be.
There is no value in discrediting the convention by calling it
'arbitrary' when it is not.
