On 24.06.2016 08:49, H. S. Teoh via Digitalmars-d wrote:
...How do you define "number of functions" when m and n are
non-integer?
...
I don't. But even when n is an arbitrary real number, I still want empty
products to be 1.
...
Have a look at this plot: http://www.wolframalpha.com/input/?i=x%5E-y
Can you even spot the discontinuity? (I can't.)
That's because your graph is of the function x^(-y), which is a
completely different beast from the function x^y.
It's a mirror image.
If you look at the
graph of the latter,
Then the point (0,0) is hidden.
you can see how the manifold curves around x=0 in
such a way that the curvature becomes extreme around (0,0), a sign of an
inherent discontinuity.
...
Well, there's no question that the discontinuity is there. It's just
that, if you want to expose the discontinuity by taking limits along
some path not intersecting the y axis, you need to choose it in a
somewhat clever way in order not to end up with the value 1. Of course,
this is not really a strong argument for assigning any specific value at
(0,0).
[...]
Why do you think that is? Again consider my example where a+b is actually
a+b unless a=b=2, in which case it is 5.
Your example has no bearing on this discussion at all. ...
It's another example of a notational convention. I was trying to figure
out if/why you consider some well-motivated conventions more arbitrary
than others.
Besides, it's very clear from basic arithmetic what 2+2 means, whereas
the same can't be said for 0^^0.
...
That's where we disagree. Both expressions arise naturally in e.g.
combinatorics.
...
(The offset by 1 here is IMHO a real example of an unfortunate and
arbitrary choice of notation, but I hope that does not take away from
my real point.)
This "unfortunate and arbitrary" choice was precisely due to the same
idea of aesthetically simplifying the integral that defines the
function. ...
n! = ∫dx [0≤x]xⁿ·e⁻ˣ.
Γ(t) = ∫dx [0≤x]xᵗ⁻¹·e⁻ˣ.
One of those is simpler, and it is not Γ.
...
Defining 0^0=1 in like manner makes certain definitions and use cases
"nicer", but not as nice in other cases. Why not just face the fact that
it's an essential discontinuity that is best left undefined?
...
Because you don't actually care about continuity properties of x^y as a
function in (x,y) in the cases when you encounter 0^0 in practice.
I agree that you sometimes don't want to consider (0,0). If you want to
study x^y as a continuous function, just say something to the effect of:
"Consider the continuous map ℝ⁺×ℝ ∋ (x,y) ↦ xʸ ∈ ℝ". This excludes (0,0)
from consideration in the way you want (it also excludes the case that x
is a negative integer and y is an integer, for example).
There is no good reason to complicate e.g. polynomial and power series
notations by default. Either x or y often (or even, usually) does not
vary continuously (and if y varies, x is usually not 0).
Anyway, 2+2=4 because this makes the definition of + "nicer". It is
not an arbitrary choice. There is /a reason/ why it is "nicer".
This is a totally backwards argument. 2+2=4 because that's how counting
in the real world works,
Yes, and 0^0 = 1 because that is how counting in the real world works.
...
Whereas 0^0 does not reflect real-world counting of any sort,
Yes it does. It counts the number of empty sequences of nothing.
but is a
concept that came about as a generalization of repeated multiplication.
That's one way to think about it, and then you would expect it to
actually generalize repeated multiplication, would you not?
BTW: I found this funny:
http://www.wolframalpha.com/input/?i=product+x+for+i%3D1+to+n
http://www.wolframalpha.com/input/?i=product+0*i+from+i%3D1+to+0
('*i' needed to pass their parser for some reason.)
http://www.wolframalpha.com/input/?i=0%5E0
By treating 0^0 consistently as 1, you never run into this kind of
problem. Doesn't this demonstrate that they are doing it wrong? How
would you design the notation?
