Re: [Haskell-cafe] Re: 0/0 1 == False
On 19 Jan 2008, at 2:52 AM, Kalman Noel wrote:
Jonathan Cast wrote:
On 12 Jan 2008, at 3:23 AM, Kalman Noel wrote:
(2) lim a_n = ∞
[...]
(2) means that the sequence does not converge,
To a value in R. Again, inf is a perfectly well defined extended
real number, and behaves like any other element of R u {-inf, inf}.
(Although that structure isn't quite a field --- 0 * inf isn't
defined, nor is inf - inf).
Out of curiosity, is there some typical application domain for
extended real
numbers?
In calculus and elementary analysis, the notion of limits at/to
infinity, improper Riemann integrals, etc., are introduced by a
succession of `special notations'. Taking the extended real numbers
as the underlying space permits these notations to be defined more
compositionally, because ∞ is now an ordinary mathematical object.
For example, if f is a nonnegative measurable function, ∫f on a
measurable set is /always/ defined (as an extended real number) and
the special case of an `integrable' function is simply one where the
integral (which is an actual mathematical value) is an element of R.
So, when we say ∫f ∈ R, that notation is compositional --- that's
real set membership there. Similarly, ∫_{-∞}^∞ f is defined (as
a Lebesgue integral) the same way any other integral is, because the
interval [-∞, ∞] is a perfectly good mathematical object.
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[Haskell-cafe] Re: 0/0 1 == False
Benja Fallenstein wrote: On Jan 10, 2008 9:22 AM, Mitar [EMAIL PROTECTED] wrote: I understand that proper mathematical behavior would be that as 0/0 is mathematically undefined that 0/0 cannot be even compared to 1. My understanding is that common mathematical practice is that comparing an undefined value to anything (including itself) always yields false; x /= x is sometimes used to formalize x is undefined. In Mathematics one usually avoids comparing undefined values. If one does the result is undefined, too, of course. IOW, the whole calculation is worthless scribble. Cheers Ben ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On Fri, Jan 11, 2008 at 07:10:20PM -0800, Jonathan Cast wrote: On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. Didn't catch this the first time around, but: only to a physicist. I don't understand what you're saying below. Do you mean that the true answer is not 1.0, or that it's not reasonable for the computer to call it NaN? Since 1.0 is the answer you get for high-precision computations, it's hard to see how you can argue that it's a wrong answer. (I mean no disrespect to the author of darcs, but nevertheless the point stands). Back in the real world, 0 / 0 may be defined arbitrarily, or left undefined. (Defining it breaks the wonderful property that, if lim (xn) = x, lim (yn) = y, and x/y = z, then lim (xn / yn) = z. This is considered a Bad Thing by real mathematicians). In fact, in integration theory 0 * inf = 0 for certain 'multiplications', which gives the lie to 0 / 0. -- David Roundy Department of Physics Oregon State University ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On 14 Jan 2008, at 9:56 AM, David Roundy wrote: On Fri, Jan 11, 2008 at 07:10:20PM -0800, Jonathan Cast wrote: On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. Ah. My apologies. Must remember to read first, then flame... You are right that, in Ratio Integer, you get this result. But you can't get that answer in Double. jcc ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On 11 Jan 2008, at 11:25 PM, Achim Schneider wrote: Jonathan Cast [EMAIL PROTECTED] wrote: On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. Didn't catch this the first time around, but: only to a physicist. (I mean no disrespect to the author of darcs, but nevertheless the point stands). Back in the real world, 0 / 0 may be defined arbitrarily, or left undefined. (Defining it breaks the wonderful property that, if lim (xn) = x, lim (yn) = y, and x/y = z, then lim (xn / yn) = z. This is considered a Bad Thing by real mathematicians). In fact, in integration theory 0 * inf = 0 for certain 'multiplications', which gives the lie to 0 / 0. whereas lim( 0 ) * lim( inf ) is anything you want, or, more precisely, the area of the thing you started with. I think you're thinking of the Riemann integral (I was thinking of the Lebesgue integral, and in particular the definition of the integral for step functions). That integral can be informally characterized as the sum of infinitely many terms, each zero (it's a limit of finite sums, where the largest (in the sense of absolute value) term in each sum goes to 0 as the sequence proceeds). That's scarcely a rigorous characterization, and it overlooks a ton of issues (like the fact that neither an infinite sum nor a sum identically 0 terms is actually involved), and it's scarcely relevant to the numerical analysis question of what 0/0 should return except to point out that any definition is going to make *somebody*'s algorithm silently go wrong... jcc ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On 12 Jan 2008, at 3:23 AM, Kalman Noel wrote:
Achim Schneider wrote:
Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
the anything is defined to one (or, rather, is _one_ anything) to be
able to use the abstraction. It's a bit like the difference between
eight pens and a box of pens. If someone knows how to properly
formalise n = 1, please speak up.
Sorry if I still don't follow at all.
That's ok, neither does he.
Here is how I understand (i. e.
have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor
terminology, I have to translate this in my mind from German maths
language ;-).
OK so far. I actually had a professor in uni who would randomly
switch to German during lectures; I'll do my best to follow your
notation :)
My point of posting this is that I don't see how to
accommodate the lim notation as I know it with your term. The limit of
infinity? What is the limit of infinity,
If you extend your concept of `sequence' to include sequences of
extended real numbers, the limit of the sequence that is identically
inf is inf. Otherwise, the notation isn't really meaningful.
and why should I multiplicate
it with 0? Why should I get 1?
(1) lim a_n = a(where a ∈ R)
(2) lim a_n = ∞
(3) lim a_n = − ∞
(4) lim { x → x0 } f(x) = y (where f is a function into R)
(1) means that the sequence of reals a_n converges towards a.
(2) means that the sequence does not converge,
To a value in R. Again, inf is a perfectly well defined extended
real number, and behaves like any other element of R u {-inf, inf}.
(Although that structure isn't quite a field --- 0 * inf isn't
defined, nor is inf - inf).
because you can
always find a value that is /larger/ than what you hoped might
be the limit.
(3) means that the sequence does not converge, because you can
always find a value that is /smaller/ than what you hoped
might
be the limit.
(4) means that for any sequence of reals (x_n ∈ dom f)
converging
towards x0, we have lim f(x_n) = y. For this equation
again, we
have the three cases above.
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Re: [Haskell-cafe] Re: 0/0 1 == False
On 12 Jan 2008, at 3:33 AM, Cristian Baboi wrote:
On Sat, 12 Jan 2008 13:23:41 +0200, Kalman Noel
[EMAIL PROTECTED] wrote:
Achim Schneider wrote:
Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
the anything is defined to one (or, rather, is _one_ anything) to be
able to use the abstraction. It's a bit like the difference between
eight pens and a box of pens. If someone knows how to properly
formalise n = 1, please speak up.
Sorry if I still don't follow at all. Here is how I understand
(i. e.
have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor
terminology, I have to translate this in my mind from German maths
language ;-). My point of posting this is that I don't see how to
accommodate the lim notation as I know it with your term. The
limit of
infinity? What is the limit of infinity, and why should I
multiplicate
it with 0? Why should I get 1?
(1) lim a_n = a(where a ∈ R)
(2) lim a_n = ∞
(3) lim a_n = − ∞
(4) lim { x → x0 } f(x) = y (where f is a function into R)
(1) means that the sequence of reals a_n converges towards a.
(2) means that the sequence does not converge, because you can
always find a value that is /larger/ than what you hoped
might
be the limit.
(3) means that the sequence does not converge, because you can
always find a value that is /smaller/ than what you hoped
might
be the limit.
(4) means that for any sequence of reals (x_n ∈ dom f)
converging
towards x0, we have lim f(x_n) = y. For this equation
again, we
have the three cases above.
Suppose lim a_n = a , lim b_n = b, c_2n = a_n, c_2n+1 = b_n.
What is lim c_n ?
If a = b, lim c_n = a = b. Otherwise lim c_n is undefined, in the
sense that there is no number c such that the assertion lim c_n = c
is true. The `=' in this assertion isn't an equality; it's just a
bit of math syntax, used in the special case where it isn't overly
confusing.
jcc
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Re: [Haskell-cafe] Re: 0/0 1 == False
On 12 Jan 2008, at 4:06 AM, Achim Schneider wrote: Kalman Noel [EMAIL PROTECTED] wrote: Achim Schneider wrote: Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and the anything is defined to one (or, rather, is _one_ anything) to be able to use the abstraction. It's a bit like the difference between eight pens and a box of pens. If someone knows how to properly formalise n = 1, please speak up. Sorry if I still don't follow at all. Here is how I understand (i. e. have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor terminology, I have to translate this in my mind from German maths language ;-). My point of posting this is that I don't see how to accommodate the lim notation as I know it with your term. The limit of infinity? What is the limit of infinity, and why should I multiplicate it with 0? Why should I get 1? n * n = 1 where lim lim n - 0 n - oo wtf? You don't get 1, you start off with it. If you start off with anything, you can often end up with it as well. Enlightenment comes when you realize that sometimes you don't, and you acquire the ability to change your mind. If you want to find the area of a function, Under, not of. you slice 1^2 into infinitely many parts and then look how much every single slice differs from lim( 0 ) * 1 Beg pardon? lim(0) = 0. lim(0) * 1 = 0. , all that lim( inf ) many times. You can't do this. lim(inf) = inf is an extended real number, and is completely unrelated to aleph_0 = the least upper bound of N. inf is not a cardinality and cannot be a number of times. When you've finished counting pebble, you know how to scale this 1^2 to match it with your normal value of 1. n = 12 n = 1 * n now, 1 is twelve. QED: The wrath of algebra. `Algebra' One as a pure concept is a very strange beast, as it can mean anything. To you. Mathematicians assign a different meaning to the concept. jcc ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
Jonathan Cast [EMAIL PROTECTED] wrote: You don't get 1, you start off with it. If you start off with anything, you can often end up with it as well. Enlightenment comes when you realize that sometimes you don't, and you acquire the ability to change your mind. n = 12 n = 1 * n now, 1 is twelve. QED: The wrath of algebra. `Algebra' Damnit, this is exactly what I meant. It's all about analysing some 1, until you found something interesting, and then continue all over with that new interesting thing that is now also a 1, 'cos you want to re-use your framework. I'll better stop right here. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
Achim Schneider wrote: whereas lim( 0 ) * lim( inf ) is anything you want Indeed I suppose that »lim inf«, which is a notation I'm not familiar with, is not actually defined to mean anything? Kalman -- Find out how you can get spam free email. http://www.bluebottle.com/tag/3 ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
Kalman Noel [EMAIL PROTECTED] wrote: Achim Schneider wrote: whereas lim( 0 ) * lim( inf ) is anything you want Indeed I suppose that »lim inf«, which is a notation I'm not familiar with, is not actually defined to mean anything? It's an ad-hoc expression of as the slices approach zero size, their number approaches infinity. It's more an observation than anything else. I have no idea how a professional mathematician would formalise it. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On Sat, 12 Jan 2008, Achim Schneider wrote: Kalman Noel [EMAIL PROTECTED] wrote: Achim Schneider wrote: whereas lim( 0 ) * lim( inf ) is anything you want Indeed I suppose that »lim inf«, which is a notation I'm not familiar with, is not actually defined to mean anything? It's an ad-hoc expression of as the slices approach zero size, their number approaches infinity. It's more an observation than anything else. I have no idea how a professional mathematician would formalise it. In Haskell you could professionally formalize it, if you would have a function, which computes the limit of a sequence: limit :: [Real] - Real (Cf. Why functional progrmaming matters http://www.math.chalmers.se/~rjmh/Papers/whyfp.html) let xs = iterate (1+) 1-- example for lim (inf) ys = map recip xs -- example for lim (0) in limit (zipWith (*) xs ys) If 'limit xs' and 'limit ys' are defined, then it holds limit (zipWith (*) xs ys) == limit xs * limit ys but in your case 'limit xs' does not exist. You can imagine it as infinity, but to make it formally safe, you need something non-standard analysis. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
Achim Schneider [EMAIL PROTECTED] wrote: Kalman Noel [EMAIL PROTECTED] wrote: Achim Schneider wrote: whereas lim( 0 ) * lim( inf ) is anything you want Indeed I suppose that »lim inf«, which is a notation I'm not familiar with, is not actually defined to mean anything? It's an ad-hoc expression of as the slices approach zero size, their number approaches infinity. It's more an observation than anything else. I have no idea how a professional mathematician would formalise it. Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and the anything is defined to one (or, rather, is _one_ anything) to be able to use the abstraction. It's a bit like the difference between eight pens and a box of pens. If someone knows how to properly formalise n = 1, please speak up. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
Achim Schneider wrote:
Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
the anything is defined to one (or, rather, is _one_ anything) to be
able to use the abstraction. It's a bit like the difference between
eight pens and a box of pens. If someone knows how to properly
formalise n = 1, please speak up.
Sorry if I still don't follow at all. Here is how I understand (i. e.
have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor
terminology, I have to translate this in my mind from German maths
language ;-). My point of posting this is that I don't see how to
accommodate the lim notation as I know it with your term. The limit of
infinity? What is the limit of infinity, and why should I multiplicate
it with 0? Why should I get 1?
(1) lim a_n = a(where a ∈ R)
(2) lim a_n = ∞
(3) lim a_n = − ∞
(4) lim { x → x0 } f(x) = y (where f is a function into R)
(1) means that the sequence of reals a_n converges towards a.
(2) means that the sequence does not converge, because you can
always find a value that is /larger/ than what you hoped might
be the limit.
(3) means that the sequence does not converge, because you can
always find a value that is /smaller/ than what you hoped might
be the limit.
(4) means that for any sequence of reals (x_n ∈ dom f) converging
towards x0, we have lim f(x_n) = y. For this equation again, we
have the three cases above.
Kalman
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Re: [Haskell-cafe] Re: 0/0 1 == False
On Sat, 12 Jan 2008 13:23:41 +0200, Kalman Noel
[EMAIL PROTECTED] wrote:
Achim Schneider wrote:
Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
the anything is defined to one (or, rather, is _one_ anything) to be
able to use the abstraction. It's a bit like the difference between
eight pens and a box of pens. If someone knows how to properly
formalise n = 1, please speak up.
Sorry if I still don't follow at all. Here is how I understand (i. e.
have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor
terminology, I have to translate this in my mind from German maths
language ;-). My point of posting this is that I don't see how to
accommodate the lim notation as I know it with your term. The limit of
infinity? What is the limit of infinity, and why should I multiplicate
it with 0? Why should I get 1?
(1) lim a_n = a(where a ∈ R)
(2) lim a_n = ∞
(3) lim a_n = − ∞
(4) lim { x → x0 } f(x) = y (where f is a function into R)
(1) means that the sequence of reals a_n converges towards a.
(2) means that the sequence does not converge, because you can
always find a value that is /larger/ than what you hoped might
be the limit.
(3) means that the sequence does not converge, because you can
always find a value that is /smaller/ than what you hoped might
be the limit.
(4) means that for any sequence of reals (x_n ∈ dom f) converging
towards x0, we have lim f(x_n) = y. For this equation again, we
have the three cases above.
Suppose lim a_n = a , lim b_n = b, c_2n = a_n, c_2n+1 = b_n.
What is lim c_n ?
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[Haskell-cafe] Re: 0/0 1 == False
Kalman Noel [EMAIL PROTECTED] wrote: Achim Schneider wrote: Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and the anything is defined to one (or, rather, is _one_ anything) to be able to use the abstraction. It's a bit like the difference between eight pens and a box of pens. If someone knows how to properly formalise n = 1, please speak up. Sorry if I still don't follow at all. Here is how I understand (i. e. have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor terminology, I have to translate this in my mind from German maths language ;-). My point of posting this is that I don't see how to accommodate the lim notation as I know it with your term. The limit of infinity? What is the limit of infinity, and why should I multiplicate it with 0? Why should I get 1? n * n = 1 where lim lim n - 0 n - oo You don't get 1, you start off with it. If you want to find the area of a function, you slice 1^2 into infinitely many parts and then look how much every single slice differs from lim( 0 ) * 1, all that lim( inf ) many times. When you've finished counting pebble, you know how to scale this 1^2 to match it with your normal value of 1. n = 12 n = 1 * n now, 1 is twelve. QED: The wrath of algebra. One as a pure concept is a very strange beast, as it can mean anything. Like, if you take something and try to understand it by dividing it successively into infinitely many parts, the meaning of each part will approach zero, as you don't change the thing you're analysing but the nature of your lenses. If you ever want to watch a zen master frowning or despair, tell him exactly that. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
2008/1/12, Cristian Baboi [EMAIL PROTECTED]:
Suppose lim a_n = a , lim b_n = b, c_2n = a_n, c_2n+1 = b_n.
What is lim c_n ?
This reminds me of these good old days in Classe prépa (something
typically french) doing these silly things with sequences and
Epsilons...
So let's try to do it :
First, if we assume that a = b, that's ok : lim c_n = a = b
Second, we consider that a \neq b. We will show that c_n do not have a limit.
By contraction, we assume that c_n has a limit c.
Then, by definition of limit, we have :
\exists N, \forall n \geq N, | c_n - c | \leq \epsilon
Therefore, this result holds for the suits (c_2n) and (c_{2n+1}), ie. :
\exists N, \forall n \geq N, | c_2n - c | \leq \epsilon and
\exists N, \forall n \geq N, | c_{2n+1} - c | \leq \epsilon
By definition of c_2n and c_{2n+1}, we can replace them by a_n and b_n :
\exists N, \forall n \geq N, | a_n - c | \leq \epsilon and
\exists N, \forall n \geq N, | b_n -c | \leq \epsilon
By definition of limit, this means that :
lim a_n = c andlim b_n = c
By unicity of the limit, this means that :
a = c and b = c
Leading to a contradiction as we have assumed that a \neq b.
Q.E.D.
But I'm still wondering about the relation between this and Haskell...
--
Pierre-Evariste DAGAND
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Re: [Haskell-cafe] Re: 0/0 1 == False
On 12 Jan 2008, at 1:45 AM, Achim Schneider wrote: Kalman Noel [EMAIL PROTECTED] wrote: Achim Schneider wrote: whereas lim( 0 ) * lim( inf ) is anything you want Indeed I suppose that »lim inf«, which is a notation I'm not familiar with, is not actually defined to mean anything? It's an ad-hoc expression of as the slices approach zero size, their number approaches infinity. It's more an observation than anything else. I have no idea how a professional mathematician would formalise it. I suspected as much. You mean that, for any extended real number z, two sequences of real numbers xn and yn may be chosen so that lim(xn) = 0, lim(yn) = inf, and lim(xn * yn) = z. jcc -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On Sat, 12 Jan 2008 13:55:19 +0200, Kalman Noel [EMAIL PROTECTED] wrote: Cristian Baboi: Suppose lim a_n = a , lim b_n = b, c_2n = a_n, c_2n+1 = b_n. What is lim c_n ? If my intuition was of any importance here, it would claim that c_n diverges, because if I roughly approximate c_n by the sequence c' = ⟨a,b,a,b,...⟩, then I note that c' oscillates, so c_n »roughly oscillates«. Kalman You mean something like this x=a:b:x ? ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
Cristian Baboi wrote: Cristian Baboi: Suppose lim a_n = a , lim b_n = b, c_2n = a_n, c_2n+1 = b_n. What is lim c_n ? If my intuition was of any importance here, it would claim that c_n diverges, because if I roughly approximate c_n by the sequence c' = ⟨a,b,a,b,...⟩, then I note that c' oscillates, so c_n »roughly oscillates«. You mean something like this x=a:b:x ? Ah, you try to be on-topic by using Haskell syntax :) Obviously though, I really forgot to consider the case a = b... But that's enough OT for today, I guess. Kalman -- Get a free email account with anti spam protection. http://www.bluebottle.com/tag/2 ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
Achim Schneider [EMAIL PROTECTED] writes: You need to use the / operator, if you want to do floating-point division. Yes, exactly, integers don't have +-0 and +-infinity... only (obviously) a kind of nan. No, failure (exception, bottom) is different from NaN, which is just another value in the domain - admittedly one which behaves rather strangely. Said differently: I don't know a thing about floats or numerics. Perhaps it helps to think of floating point values as intervals? If +0 means some number between 0 and the next possible representable number (and similar for -0), it may make more sense to have 1/+0 and 1/-0 behave differently. -k -- If I haven't seen further, it is by standing in the footprints of giants ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On Fri, Jan 11, 2008 at 02:54:20PM +0100, Achim Schneider wrote: +-0 / +-0 is always NaN 'cos you can't tell which one is bigger and thus can't decide between positive and negative Infinity, and it isn't both, either. But then there's +0/0 and -0/0, which would be +Infinity and -Infinity, and +0 0 -0. AFAIK there are no floats with three zero values, though. No, +0/0 might be 0 or finite instead of +Infinity, so it's a NaN. e.g. consider Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. -- David Roundy Department of Physics Oregon State University ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
Ketil Malde [EMAIL PROTECTED] wrote: Achim Schneider [EMAIL PROTECTED] writes: You need to use the / operator, if you want to do floating-point division. Yes, exactly, integers don't have +-0 and +-infinity... only (obviously) a kind of nan. No, failure (exception, bottom) is different from NaN, which is just another value in the domain - admittedly one which behaves rather strangely. s/a kind of/not entirely unlike a/ Said differently: I don't know a thing about floats or numerics. Perhaps it helps to think of floating point values as intervals? If +0 means some number between 0 and the next possible representable number (and similar for -0), it may make more sense to have 1/+0 and 1/-0 behave differently. Hmmm... ah. +-0 / +-0 is always NaN 'cos you can't tell which one is bigger and thus can't decide between positive and negative Infinity, and it isn't both, either. But then there's +0/0 and -0/0, which would be +Infinity and -Infinity, and +0 0 -0. AFAIK there are no floats with three zero values, though. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
David Roundy [EMAIL PROTECTED] wrote: Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. Wl.. math philosophy, Ok. You can't divide something in a way that uses no slices. You just don't cut, if you cut zero times. Which is what you do when you divide by one, mind you, not when you divide by zero. Division by [1..0] equals multiplication by [1..]. You can't get to the end of either spectrum, just axiomatically dodge around the singularities to axiomatically connect the loose ends. There is no true answer here, the question is wrong. And you're using floats where only rationals can tackle the problem. I was wrong in claiming that But then there's +0/0 and -0/0, which would be +Infinity and -Infinity, and +0 0 -0. AFAIK there are no floats with three zero values, though. , both are NaN. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
Also, there are only two zeros, +0 and -0 and they compare equal. On Jan 11, 2008 10:12 AM, Achim Schneider [EMAIL PROTECTED] wrote: I was wrong in claiming that But then there's +0/0 and -0/0, which would be +Infinity and -Infinity, and +0 0 -0. AFAIK there are no floats with three zero values, though. , both are NaN. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. Didn't catch this the first time around, but: only to a physicist. (I mean no disrespect to the author of darcs, but nevertheless the point stands). Back in the real world, 0 / 0 may be defined arbitrarily, or left undefined. (Defining it breaks the wonderful property that, if lim (xn) = x, lim (yn) = y, and x/y = z, then lim (xn / yn) = z. This is considered a Bad Thing by real mathematicians). In fact, in integration theory 0 * inf = 0 for certain 'multiplications', which gives the lie to 0 / 0. Wl.. math philosophy, Ok. You can't divide something in a way that uses no slices. You just don't cut, if you cut zero times. Which is what you do when you divide by one, mind you, not when you divide by zero. Division by [1..0] equals multiplication by [1..]. Right. (Although 0 * inf is defined by fiat, as noted above). You can't get to the end of either spectrum, just axiomatically dodge around the singularities to axiomatically connect the loose ends. `Axiomatically' --- you mean by re-defining standard notation like * and / to mean what you need in this case. I think this is a different thing than setting up ZFC so everyone agrees on what a `set' is from henceforth. There is no true answer here, the question is wrong. Exactly. jcc ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
Jonathan Cast [EMAIL PROTECTED] wrote: On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: Prelude let x=1e-300/1e300 Prelude x 0.0 Prelude x/x NaN The true answer here is that x/x == 1.0 (not 0 or +Infinity), but there's no way for the computer to know this, so it's NaN. Didn't catch this the first time around, but: only to a physicist. (I mean no disrespect to the author of darcs, but nevertheless the point stands). Back in the real world, 0 / 0 may be defined arbitrarily, or left undefined. (Defining it breaks the wonderful property that, if lim (xn) = x, lim (yn) = y, and x/y = z, then lim (xn / yn) = z. This is considered a Bad Thing by real mathematicians). In fact, in integration theory 0 * inf = 0 for certain 'multiplications', which gives the lie to 0 / 0. whereas lim( 0 ) * lim( inf ) is anything you want, or, more precisely, the area of the thing you started with. It's like taking seven balls, assembling them into a hexagon and claiming it's a circle. Such things just happen if you are working with pure abstractions. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
John Meacham [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 11:17:18AM +0200, Yitzchak Gale wrote: The special case of 1/0 is less clear, though. One might decide that it should be an error rather than NaN, as some languages have. It is neither, 1/0 = Infinity -1/0 = -Infinity Just out of curiosity: 1/-0 = -Infinity? -1/-0 = Infinity? If you don't have two separate values for nothing, one approaching from negativeness and one from positiveness, defining the result to be infinite instead of NaN makes no sense IMHO. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On Thu, Jan 10, 2008 at 09:24:34PM +0100, Achim Schneider wrote: John Meacham [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 11:17:18AM +0200, Yitzchak Gale wrote: The special case of 1/0 is less clear, though. One might decide that it should be an error rather than NaN, as some languages have. It is neither, 1/0 = Infinity -1/0 = -Infinity Just out of curiosity: 1/-0 = -Infinity? -1/-0 = Infinity? Yes. (You could have tried this for yourself, you know... but I suppose haskell-cafe isn't a bad interactive Haskell interpreter, perhaps more user friendly than ghci.) -- David Roundy Department of Physics Oregon State University ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
David Roundy [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 09:24:34PM +0100, Achim Schneider wrote: John Meacham [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 11:17:18AM +0200, Yitzchak Gale wrote: The special case of 1/0 is less clear, though. One might decide that it should be an error rather than NaN, as some languages have. It is neither, 1/0 = Infinity -1/0 = -Infinity Just out of curiosity: 1/-0 = -Infinity? -1/-0 = Infinity? Yes. (You could have tried this for yourself, you know... but I suppose haskell-cafe isn't a bad interactive Haskell interpreter, perhaps more user friendly than ghci.) Prelude 1 `div` 0 *** Exception: divide by zero That's it. One just shouldn't just extrapolate and think you didn't mean GHC but IEEE... -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
Re: [Haskell-cafe] Re: 0/0 1 == False
On Thu, Jan 10, 2008 at 09:41:53PM +0100, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 09:24:34PM +0100, Achim Schneider wrote: John Meacham [EMAIL PROTECTED] wrote: 1/0 = Infinity -1/0 = -Infinity Just out of curiosity: 1/-0 = -Infinity? -1/-0 = Infinity? Yes. (You could have tried this for yourself, you know... but I suppose haskell-cafe isn't a bad interactive Haskell interpreter, perhaps more user friendly than ghci.) Prelude 1 `div` 0 *** Exception: divide by zero That's it. One just shouldn't just extrapolate and think you didn't mean GHC but IEEE... Prelude 1/(-0) -Infinity You need to use the / operator, if you want to do floating-point division. -- David Roundy Department of Physics Oregon State University ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
[Haskell-cafe] Re: 0/0 1 == False
David Roundy [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 09:41:53PM +0100, Achim Schneider wrote: David Roundy [EMAIL PROTECTED] wrote: On Thu, Jan 10, 2008 at 09:24:34PM +0100, Achim Schneider wrote: John Meacham [EMAIL PROTECTED] wrote: 1/0 = Infinity -1/0 = -Infinity Just out of curiosity: 1/-0 = -Infinity? -1/-0 = Infinity? Yes. (You could have tried this for yourself, you know... but I suppose haskell-cafe isn't a bad interactive Haskell interpreter, perhaps more user friendly than ghci.) Prelude 1 `div` 0 *** Exception: divide by zero That's it. One just shouldn't just extrapolate and think you didn't mean GHC but IEEE... Prelude 1/(-0) -Infinity You need to use the / operator, if you want to do floating-point division. Yes, exactly, integers don't have +-0 and +-infinity... only (obviously) a kind of nan. It's just that with the stuff I do I know I have some logical problem in my formulas when I get any special floating point value anywhere, and using --excess-precision can only make the numbers more precise. Said differently: I don't know a thing about floats or numerics. -- (c) this sig last receiving data processing entity. Inspect headers for past copyright information. All rights reserved. Unauthorised copying, hiring, renting, public performance and/or broadcasting of this signature prohibited. ___ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe
