[sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Monday, July 13, 2015 at 7:15:40 PM UTC+2, Simon King wrote: So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. +1 -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Mon, 13 Jul 2015, mmarco wrote: Related to that, i think the current 17? digits are way too much for a visually nice representation. I would prefear to see 1.4142... than 1.414213562373095? inside an expression. OK for me. But I would like to have a setting for default length; I have done some computations where differences arise, for example, about tenth digit. Not a big deal, thought. -- Jori Mäntysalo
[sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
Hi! On 2015-07-13, Nathann Cohen nathann.co...@gmail.com wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. But it seems like an appealing idea to show a nice expression for algebraic numbers of degree up to 4. If we cannot get rid totally of this numerical representation, what would you think of replacing this '?' by a 'alg', which would be (slightly) more informative, e.g.: 1.4142134... looks exact to me: ... seems to suggest that Sage knows all (potentially infinitely many) digits but can't show them all, whereas ? seems to suggest that the last shown digit is questionable (i.e., subject to rounding errors), i.e., ? seems to suggest that Sage doesn't know the exact value. So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
I agree with Simon, although finding a nice expression like sqrt(2)+3^(1/3) can be very costly deppending on how the algebraic number was constructed. Anyways we could have such an expression for the cases where it is evident from the number construction. Related to that, i think the current 17? digits are way too much for a visually nice representation. I would prefear to see 1.4142... than 1.414213562373095? inside an expression. El lunes, 13 de julio de 2015, 19:15:40 (UTC+2), Simon King escribió: Hi! On 2015-07-13, Nathann Cohen nathan...@gmail.com javascript: wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. But it seems like an appealing idea to show a nice expression for algebraic numbers of degree up to 4. If we cannot get rid totally of this numerical representation, what would you think of replacing this '?' by a 'alg', which would be (slightly) more informative, e.g.: 1.4142134... looks exact to me: ... seems to suggest that Sage knows all (potentially infinitely many) digits but can't show them all, whereas ? seems to suggest that the last shown digit is questionable (i.e., subject to rounding errors), i.e., ? seems to suggest that Sage doesn't know the exact value. So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
I agree with Simon, although finding a nice expression like sqrt(2)+3^(1/3) can be very costly deppending on how the algebraic number was constructed. Yepyep. As Simon said we cannot always express algebraic numbers in such a nice way, though. Well, if you want to build such an object *in Sage* then you must describe your value somehow, and it is also stored internally somewhere.. And I wonder how, and whether we can base the representation on this internal version of the value :-) Anyways we could have such an expression for the cases where it is evident from the number construction. +1 Related to that, i think the current 17? digits are way too much for a visually nice representation. I would prefear to see 1.4142... than 1.414213562373095? inside an expression. +1. It would also lessen the odds of having people believe that it is stored as a float (with possibly many, but finitely many, digits) Nathann -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
Jonas -- the implementation of QQbar elements is precisely via real intervals and a polynomial satisfied by the number (not always the min poly unless that is forced, since that can be expensive). This I think that the only issue is in how real interval field elements are represented. No-one (or almost) has ever liked the current form ending in a ? which does of course have a logic behind it but is *very* hard to explain (and to remember). The sqrt() - style representation is only really practical for number of degree 2 -- in principle one could do more general radical expressions, and sqrt(2)+sqrt(3) looks rather nicer than sage: QQbar(2).sqrt()+QQbar(3).sqrt() 3.146264369941973? but that is (a) expensive and (b) only applies to rather special algebraic numbers. John On 13 July 2015 at 18:49, William Stein wst...@gmail.com wrote: On Mon, Jul 13, 2015 at 10:15 AM, Simon King simon.k...@uni-jena.de wrote: Hi! On 2015-07-13, Nathann Cohen nathann.co...@gmail.com wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. But it seems like an appealing idea to show a nice expression for algebraic numbers of degree up to 4. If we cannot get rid totally of this numerical representation, what would you think of replacing this '?' by a 'alg', which would be (slightly) more informative, e.g.: 1.4142134... looks exact to me: ... seems to suggest that Sage knows all (potentially infinitely many) digits but can't show them all, whereas ? seems to suggest that the last shown digit is questionable (i.e., subject to rounding errors), i.e., ? seems to suggest that Sage doesn't know the exact value. Yes. Also, with interval arithmetic, that is precisely what it means: sage: RealIntervalField(53)(sqrt(2)) 1.414213562373095? sage: QQbar(sqrt(2)) 1.414213562373095? It's certainly not good that the above two print in the same way. The (mysterious [1]) person who made those design choices -- Carl Witty -- isn't contributing anymore, or I'd ask his opinion. [1] http://stackoverflow.com/users/684532/carl-witty So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout. -- William (http://wstein.org) -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Monday, 13 July 2015 18:49:48 UTC+1, William wrote: On Mon, Jul 13, 2015 at 10:15 AM, Simon King simon...@uni-jena.de javascript: wrote: Hi! On 2015-07-13, Nathann Cohen nathan...@gmail.com javascript: wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. But it seems like an appealing idea to show a nice expression for algebraic numbers of degree up to 4. If we cannot get rid totally of this numerical representation, what would you think of replacing this '?' by a 'alg', which would be (slightly) more informative, e.g.: 1.4142134... looks exact to me: ... seems to suggest that Sage knows all (potentially infinitely many) digits but can't show them all, whereas ? seems to suggest that the last shown digit is questionable (i.e., subject to rounding errors), i.e., ? seems to suggest that Sage doesn't know the exact value. Yes. Also, with interval arithmetic, that is precisely what it means: sage: RealIntervalField(53)(sqrt(2)) 1.414213562373095? sage: QQbar(sqrt(2)) 1.414213562373095? It's certainly not good that the above two print in the same way. The (mysterious [1]) person who made those design choices -- Carl Witty -- isn't contributing anymore, or I'd ask his opinion. As we now allow unicode, we can probably find a better character than '?'. Say ⇘, or ↴, or even ¿. [1] http://stackoverflow.com/users/684532/carl-witty So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+...@googlegroups.com javascript:. To post to this group, send email to sage-...@googlegroups.com javascript:. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout. -- William (http://wstein.org) -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
¿¡Qué estás diciendo!? On Monday, July 13, 2015 at 9:14:51 PM UTC+2, Dima Pasechnik wrote: On Monday, 13 July 2015 18:49:48 UTC+1, William wrote: On Mon, Jul 13, 2015 at 10:15 AM, Simon King simon...@uni-jena.de wrote: Hi! On 2015-07-13, Nathann Cohen nathan...@gmail.com wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. But it seems like an appealing idea to show a nice expression for algebraic numbers of degree up to 4. If we cannot get rid totally of this numerical representation, what would you think of replacing this '?' by a 'alg', which would be (slightly) more informative, e.g.: 1.4142134... looks exact to me: ... seems to suggest that Sage knows all (potentially infinitely many) digits but can't show them all, whereas ? seems to suggest that the last shown digit is questionable (i.e., subject to rounding errors), i.e., ? seems to suggest that Sage doesn't know the exact value. Yes. Also, with interval arithmetic, that is precisely what it means: sage: RealIntervalField(53)(sqrt(2)) 1.414213562373095? sage: QQbar(sqrt(2)) 1.414213562373095? It's certainly not good that the above two print in the same way. The (mysterious [1]) person who made those design choices -- Carl Witty -- isn't contributing anymore, or I'd ask his opinion. As we now allow unicode, we can probably find a better character than '?'. Say ⇘, or ↴, or even ¿. [1] http://stackoverflow.com/users/684532/carl-witty So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+...@googlegroups.com. To post to this group, send email to sage-...@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout. -- William (http://wstein.org) -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On 13/07/15 19:15, Simon King wrote:
Hi!
On 2015-07-13, Nathann Cohen nathann.co...@gmail.com wrote:
sage: sqrt(2) # a symbolic ring element
sqrt(2)
sage: QQbar(sqrt(2)) # an algebraic value
1.414213562373095?
It is true that this final '?' sounds more like a '...', as if some additional
digits were hidden in a value stored as a float/double. Yet it is exact.
How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always
provide such a representation cheaply? Could we display it as 'sqrt(2)' at least
when it is free to do so?
The elements of QQbar are the solutions of algebraic equations. As you
probably know, the solutions of algebraic equations of degree 4 can, in
general, not be expressed that nicely. But it seems like an appealing
idea to show a nice expression for algebraic numbers of degree up to 4.
If we cannot get rid totally of this numerical representation, what would you
think of replacing this '?' by a 'alg', which would be (slightly) more
informative, e.g.:
1.4142134... looks exact to me: ... seems to suggest that Sage knows all
(potentially infinitely many) digits but can't show them all, whereas ? seems
to suggest that the last shown digit is questionable (i.e., subject to rounding
errors), i.e., ? seems to suggest that Sage doesn't know the exact value.
+1
It is consistent with
sage: words.FibonacciWord([0,1])
word: 0100101001001010010100100101001001010010...
sage: continued_fraction(pi)
[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, 2, 2, 2, 2, ...]
sage: IntegerRange(0,+Infinity,2)
{0, 2, ...}
Vincent
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Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On 13.07.2015 19:33, Nathann Cohen wrote: I agree with Simon, although finding a nice expression like sqrt(2)+3^(1/3) can be very costly deppending on how the algebraic number was constructed. Yepyep. As Simon said we cannot always express algebraic numbers in such a nice way, though. Well, if you want to build such an object *in Sage* then you must describe your value somehow, and it is also stored internally somewhere.. And I wonder how, and whether we can base the representation on this internal version of the value :-) Anyways we could have such an expression for the cases where it is evident from the number construction. -1 I definitely prefer a uniform output. If you want a symbolic representation then convert the element to one from SR (symbolic ring). This is basically what you are asking with where it is evident anyway... Of course you could add a function which tries to do exactly that but in my opinion the default output should have a uniform look for a given parent. That way you also realize (better) over what parent you are working with currently. Sidenote: Another useful representation is the minimal polynomial + exact bounds (maybe interval field element?) for a root location. I implemented a proof of concept of such a field once where all elements were displayed as their minimal polynomial + bounds for the root location. Arithmetic operations like addition or multiplication can be defined for the root polynomial. What is harder are the root location bounds (+ irreducible parts of the polynomial) which have to be recalculated I suppose. Best Jonas -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Mon, Jul 13, 2015 at 10:15 AM, Simon King simon.k...@uni-jena.de wrote: Hi! On 2015-07-13, Nathann Cohen nathann.co...@gmail.com wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. But it seems like an appealing idea to show a nice expression for algebraic numbers of degree up to 4. If we cannot get rid totally of this numerical representation, what would you think of replacing this '?' by a 'alg', which would be (slightly) more informative, e.g.: 1.4142134... looks exact to me: ... seems to suggest that Sage knows all (potentially infinitely many) digits but can't show them all, whereas ? seems to suggest that the last shown digit is questionable (i.e., subject to rounding errors), i.e., ? seems to suggest that Sage doesn't know the exact value. Yes. Also, with interval arithmetic, that is precisely what it means: sage: RealIntervalField(53)(sqrt(2)) 1.414213562373095? sage: QQbar(sqrt(2)) 1.414213562373095? It's certainly not good that the above two print in the same way. The (mysterious [1]) person who made those design choices -- Carl Witty -- isn't contributing anymore, or I'd ask his opinion. [1] http://stackoverflow.com/users/684532/carl-witty So, I'd prefer to display elements of QQbar as floating point numbers (what default precision?), always rounded DOWN to the last digit that is displayed, and followed by ... (not ?) unless the displayed value is exact. So, what is displayed is an initial part of the potentially infinite sequence of digits. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout. -- William (http://wstein.org) -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On 2015-07-13 21:55, Bill Hart wrote: The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. This is slightly incorrect. The general quintic can be solved in terms of Jacobi theta functions, the general sextic in terms of Kampe de Feriet functions, amongst others. In general Mellin integrals, Fuchsian functions and theta functions can be used to solve general equations of degree n. ...which proves the original poster's point about not be expressed that nicely. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Monday, 13 July 2015 22:07:46 UTC+2, Jeroen Demeyer wrote: On 2015-07-13 21:55, Bill Hart wrote: The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. This is slightly incorrect. The general quintic can be solved in terms of Jacobi theta functions, the general sextic in terms of Kampe de Feriet functions, amongst others. In general Mellin integrals, Fuchsian functions and theta functions can be used to solve general equations of degree n. ...which proves the original poster's point about not be expressed that nicely. The formulas for the quintic in terms of Jacobi functions, for example, don't look that much more frightening than those for the quartic in terms of radicals. I don't see any explosion in complexity. Perhaps it is a matter of taste. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Monday, 13 July 2015 19:15:40 UTC+2, Simon King wrote: Hi! On 2015-07-13, Nathann Cohen nathan...@gmail.com javascript: wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. This is slightly incorrect. The general quintic can be solved in terms of Jacobi theta functions, the general sextic in terms of Kampe de Feriet functions, amongst others. In general Mellin integrals, Fuchsian functions and theta functions can be used to solve general equations of degree n. Bill. -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On 13/07/15 19:48, Jonas Jermann wrote: On 13.07.2015 19:33, Nathann Cohen wrote: I agree with Simon, although finding a nice expression like sqrt(2)+3^(1/3) can be very costly deppending on how the algebraic number was constructed. Yepyep. As Simon said we cannot always express algebraic numbers in such a nice way, though. Well, if you want to build such an object *in Sage* then you must describe your value somehow, and it is also stored internally somewhere.. And I wonder how, and whether we can base the representation on this internal version of the value :-) Anyways we could have such an expression for the cases where it is evident from the number construction. -1 I definitely prefer a uniform output. Me too. I would really much prefer uniform output and uniform storage... If somebody input sum(QQbar(i).sqrt() for i in range(100)) do we really want to output 1 + sqrt(2) + sqrt(3) + etc I guess not. And we do not want to slow down the computation of numbers in QQbar because one can potentially have a nicer output form. If you want a symbolic representation then convert the element to one from SR (symbolic ring). This is basically what you are asking with where it is evident anyway... Of course you could add a function which tries to do exactly that but in my opinion the default output should have a uniform look for a given parent. That way you also realize (better) over what parent you are working with currently. That is a different matter and we already have #17516 for that. Vincent -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
Re: [sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
On Monday, 13 July 2015 20:30:34 UTC+1, Volker Braun wrote: ¿¡Qué estás diciendo!? So that '¿' indicates the beginning of the questionable part! -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
[sage-devel] Re: Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)
Hi Bill, On 2015-07-13, Bill Hart goodwillh...@googlemail.com wrote: On Monday, 13 July 2015 19:15:40 UTC+2, Simon King wrote: On 2015-07-13, Nathann Cohen nathan...@gmail.com javascript: wrote: sage: sqrt(2) # a symbolic ring element sqrt(2) sage: QQbar(sqrt(2)) # an algebraic value 1.414213562373095? It is true that this final '?' sounds more like a '...', as if some additional digits were hidden in a value stored as a float/double. Yet it is exact. How could we replace it? Ideally, that would be a 'sqrt(2)' but can we always provide such a representation cheaply? Could we display it as 'sqrt(2)' at least when it is free to do so? The elements of QQbar are the solutions of algebraic equations. As you probably know, the solutions of algebraic equations of degree 4 can, in general, not be expressed that nicely. This is slightly incorrect. For an appropriate notion of such a representation resp. that nicely, my statement is correct :-) Indeed, I replied to a message that was about representation of elements of QQbar in terms of square (generally n-th) roots. Best regards, Simon -- You received this message because you are subscribed to the Google Groups sage-devel group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
