cube roots in Sage
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Gregory Bard
Jun 18, 2014, 2:37:21 AM6/18/14
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This has been brought up many times before, but I'd like to bring up
the possibility of adding two commands to Sage: cuberoot(x) and
nthroot(x, n)
The reason is that currently plot( x^(1/3), -5, 5) only shows values for x>0,
and not for x<0. The current work-around recommended is
plot(sign(x)*abs(x)^(1/3), -5, 5)
(Track ticket #11458)
which replaced the hideous
plot(lambda x: RR(x).nth_root(3), -5, 5, plot_points=20)
This is true in classes like Precalculus and Calculus 1, where trying to explain
workarounds like these would just really a headache for the instructor and
extremely fragile students.
Also the cube root can occur in many other situations and applications.
This is urgent, because my textbook "Sage for Undergraduates" is due
at the American Mathematical Society on June 30th.
The namespace is so huge, can't we just add two more commands?
I suggest:
def cuberoot(x):
return sign(x)*((x*sign(x))^(1/3))
Last but not least, links to previous demands for exactly this problem:
https://www.mail-archive.com/sage-s...@googlegroups.com/msg11563.html
https://groups.google.com/forum/#!msg/sage-devel/_JeSMD-Kvfk/xeNstGrcvXQJ
https://groups.google.com/forum/#!topic/sage-support/icZ8ekC_P4Y
https://groups.google.com/forum/#!topic/sage-devel/_JeSMD-Kvfk
http://trac.sagemath.org/ticket/11458
https://sage.uwstout.edu/home/pub/32/
https://groups.google.com/forum/#!topic/sage-support/ZtRWScqMHMM
https://groups.google.com/forum/#!topic/sage-devel/_JeSMD-Kvfk
the possibility of adding two commands to Sage: cuberoot(x) and
nthroot(x, n)
The reason is that currently plot( x^(1/3), -5, 5) only shows values for x>0,
and not for x<0. The current work-around recommended is
plot(sign(x)*abs(x)^(1/3), -5, 5)
(Track ticket #11458)
which replaced the hideous
plot(lambda x: RR(x).nth_root(3), -5, 5, plot_points=20)
This is true in classes like Precalculus and Calculus 1, where trying to explain
workarounds like these would just really a headache for the instructor and
extremely fragile students.
Also the cube root can occur in many other situations and applications.
This is urgent, because my textbook "Sage for Undergraduates" is due
at the American Mathematical Society on June 30th.
The namespace is so huge, can't we just add two more commands?
I suggest:
def cuberoot(x):
return sign(x)*((x*sign(x))^(1/3))
Last but not least, links to previous demands for exactly this problem:
https://www.mail-archive.com/sage-s...@googlegroups.com/msg11563.html
https://groups.google.com/forum/#!msg/sage-devel/_JeSMD-Kvfk/xeNstGrcvXQJ
https://groups.google.com/forum/#!topic/sage-support/icZ8ekC_P4Y
https://groups.google.com/forum/#!topic/sage-devel/_JeSMD-Kvfk
http://trac.sagemath.org/ticket/11458
https://sage.uwstout.edu/home/pub/32/
https://groups.google.com/forum/#!topic/sage-support/ZtRWScqMHMM
https://groups.google.com/forum/#!topic/sage-devel/_JeSMD-Kvfk
Jori Mantysalo
Jun 18, 2014, 3:02:27 AM6/18/14
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On Tue, 17 Jun 2014, Gregory Bard wrote:
> This has been brought up many times before, but I'd like to bring up
> the possibility of adding two commands to Sage: cuberoot(x) and
> nthroot(x, n)
Could you summarize previos discussion(s)?
> This has been brought up many times before, but I'd like to bring up
> the possibility of adding two commands to Sage: cuberoot(x) and
> nthroot(x, n)
For example what should cuberoot(1)^2 be? Different from cuberoot(1)? How
about "realroot(x, n)"?
--
Jori Mäntysalo
Nils Bruin
Jun 18, 2014, 8:44:22 PM6/18/14
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On Wednesday, June 18, 2014 2:37:21 AM UTC-4, Gregory Bard wrote:
This has been brought up many times before, but I'd like to bring up
the possibility of adding two commands to Sage: cuberoot(x) and
nthroot(x, n)
+1 for nthroot. Once we have that, I don't think we need cuberoot. The function should only accept (positive?) integer arguments n. For even n it could have the usual behaviour, for odd n it could do the sign thing. It's possibly nicer to implement it as a symbolic function with custom numerical evaluation rather than a python function that produces a relatively cryptic symbolic expression. It should be documented as the "real-valued n-th root of real-valued expressions".
Please look whether we should have nthroot(n,x) or nthroot(x,n), or perhaps even nthroot[n](x) [I hope not].
Please look whether we should have nthroot(n,x) or nthroot(x,n), or perhaps even nthroot[n](x) [I hope not].
Vincent Delecroix
Jun 19, 2014, 2:38:11 AM6/19/14
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Note that there is already a method "nth_root" on several elements
(ZZ, finite fields, etc). So I would rather go for "real_nth_root"
which makes things clearer.
Vincent
2014-06-19 2:44 UTC+02:00, Nils Bruin <nbr...@sfu.ca>:
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(ZZ, finite fields, etc). So I would rather go for "real_nth_root"
which makes things clearer.
Vincent
2014-06-19 2:44 UTC+02:00, Nils Bruin <nbr...@sfu.ca>:
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Nils Bruin
Jun 19, 2014, 9:13:25 AM6/19/14
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On Thursday, June 19, 2014 2:38:11 AM UTC-4, vdelecroix wrote:
Note that there is already a method "nth_root" on several elements
(ZZ, finite fields, etc). So I would rather go for "real_nth_root"
which makes things clearer.
Perhaps we can then just get away with a "nth_root" symbolic function that uses nth_root on its argument, making use of the fact that if the numerical evaluation of the argument produces a real number, then the real number itself will know what its nth root is? Something along the lines of:
from sage.symbolic.function import SymbolicFunction
class nth_root(SymbolicFunction):
def __init__(self):
SymbolicFunction.__init__(self, 'nth_root', nargs=2)
def _evalf_(self, n, x, parent=None):
return parent(x).nth_root(n)
(this needs refinement, but numerical evaluation does seem to be able to produce sometimes real elements and sometimes complex numbers, so perhaps we can utilize that)
from sage.symbolic.function import SymbolicFunction
class nth_root(SymbolicFunction):
def __init__(self):
SymbolicFunction.__init__(self, 'nth_root', nargs=2)
def _evalf_(self, n, x, parent=None):
return parent(x).nth_root(n)
(this needs refinement, but numerical evaluation does seem to be able to produce sometimes real elements and sometimes complex numbers, so perhaps we can utilize that)
Gregory Bard
Jun 20, 2014, 9:33:55 PM6/20/14
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It seems that the consensus on both Sage-devel and Sage-edu is to go
with some sort of nth_real_root function. I propose the following,
which I have tested for evaluation, plotting, differentiation, and
integration. Sadly, the derivative has a Dirac delta in it, which is
... perhaps unavoidable because of the vertical tangency of the
cuberoot function at x=0. (Naturally, we can remove the asserts once
testing is completed.
---Greg
def nth_real_root( x, n ):
"""Note: n must be a positive integer. However, x is any real number.
(Otherwise this explanation will make no sense.)
For odd values of n, we return the unique real number y such
that y^n = x.
For even values of n, if x<0, there is no real number y such
that y^n = x and so we throw an exception.
For even values of n, if x=>0, then we return the unique real
number y such that y^n = x."""
if ((n in ZZ)==false):
raise RuntimeError('nth_real_root(x,n) requires n to be a
positive integer. Your n is not an integer.')
if (n<=0):
raise RuntimeError('nth_real_root(x,n) requires n to be a
positive integer. Your n is not positive.')
assert (n in ZZ)
assert (n>0)
if ((n%2) == 0):
# n is even
if (x<0):
raise RuntimeError('There is no nth real root (of a
negative number x) when n is even.')
else:
return x^(1/n)
assert ((n%2)==1)
# n is odd
return sign(x)*(abs(x))^(1/n)
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with some sort of nth_real_root function. I propose the following,
which I have tested for evaluation, plotting, differentiation, and
integration. Sadly, the derivative has a Dirac delta in it, which is
... perhaps unavoidable because of the vertical tangency of the
cuberoot function at x=0. (Naturally, we can remove the asserts once
testing is completed.
---Greg
def nth_real_root( x, n ):
"""Note: n must be a positive integer. However, x is any real number.
(Otherwise this explanation will make no sense.)
For odd values of n, we return the unique real number y such
that y^n = x.
For even values of n, if x<0, there is no real number y such
that y^n = x and so we throw an exception.
For even values of n, if x=>0, then we return the unique real
number y such that y^n = x."""
if ((n in ZZ)==false):
raise RuntimeError('nth_real_root(x,n) requires n to be a
positive integer. Your n is not an integer.')
if (n<=0):
raise RuntimeError('nth_real_root(x,n) requires n to be a
positive integer. Your n is not positive.')
assert (n in ZZ)
assert (n>0)
if ((n%2) == 0):
# n is even
if (x<0):
raise RuntimeError('There is no nth real root (of a
negative number x) when n is even.')
else:
return x^(1/n)
assert ((n%2)==1)
# n is odd
return sign(x)*(abs(x))^(1/n)
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Nicolas M. Thiery
Jun 22, 2014, 11:27:16 AM6/22/14
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On Fri, Jun 20, 2014 at 06:33:52PM -0700, Gregory Bard wrote:
> It seems that the consensus on both Sage-devel and Sage-edu is to go
> with some sort of nth_real_root function. I propose the following,
> which I have tested for evaluation, plotting, differentiation, and
> integration. Sadly, the derivative has a Dirac delta in it, which is
> ... perhaps unavoidable because of the vertical tangency of the
> cuberoot function at x=0. (Naturally, we can remove the asserts once
> testing is completed.
> ---Greg
>
> def nth_real_root( x, n ):
Just 2 cents of outsider feedback since I have not followed the
> It seems that the consensus on both Sage-devel and Sage-edu is to go
> with some sort of nth_real_root function. I propose the following,
> which I have tested for evaluation, plotting, differentiation, and
> integration. Sadly, the derivative has a Dirac delta in it, which is
> ... perhaps unavoidable because of the vertical tangency of the
> cuberoot function at x=0. (Naturally, we can remove the asserts once
> testing is completed.
> ---Greg
>
> def nth_real_root( x, n ):
discussion, and am not knowledgeable on the topic. This names suggests
to me that we look at all the real roots of n (for whatever this
means), and then take the n-th one. So maybe real_nth_root instead?
Cheers,
Nicolas
--
Nicolas M. Thiéry "Isil" <nth...@users.sf.net>
http://Nicolas.Thiery.name/
William Stein
Jun 22, 2014, 12:03:02 PM6/22/14
to sage-devel
On Sun, Jun 22, 2014 at 8:27 AM, Nicolas M. Thiery
<Nicolas...@u-psud.fr> wrote:
> On Fri, Jun 20, 2014 at 06:33:52PM -0700, Gregory Bard wrote:
>> It seems that the consensus on both Sage-devel and Sage-edu is to go
>> with some sort of nth_real_root function. I propose the following,
>> which I have tested for evaluation, plotting, differentiation, and
>> integration. Sadly, the derivative has a Dirac delta in it, which is
>> ... perhaps unavoidable because of the vertical tangency of the
>> cuberoot function at x=0. (Naturally, we can remove the asserts once
>> testing is completed.
>> ---Greg
>>
>> def nth_real_root( x, n ):
>
> Just 2 cents of outsider feedback since I have not followed the
> discussion, and am not knowledgeable on the topic. This names suggests
> to me that we look at all the real roots of n (for whatever this
> means), and then take the n-th one. So maybe real_nth_root instead?
+1
<Nicolas...@u-psud.fr> wrote:
> On Fri, Jun 20, 2014 at 06:33:52PM -0700, Gregory Bard wrote:
>> It seems that the consensus on both Sage-devel and Sage-edu is to go
>> with some sort of nth_real_root function. I propose the following,
>> which I have tested for evaluation, plotting, differentiation, and
>> integration. Sadly, the derivative has a Dirac delta in it, which is
>> ... perhaps unavoidable because of the vertical tangency of the
>> cuberoot function at x=0. (Naturally, we can remove the asserts once
>> testing is completed.
>> ---Greg
>>
>> def nth_real_root( x, n ):
>
> Just 2 cents of outsider feedback since I have not followed the
> discussion, and am not knowledgeable on the topic. This names suggests
> to me that we look at all the real roots of n (for whatever this
> means), and then take the n-th one. So maybe real_nth_root instead?
>
> Cheers,
> Nicolas
> --
> Nicolas M. Thiéry "Isil" <nth...@users.sf.net>
> http://Nicolas.Thiery.name/
>
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William Stein
Professor of Mathematics
University of Washington
http://wstein.org
Gregory Bard
Jun 22, 2014, 4:36:29 PM6/22/14
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Yes, that is reasonable. Let us call it "real_nth_root" instead, as
suggested by Nicolas Thiery. Any other requests/comments?
It would be superb if this could be resolved by June 30th, when my
book goes to the American Mathematical Society for publication...
---Greg
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suggested by Nicolas Thiery. Any other requests/comments?
It would be superb if this could be resolved by June 30th, when my
book goes to the American Mathematical Society for publication...
---Greg
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Dima Pasechnik
Jun 22, 2014, 5:56:34 PM6/22/14
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On 2014-06-22, Gregory Bard <gregory....@gmail.com> wrote:
> Yes, that is reasonable. Let us call it "real_nth_root" instead, as
> suggested by Nicolas Thiery. Any other requests/comments?
it's better talk about "n-ic" root rather than "n-th" root, IMHO.
> Yes, that is reasonable. Let us call it "real_nth_root" instead, as
> suggested by Nicolas Thiery. Any other requests/comments?
Vincent Delecroix
Jun 22, 2014, 6:00:37 PM6/22/14
to sage-...@googlegroups.com
As Niles already said it would be better to have it as a symbolic function
sage: f(x) = real_nth_root(x, 5)
sage: f
x |--> real_nth_root(x,5)
2014-06-22 22:36 UTC+02:00, Gregory Bard <gregory....@gmail.com>:
sage: f(x) = real_nth_root(x, 5)
sage: f
x |--> real_nth_root(x,5)
2014-06-22 22:36 UTC+02:00, Gregory Bard <gregory....@gmail.com>:
Gregory Bard
Jun 25, 2014, 1:10:06 AM6/25/14
to sage-...@googlegroups.com
As Vincent and Niles have brought up, there might be advantages to it
being a symbolic function. How does one actually go about making that
happen? Is this an intrusive change, or an easy one? I really have no
idea...
---Greg
On Sun, Jun 22, 2014 at 5:00 PM, Vincent Delecroix
being a symbolic function. How does one actually go about making that
happen? Is this an intrusive change, or an easy one? I really have no
idea...
---Greg
On Sun, Jun 22, 2014 at 5:00 PM, Vincent Delecroix
Gregory Bard
Jul 10, 2014, 6:23:21 PM7/10/14
to sage-...@googlegroups.com
I think we have a consensus that
we should do *something* but unless I am very much mistaken, the
suggestion from Vincent Delecroix and Nils Bruin that we make a symbolic
function has advantages.
I was looking at Nils's code, but I have to confess that I don't understand that code. Actually, I wasn't able to get the following to work, but I have a feeling that I am making a minor and stupid mistake with the syntax.
################
I was looking at Nils's code, but I have to confess that I don't understand that code. Actually, I wasn't able to get the following to work, but I have a feeling that I am making a minor and stupid mistake with the syntax.
################
from sage.symbolic.function import SymbolicFunction
class nth_root(SymbolicFunction):
def __init__(self):
SymbolicFunction.__init__(self, 'nth_root', nargs=2)
def _evalf_(self, n, x, parent=None):
return parent(x).nth_root(n)
plot( nth_root( x, 3), -5, 5 )
################
Maybe someone here could explain exactly what the distinction is or what the pros/cons are for my approach versus Nils's? I have no idea.
I have also posted this on Sage-Edu, but perhaps all future replies should be restricted to Sage-Devel and not Sage-Edu?
---Greg
################
Maybe someone here could explain exactly what the distinction is or what the pros/cons are for my approach versus Nils's? I have no idea.
I have also posted this on Sage-Edu, but perhaps all future replies should be restricted to Sage-Devel and not Sage-Edu?
---Greg
>>>> email to sage-devel+unsubscribe@googlegroups.com.
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>>>
>>>
>>>
>>> --
>>> William Stein
>>> Professor of Mathematics
>>> University of Washington
>>> http://wstein.org
>>>
>>> --
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>>
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>
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P Purkayastha
Jul 10, 2014, 6:47:37 PM7/10/14
to sage-...@googlegroups.com, Gregory Bard
Hi Greg,
Hope the following code helps you in implementing your function:
from sage.symbolic.function import SymbolicFunction
class real_nth_root_class(SymbolicFunction):
def __init__(self):
SymbolicFunction.__init__(self, 'real_nth_root', nargs=2)
def _evalf_(self, x, n, parent=None):
# Handle python floats which do not have the 'nth_root' method
if isinstance(x, float):
return float(RDF(x).nth_root(n))
return parent(x).nth_root(n)
real_nth_root = real_nth_root_class()
In actual usage:
sage: real_nth_root( x, 3)
real_nth_root(x, 3)
sage: f(x) = real_nth_root( x, 3)
sage: f
x |--> real_nth_root(x, 3)
sage: plot( real_nth_root( x, 3), -5, 5 )
# plots the graph.
Regards,
basu.
> <mailto:nth...@users.sf.net>>
> <mailto:sage-devel%2Bunsu...@googlegroups.com>.
> <mailto:sage-devel%2Bunsu...@googlegroups.com>.
> <mailto:sage-devel%2Bunsu...@googlegroups.com>.
> <mailto:sage-devel%2Bunsu...@googlegroups.com>.
> <mailto:sage-devel+...@googlegroups.com>.
Hope the following code helps you in implementing your function:
from sage.symbolic.function import SymbolicFunction
class real_nth_root_class(SymbolicFunction):
def __init__(self):
SymbolicFunction.__init__(self, 'real_nth_root', nargs=2)
def _evalf_(self, x, n, parent=None):
# Handle python floats which do not have the 'nth_root' method
if isinstance(x, float):
return float(RDF(x).nth_root(n))
return parent(x).nth_root(n)
real_nth_root = real_nth_root_class()
In actual usage:
sage: real_nth_root( x, 3)
real_nth_root(x, 3)
sage: f(x) = real_nth_root( x, 3)
sage: f
x |--> real_nth_root(x, 3)
sage: plot( real_nth_root( x, 3), -5, 5 )
# plots the graph.
Regards,
basu.
> <20100.d...@gmail.com <mailto:20100.d...@gmail.com>> wrote:
> > As Niles already said it would be better to have it as a
> symbolic function
> >
> > sage: f(x) = real_nth_root(x, 5)
> > sage: f
> > x |--> real_nth_root(x,5)
> >
> > 2014-06-22 22:36 UTC+02:00, Gregory Bard
> <gregory....@gmail.com <mailto:gregory....@gmail.com>>:
> > As Niles already said it would be better to have it as a
> symbolic function
> >
> > sage: f(x) = real_nth_root(x, 5)
> > sage: f
> > x |--> real_nth_root(x,5)
> >
> > 2014-06-22 22:36 UTC+02:00, Gregory Bard
> >> Yes, that is reasonable. Let us call it "real_nth_root"
> instead, as
> >> suggested by Nicolas Thiery. Any other requests/comments?
> >>
> >> It would be superb if this could be resolved by June 30th, when my
> >> book goes to the American Mathematical Society for publication...
> >> ---Greg
> >>
> >> On Sun, Jun 22, 2014 at 9:02 AM, William Stein
> <wst...@gmail.com <mailto:wst...@gmail.com>> wrote:
> >>> On Sun, Jun 22, 2014 at 8:27 AM, Nicolas M. Thiery
> >>> <Nicolas...@u-psud.fr <mailto:Nicolas...@u-psud.fr>>
> instead, as
> >> suggested by Nicolas Thiery. Any other requests/comments?
> >>
> >> It would be superb if this could be resolved by June 30th, when my
> >> book goes to the American Mathematical Society for publication...
> >> ---Greg
> >>
> >> On Sun, Jun 22, 2014 at 9:02 AM, William Stein
> <wst...@gmail.com <mailto:wst...@gmail.com>> wrote:
> >>> On Sun, Jun 22, 2014 at 8:27 AM, Nicolas M. Thiery
> >>>> http://Nicolas.Thiery.name/
> >>>>
> >>>> --
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Gregory Bard
Jun 11, 2020, 2:24:01 PM6/11/20
to sage-devel
Dear Sage developers,
I'm currently working on a 2nd edition of my book Sage for Undergraduates,
which adds some new material but mostly updates the book to reflect syntax
changes in Sage since mid-2014, especially relating to the Python2 to Python3
transition.
Some of you might remember a spirited discussion in June of 2014 regarding
cube roots, and real values of nth roots of negative reals, especially as
concerns plotting. For example, compare the two plots:
The eventual consensus was that a command, named real_nth_root( ),
was a good way to resolve this matter by adding only one command. Sadly,
the trac ticket has stalled. https://trac.sagemath.org/ticket/12074
It would be really wonderful if some knowledgeable person could jump on
this and code something up, for possible inclusion in a soon-to-be-released
version of Sage. It would be superb if this could be in Sage by the start of
the Fall 2020 semester in the USA, which is in late August. However, since
the ticket has stalled since June of 2014, I'd be happy with any timeline
of any kind. Of course, I'd do the work myself if I knew how, and if I had time,
but I'm currently up to my eyeballs in typo corrections and updating of code,
plus adding some new examples, challenges, and projects.
I cannot emphasize enough how useful this change would be for those of
us who have to teach pre-calculus and differential calculus. The cube root
function is the best example of a function that is continuous everywhere,
but not differentiable everywhere, perhaps tied with f(x) = abs(x).
Please help,
---Greg
p.s. For those unfamiliar with Sage for Undergraduates, it is available for
free as a pdf file, and the American Mathematical Society has published
a print version at a very reasonable price. Diego Sejas has translated it
recently into Spanish, and the Spanish translation uses the up-to-date
Python3 syntax. (http://www.sage-para-estudiantes.com/)
Message has been deleted
Diego Sejas
Jun 11, 2020, 8:38:59 PM6/11/20
to sage-devel
Hello, Sage Community!
I think the
addition of a real_nth_root() function could be very beneficial,
specially in the teaching context, where using Sage with odd roots of
negative numbers implies the use of some programming tweaks. In my case,
I lost the count of how many times I forgot (in the middle of a class)
about how Sage treats those roots, and wrote "plot(x^(1/3), -2, 2)", just to
discover to my great embarrassment that the plot doesn't look as I told
my students. (Of course, this is a minor problem, but a quite annoying
one, if you ask me.)
The addition of a "real_nth_root" function has the following benefits:
- It would make Sage a little bit more flexible for (pre-)calculus, real analysis, and other classes.
- It
will definitely make the use of Sage more comfortable for plotting and
obtaining results in RR. (Plotting the cube root is a simple school
exercise, with a simple statement, and a simple result. It shouldn't be
much harder to do in Sage.)
- It would amplify a little bit the "ready-to-use" scope of Sage.
- In the end, this is a much requested feature, so maybe we should respond in some satisfactory way to the community.
- We wouldn't be that behind with respect to Mathematica, which plots x^(1/3) as you would expect it. However, we would also have the current mechanism of obtaining complex roots (if required).
kcrisman
Jun 12, 2020, 10:04:17 AM6/12/20
to sage-devel
Just want to thank Kwankyu for getting this on a proper branch. I've made a few comments, and encourage anyone with feedback to do the same.
Gregory Bard
Aug 19, 2021, 2:58:00 PM8/19/21
to sage-devel
I'm so grateful that this issue of cube roots (and other real nth roots)
has been solved so successfully. It's a huge help to those of us who
teach calculus, and it's a good story to showcase the cooperative
atmosphere of community developed open-source software.
I'd like to thank those responsible in the acknowledgements of the
2nd edition of Sage for Undergraduates. Would everyone who
contributed a medium or large amount to this effort please email
me, and provide the exact spelling of their name, as they would
like to see it appear in print?
With respect and gratitude,
---Greg
has been solved so successfully. It's a huge help to those of us who
teach calculus, and it's a good story to showcase the cooperative
atmosphere of community developed open-source software.
I'd like to thank those responsible in the acknowledgements of the
2nd edition of Sage for Undergraduates. Would everyone who
contributed a medium or large amount to this effort please email
me, and provide the exact spelling of their name, as they would
like to see it appear in print?
With respect and gratitude,
---Greg
Kwankyu Lee
Aug 20, 2021, 9:27:46 AM8/20/21
to sage-devel
On Friday, August 20, 2021 at 3:58:00 AM UTC+9 Gregory Bard wrote:
I'm so grateful that this issue of cube roots (and other real nth roots)
has been solved so successfully. It's a huge help to those of us who
teach calculus, and it's a good story to showcase the cooperative
atmosphere of community developed open-source software.
I'd like to thank those responsible in the acknowledgements of the
2nd edition of Sage for Undergraduates. Would everyone who
contributed a medium or large amount to this effort please email
me, and provide the exact spelling of their name, as they would
like to see it appear in print?
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