[sage-devel] Re: base_ring()
Hi William On Wed, Sep 02, 2009 at 10:31:01PM -0700, William Stein wrote: Is this the intended behaviour? sage: z=1.+sqrt(-1); print z; z.base_ring() 1.00 + 1.00*I Symbolic Ring sage: z=1.+sqrt(-1.); print z; z.base_ring() 1.00 + 1.00*I Real Field with 53 bits of precision note the sqrt(-1) versus sqrt(-1.) Yes, this is definitely the intended behavior. Why do you think either one is wrong? Uhm, I have asked the originator to join the thread. The first one, I is not a symbolic variable, it is sqrt(-1). I am not sure what something like integers with I adjoined is? The second is not a real field, it is the complex field? regards, Jan -- .~. /V\ Jan Groenewald /( )\www.aims.ac.za ^^-^^ --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
2009/9/2 Jan Groenewald j...@aims.ac.za Hi William On Wed, Sep 02, 2009 at 10:31:01PM -0700, William Stein wrote: Is this the intended behaviour? sage: z=1.+sqrt(-1); print z; z.base_ring() 1.00 + 1.00*I Symbolic Ring sage: z=1.+sqrt(-1.); print z; z.base_ring() 1.00 + 1.00*I Real Field with 53 bits of precision note the sqrt(-1) versus sqrt(-1.) Yes, this is definitely the intended behavior. Why do you think either one is wrong? Uhm, I have asked the originator to join the thread. The first one, I is not a symbolic variable, it is sqrt(-1). I am not sure what something like integers with I adjoined is? If you take any integer (or rational) alpha such that alpha is not a perfect square, and try to compute sqrt(alpha), Sage promotes alpha to the symbolic ring (SR) and takes the square root there. Thus the first is correct, since sqrt(-1) is not in ZZ, so the square root is instead taken in the symbolic ring, which yields I. In the second case, the expression z=1.+sqrt(-1.) is in the complex real field with 53 bits precision. The *base ring* of that field is the real field with 53 bits precision. Maybe you were instead thinking about the parent of z? William The second is not a real field, it is the complex field? regards, Jan -- .~. /V\ Jan Groenewald /( )\www.aims.ac.za ^^-^^ -- William Stein Associate Professor of Mathematics University of Washington http://wstein.org --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
Hi William On Wed, Sep 02, 2009 at 11:18:40PM -0700, William Stein wrote: If you take any integer (or rational) alpha such that alpha is not a perfect square, and try to compute sqrt(alpha), Sage promotes alpha to the symbolic ring (SR) and takes the square root there. Thus the first is correct, since sqrt(-1) is not in ZZ, so the square root is instead taken in the symbolic ring, which yields I. OK In the second case, the expression z=1.+sqrt(-1.) is in the complex real field with 53 bits precision. The *base ring* of that field is the real field with 53 bits precision. Maybe you were instead thinking about the parent of z? Thanks sage: z=1.+sqrt(-1); print z; z.parent() 1.00 + 1.00*I Symbolic Ring sage: z=1.+sqrt(-1.); print z; z.parent() 1.00 + 1.00*I Complex Field with 53 bits of precision sage: regards, Jan -- .~. /V\ Jan Groenewald /( )\www.aims.ac.za ^^-^^ --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
I'm the originator. In fairness to Jan, I must say that I only showed him the code and output with no other comment than that I hoped that the students would not ask me to explain it. I've in the meantime found a way of illustrating the point I meant to make more clearly. sage: z=1.+sqrt(-1) sage: base_ring(z) Symbolic Ring sage: base_ring(real(z)) Symbolic Ring sage: base_ring(imag(z)) Real Field with 53 bits of precision Dirk --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
2009/9/2 Dirk dirk.lau...@gmail.com I'm the originator. In fairness to Jan, I must say that I only showed him the code and output with no other comment than that I hoped that the students would not ask me to explain it. I've in the meantime found a way of illustrating the point I meant to make more clearly. sage: z=1.+sqrt(-1) sage: base_ring(z) Symbolic Ring sage: base_ring(real(z)) Symbolic Ring sage: base_ring(imag(z)) Real Field with 53 bits of precision Dirk I see, your point is that the following is inconsistent: sage: z = 1.0+1.0*I sage: type(real(z)) type 'sage.symbolic.expression.Expression' sage: type(imag(z)) type 'sage.rings.real_mpfr.RealNumber' I agree that this definitely looks like a bug. William --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
Hi William, On Sep 3, 8:18 am, William Stein wst...@gmail.com wrote: I am not sure what something like integers with I adjoined is? I guess that means the complex numbers of the fomr a + bI with a, b integers, or Z[I] (the Gaussian Integers). Mathematica prides itself to be able to apply primality tests, factorization algorithms and the kind in this ring: http://reference.wolfram.com/mathematica/ref/GaussianIntegers.html Maybe we should add an easy way of working with it, taking advantage of Sage fast arithmetic for integers? Cheers Javier --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
2009/9/3 javier vengor...@gmail.com
Hi William,
On Sep 3, 8:18 am, William Stein wst...@gmail.com wrote:
I am not sure what something like integers with I adjoined is?
I guess that means the complex numbers of the fomr a + bI with a, b
integers, or Z[I] (the Gaussian Integers). Mathematica prides itself
to be able to apply primality tests, factorization algorithms and the
kind in this ring:
http://reference.wolfram.com/mathematica/ref/GaussianIntegers.html
Maybe we should add an easy way of working with it, taking advantage
of Sage fast arithmetic for integers?
Sage has the Gaussian integers, and I'm sure the basic arithmetic and
functionality is as good or better than Mathematica already.
sage: R.I = ZZ[sqrt(-1)]; R
Order in Number Field in I with defining polynomial x^2 + 1
sage: a = 2 + 3*I
sage: timeit('a*a')
625 loops, best of 3: 1.4 µs per loop
sage: R.ideal(3)
Fractional ideal (3)
sage: R.ideal(3).factor()
Fractional ideal (3)
sage: R.ideal(5).factor()
(Fractional ideal (-I - 2)) * (Fractional ideal (2*I + 1))
sage: R.class_group()
Class group of order 1 with structure of Number Field in I with defining
polynomial x^2 + 1
sage: P = R.ideal(11); P
Fractional ideal (11)
sage: P.is_prime()
True
sage: k = R.quotient(P,'a'); k
Quotient of Maximal Order in Number Field in I with defining polynomial x^2
+ 1 by the ideal (11)
Of course Sage can do all the above sort of stuff with any number field.
William
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[sage-devel] Re: base_ring()
On Sep 3, 9:36 am, William Stein wst...@gmail.com wrote: Sage has the Gaussian integers, and I'm sure the basic arithmetic and functionality is as good or better than Mathematica already. Sure, what I meant (sorry if I wasn't very clear) is to make an straightforward way to access it, kind of R = GaussianIntegers() in which you could factor directly the elements without needing to define the ideals generated by them. Functionality is of course equivalent to what we already have, just thought it would be nice (maybe just for marketing reasons) to be able to do something like (1 + I).is_prime() Cheers Javier --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
2009/9/3 javier vengor...@gmail.com: On Sep 3, 9:36 am, William Stein wst...@gmail.com wrote: Sage has the Gaussian integers, and I'm sure the basic arithmetic and functionality is as good or better than Mathematica already. Sure, what I meant (sorry if I wasn't very clear) is to make an straightforward way to access it, kind of R = GaussianIntegers() in which you could factor directly the elements without needing to define the ideals generated by them. Functionality is of course equivalent to what we already have, just thought it would be nice (maybe just for marketing reasons) to be able to do something like (1 + I).is_prime() I always thought that Maple's and Mathematica's ability to work directly with Gaussian Integers was just that, a marketing ploy, giving certain customers the impression of very fancy capabilities with algebraic numbers. But of course number theorists know that this is just one interesting ring o algebraic integers, certainly the easiest to define, but not exactly typical. A number theorist requires far more than that -- as Sage does provide -- and for others is this not just a curiosity? Maybe a useful one for teaching, though, and implementing this would certainly be possible. John Cheers Javier --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
is this not just a curiosity? Maybe a useful one for teaching, though, and implementing this would certainly be possible. Very useful. I had to resort to some annoying crutches (i.e., using the theorem in the code instead of discovering the theorem via the demonstration of the code) to do stuff with Gaussian integers last spring in my undergraduate course. In particular, (1 + I).is_prime() would have been really useful, but was not available. I realize that in the symbolic ring it is not clear what should be prime, but then again sage: is_prime(SR(3)) True yet in theory it might not be a generator of a prime ideal depending on what else is in the symbolic ring: sage: 1/3 in SR True So anything that gives easy access to this particular number field *and* its elements being treated as generators of the ideals would be very nice, for instance for things like Gaussian constellations etc. If there are any other rings of integers of number fields that show up a lot they could have shortcuts too, maybe Cyclotomic_Integers(n) or something, in the way Javier points out. - kcrisman --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
On Sep 3, 12:36 am, William Stein wst...@gmail.com wrote: Sage has the Gaussian integers, and I'm sure the basic arithmetic and functionality is as good or better than Mathematica already. sage: R.I = ZZ[sqrt(-1)]; R Order in Number Field in I with defining polynomial x^2 + 1 Okay, this looks like a bug to me: -- | Sage Version 4.1.1, Release Date: 2009-08-14 | | Type notebook() for the GUI, and license() for information.| -- sage: I I sage: R.I = ZZ[sqrt(-1)] sage: I 1 sage: I^2 1 Why is I equal to 1 all of a sudden? Same problem here: sage: reset() sage: R.a = ZZ[sqrt(-5)] sage: a 1 sage: R.1 a sage: R.1 == a False sage: (R.1)^2 -5 sage: R.inject_variables() Defining a sage: a 1 Ouch. -- John --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
On Thu, Sep 3, 2009 at 9:15 AM, John H Palmieri jhpalmier...@gmail.comwrote: On Sep 3, 12:36 am, William Stein wst...@gmail.com wrote: Sage has the Gaussian integers, and I'm sure the basic arithmetic and functionality is as good or better than Mathematica already. sage: R.I = ZZ[sqrt(-1)]; R Order in Number Field in I with defining polynomial x^2 + 1 Okay, this looks like a bug to me: -- | Sage Version 4.1.1, Release Date: 2009-08-14 | | Type notebook() for the GUI, and license() for information.| -- sage: I I sage: R.I = ZZ[sqrt(-1)] sage: I 1 sage: I^2 1 Why is I equal to 1 all of a sudden? Same problem here: sage: reset() sage: R.a = ZZ[sqrt(-5)] sage: a 1 sage: R.1 a sage: R.1 == a False sage: (R.1)^2 -5 sage: R.inject_variables() Defining a sage: a 1 Ouch. -- it's actually not a bug; it's confusing (in this particular situation) documented behavior. It's clearly confusing. What is happening is that R = ZZ[blah, blahs] constructs the smallest *order* that contains blahs. These aren't in general monogenic (generated by one element), so R.gens() is just a ZZ basis for that order. So: sage: ZZ[sqrt(-5)] Order in Number Field in a with defining polynomial x^2 + 5 sage: ZZ[sqrt(-5)].gens() [1, a] sage: R.a,b = ZZ[sqrt(-5)] sage: b^2 -5 Obviously this is confusing in this special cases, as we were both confused. Number fields are monogenic so work as expected: sage: R.I = QQ[sqrt(-1)] sage: I^2 -1 sage: R.gens() (I,) William --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
2009/9/3 John H Palmieri jhpalmier...@gmail.com: On Sep 3, 12:36 am, William Stein wst...@gmail.com wrote: Sage has the Gaussian integers, and I'm sure the basic arithmetic and functionality is as good or better than Mathematica already. sage: R.I = ZZ[sqrt(-1)]; R Order in Number Field in I with defining polynomial x^2 + 1 Okay, this looks like a bug to me: -- | Sage Version 4.1.1, Release Date: 2009-08-14 | | Type notebook() for the GUI, and license() for information. | -- sage: I I sage: R.I = ZZ[sqrt(-1)] sage: I 1 sage: I^2 1 Why is I equal to 1 all of a sudden? Same problem here: Here is the reason, which caught me out also. ZZ[sqrt(-1)] is an order, and has two gens, namely its ZZ-module generators: sage: R = ZZ[sqrt(-1)] sage: R.gens() [1, I] sage: R.one,I = ZZ[sqrt(-1)] sage: one 1 sage: I I sage: I^2 -1 So, R needs two names, you only gave it one. I guess WAS is about to say the same thing but I'll post anyway... John sage: reset() sage: R.a = ZZ[sqrt(-5)] sage: a 1 sage: R.1 a sage: R.1 == a False sage: (R.1)^2 -5 sage: R.inject_variables() Defining a sage: a 1 Ouch. -- John --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
[sage-devel] Re: base_ring()
On Wed, Sep 2, 2009 at 9:56 PM, Jan Groenewald j...@aims.ac.za wrote: Hi Sage-support did not solicit an answer. Both of these seem wrong: Is this the intended behaviour? sage: z=1.+sqrt(-1); print z; z.base_ring() 1.00 + 1.00*I Symbolic Ring sage: z=1.+sqrt(-1.); print z; z.base_ring() 1.00 + 1.00*I Real Field with 53 bits of precision note the sqrt(-1) versus sqrt(-1.) Yes, this is definitely the intended behavior. Why do you think either one is wrong? William --~--~-~--~~~---~--~~ To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel-unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URLs: http://www.sagemath.org -~--~~~~--~~--~--~---
