Algebraic numbers: the '?' could be more informative (e.g.: 1.4142?)

[Nathann Cohen]

Hello everybody!

In a recent thread [1], Nils Bruin mentionned that the default representation of
algebraic numbers does not really inspire trust:

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?

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.:

sage: QQbar(sqrt(2)) # still not very trust-inspiring
1.414213562373095alg

Thaaaaaaaaaaanks,

Nathann

P.S.: If you know a word which means "that inspires trust", please
share it (in private :-P)

[1] https://groups.google.com/d/msg/sage-support/b6OinTC9mdk/Y_SLTILRI9kJ

[Simon King]

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.

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

[mmarco]

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.

[Nathann Cohen]

> 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

[Jonas Jermann]

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

[William Stein]

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.


[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
>
>

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--
William (http://wstein.org)

[John Cremona]

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

[Dima Pasechnik]

As we now allow unicode, we can probably find a better character than '?'.

Say ⇘, or ↴, or even ¿.

[Volker Braun]

¿¡Qué estás diciendo!?

[Vincent Delecroix]

On 13/07/15 19:15, Simon King 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.

+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

[Vincent Delecroix]

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

[Bill Hart]

On Monday, 13 July 2015 19:15:40 UTC+2, Simon King 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.

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.

[Jeroen Demeyer]

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".

[Dima Pasechnik]

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!

[Bill Hart]

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.

[Simon King]

Hi Bill,

On 2015-07-13, Bill Hart <goodwi...@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

[Jori Mäntysalo]

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

[Sébastien Labbé]

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