October 2012 - mathfun@mailman.xmission.com

Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits
by Henry Baker 19 Oct '12

19 Oct '12

Maple seems to like "radical number": http://www.maplesoft.com/support/help/Maple/view.aspx?path=type/radnum "A radical number is defined as either a rational number or I, or a combination of roots of rational numbers specified in terms of radicals. A sum, product, or quotient of these is also a radical number." At 07:54 AM 10/17/2012, Schroeppel, Richard wrote: > Radical numbers? > >-----Original Message----- >From: math-fun-bounces(a)mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Michael Kleber >Sent: Tuesday, October 16, 2012 6:57 PM >To: math-fun >Subject: [EXTERNAL] Re: [math-fun] Computing pi (or anything else) to N digits > >On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> "Transcendental" means not the root of any finite polynomial with >> integer coefficients. >> >> http://en.wikipedia.org/wiki/Transcendental_number >> >> Is there a name for a number which isn't algebraic for a _solvable_ >> Galois polynomial -- i.e., a number which can't be constructed by >> rational & root operations? > >I think the most common description would be "[not] solvable/expressible by radicals". I don't know of a dedicated term for either state. > >--Michael

5 5

Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits
by Henry Baker 18 Oct '12

18 Oct '12

Thanks very much! "radical number" it is, then! At 01:46 PM 10/18/2012, W. Edwin Clark wrote: >I agree that the Maple definition is poorly worded. It does return true for recursively built >stuff. For example Maple 16 gives: > >>a:= sqrt(1+sqrt(3)): >>type(a,radnum); > true > >>b:=(a^(1/5)+a^(9/10))^(7/11) + a + 1/2: >>type(b,radnum); > true > >On Wed, Oct 17, 2012 at 4:54 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Hmmm... I speculate that this definition from Maple is poorly worded; I suspect that >> recursively built stuff like sqrt(1+sqrt2) is allowed within the Maple definition. >> >> Is anyone here from Maple, or does anyone here have a Maple system to try this out on? >> >> At 01:39 PM 10/17/2012, Schroeppel, Richard wrote: >> >This appears to exclude recursively built stuff like sqrt(1+sqrt2). >> --Rich >> > >> >-----Original Message----- >> >From: Henry Baker [mailto:hbaker1@pipeline.com] >> >Sent: Wednesday, October 17, 2012 10:37 AM >> >To: math-fun >> >Cc: rcs(a)xmission.com; Schroeppel, Richard >> >Subject: Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits >> > >> >Maple seems to like "radical number": >> > >> >http://www.maplesoft.com/support/help/Maple/view.aspx?path=type/radnum >> > >> >"A radical number is defined as either a rational number or I, or a >> combination of roots of rational numbers specified in terms of radicals. A >> sum, product, or quotient of these is also a radical number." >> > >> >At 07:54 AM 10/17/2012, Schroeppel, Richard wrote: >> >> Radical numbers? >> >> >> >>-----Original Message----- >> >>From: math-fun-bounces(a)mailman.xmission.com >> >>[mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Michael Kleber >> >>Sent: Tuesday, October 16, 2012 6:57 PM >> >>To: math-fun >> >>Subject: [EXTERNAL] Re: [math-fun] Computing pi (or anything else) to N digits >> >> >> >>On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> >> >> >>> "Transcendental" means not the root of any finite polynomial with integer coefficients. >> >>> >> >>> http://en.wikipedia.org/wiki/Transcendental_number >> >>> >> >>> Is there a name for a number which isn't algebraic for a _solvable_ >> >>> Galois polynomial -- i.e., a number which can't be constructed by >> >>> rational & root operations? >> >> >> >>I think the most common description would be "[not] solvable/expressible by radicals". I don't know of a dedicated term for either state. >> >> >> >>--Michael

1 0

Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits
by Henry Baker 18 Oct '12

18 Oct '12

Hmmm... I speculate that this definition from Maple is poorly worded; I suspect that recursively built stuff like sqrt(1+sqrt2) is allowed within the Maple definition. Is anyone here from Maple, or does anyone here have a Maple system to try this out on? At 01:39 PM 10/17/2012, Schroeppel, Richard wrote: >This appears to exclude recursively built stuff like sqrt(1+sqrt2). --Rich > >-----Original Message----- >From: Henry Baker [mailto:hbaker1@pipeline.com] >Sent: Wednesday, October 17, 2012 10:37 AM >To: math-fun >Cc: rcs(a)xmission.com; Schroeppel, Richard >Subject: Re: [math-fun] [EXTERNAL] Re: Computing pi (or anything else) to N digits > >Maple seems to like "radical number": > >http://www.maplesoft.com/support/help/Maple/view.aspx?path=type/radnum > >"A radical number is defined as either a rational number or I, or a combination of roots of rational numbers specified in terms of radicals. A sum, product, or quotient of these is also a radical number." > >At 07:54 AM 10/17/2012, Schroeppel, Richard wrote: >> Radical numbers? >> >>-----Original Message----- >>From: math-fun-bounces(a)mailman.xmission.com >>[mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Michael Kleber >>Sent: Tuesday, October 16, 2012 6:57 PM >>To: math-fun >>Subject: [EXTERNAL] Re: [math-fun] Computing pi (or anything else) to N digits >> >>On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> >>> "Transcendental" means not the root of any finite polynomial with integer coefficients. >>> >>> http://en.wikipedia.org/wiki/Transcendental_number >>> >>> Is there a name for a number which isn't algebraic for a _solvable_ >>> Galois polynomial -- i.e., a number which can't be constructed by >>> rational & root operations? >> >>I think the most common description would be "[not] solvable/expressible by radicals". I don't know of a dedicated term for either state. >> >>--Michael

2 1

[math-fun] Amateurs find planet with 4 (four) suns!
by Warren Smith 18 Oct '12

18 Oct '12

Allegedly there is a planet in a binary star system, which is actually two binary star systems orbiting each other (4 star system). And it was found by 2 amateur guys named Kian Jek and Robert Gagliano by examining public Kepler data to detect eclipses. Which seems kind of stupid, actually. I mean, it looks like they could have examined all their data by computer to spot planets, but instead built a web site to allow humans to examine the data, which since for 160000 stars is a ton of data. The humans then duly spotted the planet. http://blogs.discovermagazine.com/badastronomy/2012/10/15/amateur-planet-hu… http://arxiv.org/pdf/1210.3612.pdf see conclusions page 22

2 1

Re: [math-fun] Computing pi (or anything else) to N digits
by Henry Baker 17 Oct '12

17 Oct '12

Perhaps we can call an algebraic number that can be represented with rational ops and roots a "constructible" number. Then what I want is a non-constructible algebraic number. Is there a shorter and/or traditional name for non-constructible algebraic number? At 05:56 PM 10/16/2012, Michael Kleber wrote: >On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> "Transcendental" means not the root of any finite polynomial with integer >> coefficients. >> >> http://en.wikipedia.org/wiki/Transcendental_number >> >> Is there a name for a number which isn't algebraic for a _solvable_ Galois >> polynomial -- i.e., a number which can't be constructed by rational & root >> operations? > >I think the most common description would be "[not] solvable/expressible by >radicals". I don't know of a dedicated term for either state. > >--Michael

3 2

[math-fun] RCS's minimum-memory method for computing pi to N bits
by Warren Smith 17 Oct '12

17 Oct '12

>Victor Miller: It turns out that one can find an effective irrationality measure for pi/sqrt(3), which should suffice for Rich's algorithm (I think). Here's the title and abstract: Irrationality measures of $\log 2$ and $\pi/\sqrt{3}$ Nicolas Brisebarre Source: Experiment. Math. <http://projecteuclid.org/handle/euclid.em>Volume 10, Issue 1 (2001), 35-52. Abstract Using a class of polynomials that generalizes Legendre polynomials, we unify previous works of E. A. Rukhadze, A. K. Dubitskas, M. Hata, D. V. and G. V. Chudnovsky about irrationality measures of $\log 2$ and $\pi/\sqrt{3}$ --well, this will yield an RCS-style algorithm for pi*sqrt(3) which according to eq18 in this compendium by D.H. Bailey http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/bbp-formulas.pdf is a BBP number. However, I do not see why it yields an RCS-style algorithm for pi itself? Oh, I see, MIller's idea is that you square it and that square must be far from any rational. A large subset of the BBP numbers in Bailey's compendium are "linear forms in logarithms" in Baker's terminology, or polynomials in them, and therefore by the Baker-Wustholz explicit transcendence bound I cited will yield RCS-style min-memory algorithms. However, some of the BBP numbers in the compendium instead are "polylogarithms" like EQs 25, 29, 30, 32, 53, 55... and for them it would seem you are helpless except perhaps in the case of zeta(3) for which irrationality is known and perhaps explicit bounds. [Really, we do not need that our number be far from all rationals, only from rationals with denominator=power of 2.]

1 0

[math-fun] Add or subtract the result. Iterate.
by Eric Angelini 17 Oct '12

17 Oct '12

Hello Math-Fun, Here is another stupid iteration - which leads nowhere, probably. Start with (my date of birth) for instance: 1951 This in "p"; now make the absolute differences between p's digits, from left to right, like this: 1-9 = 8 9-5 = 4 5-1 = 4 1-1 = 0 (this is the last digit of "p" minus the first one) You then get "q" = 8440. And now: if q > p add q to p and iterate If q < p subtract q from p and iterate. Here: 1951 + 8440 ------- 10391 step 1 + 13680 ------- 24071 step 2 + 24761 ------- 48832 step 3 - 40512 ------- 8320 step 4 - 5128 ------ 3192 step 5 - 2871 ------ 321 step 6 - 112 ----- 209 step 7 + 297 ----- 506 step 8 + 561 ----- 1067 step 9 + 1616 ------ 2683 step 10 + 4251 ------ 6934 step 11 - 3612 ------ 3322 step 12 - 101 ------ 3221 step 13 - 1012 ------ 2209 step 14 - 297 ------ 1912 step 15 + 8811 ------- 10723 step 16 ... Etc. Don't know where this goes: loops? Fixed points? Patterns? Errors above? Waste of time? (yes) Best, É. [Color enhanced: http://www.cetteadressecomportecinquantesignes.com/AddOrSubtract.htm]

2 4

Re: [math-fun] Computing pi (or anything else) to N digits
by Henry Baker 17 Oct '12

17 Oct '12

Oops, instead of "roots", I meant "n-th roots". Hmmm. So "constructible" won't work for what I wanted. Nevertheless, "constructible" = "ruler&compass constructible" = "rational+sqrt" is still useful. Thanks. At 08:12 AM 10/17/2012, Michael Kleber wrote: >I would expect "constructible" to be short for "ruler-and-compass >constructible", so to only allow square roots. > >--Michael > >On Tue, Oct 16, 2012 at 9:30 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Perhaps we can call an algebraic number that can be represented with >> rational ops and roots a "constructible" number. >> >> Then what I want is a non-constructible algebraic number. >> >> Is there a shorter and/or traditional name for non-constructible algebraic number? >> >> At 05:56 PM 10/16/2012, Michael Kleber wrote: >> >On Mon, Oct 15, 2012 at 10:45 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >> > >> >> "Transcendental" means not the root of any finite polynomial with integer coefficients. >> >> >> >> http://en.wikipedia.org/wiki/Transcendental_number >> >> >> >> Is there a name for a number which isn't algebraic for a _solvable_ Galois >> >> polynomial -- i.e., a number which can't be constructed by rational & root >> >> operations? >> > >> >I think the most common description would be "[not] solvable/expressible by >> >radicals". I don't know of a dedicated term for either state. >> > >> >--Michael

1 0

Re: [math-fun] Computing pi (or anything else) to N digits
by Henry Baker 17 Oct '12

17 Oct '12

How about "radical-expressible number" or perhaps better, "Galois-expressible number" ? At 08:12 AM 10/17/2012, Michael Kleber wrote: >I would expect "constructible" to be short for "ruler-and-compass >constructible", so to only allow square roots.

1 0

[math-fun] Computing pi (or anything else) to N digits
by Henry Baker 17 Oct '12

17 Oct '12

"Transcendental" means not the root of any finite polynomial with integer coefficients. http://en.wikipedia.org/wiki/Transcendental_number Is there a name for a number which isn't algebraic for a _solvable_ Galois polynomial -- i.e., a number which can't be constructed by rational & root operations?

4 3