Solution?

Benach · **#21** by **Benach** » Jul 27, 2010 5:14 pm

Cynergy&Roger: perhaps an idea: although I can easily follow you two, it might be a bit out of leage to come up with words as transcedental functions, algebra, Liouville number etc unless you give the formal definitions.

Thommo · **#22** by **Thommo** » Jul 27, 2010 5:24 pm

This is the mathematics subforum, I don't think it's necessary to define every term when they are being used in a completely standard mathematical fashion. There are clear entries on all those things on (for example) wikipedia.

http://en.wikipedia.org/wiki/Transcendental_function
http://en.wikipedia.org/wiki/Liouville_number
http://en.wikipedia.org/wiki/Algebra

Benach · **#23** by **Benach** » Jul 27, 2010 5:30 pm

Those wiki entries might scare people of. Come on, if someone trusts Excel without hesitance while Mathematica isn't used, come on, you can figure out that the person that asks has not a PhD in maths.

Cynergy · **#24** by **Cynergy** » Jul 27, 2010 5:50 pm

Benach wrote:Those wiki entries might scare people of. Come on, if someone trusts Excel without hesitance while Mathematica isn't used, come on, you can figure out that the person that asks has not a PhD in maths.

As Thommo has noted, this is the mathematics subforum. If you want to contribute, I would expect you to have probably an undergraduate level of mathematical understanding, at least, otherwise, why are you here?

We're happy to answer mathematical questions (or at least attempt to), especially if they have some intriguing property. But we are not here to teach. To "explain" an elementary idea such as, for example, transcendental functions, how far back do you want me to go? Definition of a function? Methods of solution of quadratic equations? Definition of integers and rationals?

We frequently provide links in responses generally to illuminate some more exotic ideas, but that's about as far as it goes. If you want to tell us roughly what your current understanding of maths is, we'll make numerous suggestions for links and texts to help you.

Benach · **#25** by **Benach** » Jul 27, 2010 5:59 pm

As i said, I can follow you, but hey, I'm a graduated engineer. Only thing I tried to point out is that sometimes it is scary for people to ask questions.

Mononoke · **#26** by **Mononoke** » Jul 27, 2010 6:53 pm

Benach wrote:As i said, I can follow you, but hey, I'm a graduated engineer. Only thing I tried to point out is that sometimes it is scary for people to ask questions.

An algebraic function is a function that can be represented by a finite number of elementary operations. elementary operation being +, -, *, / and taking roots. so x^2+1=0 is algebraic. A transcendental function is one that cannot be expressed using a finite number of elementary operations. the catch here is that if a function is algebraic you can solve it with a finite number of elementary operations. x^x isn't algebraic. So if an analytic solution exists it would most likely involve infinite series and stuff like that.

Roger Cooke · **#27** by **Roger Cooke** » Jul 27, 2010 9:22 pm

Mononoke wrote:
Benach wrote:As i said, I can follow you, but hey, I'm a graduated engineer. Only thing I tried to point out is that sometimes it is scary for people to ask questions.

An algebraic function is a function that can be represented by a finite number of elementary operations. elementary operation being +, -, *, / and taking roots. so x^2+1=0 is algebraic. A transcendental function is one that cannot be expressed using a finite number of elementary operations. the catch here is that if a function is algebraic you can solve it with a finite number of elementary operations. x^x isn't algebraic. So if an analytic solution exists it would most likely involve infinite series and stuff like that.

It's probably useful to distinguish between algebraic functions and algebraic formulas. An algebraic formula is a finite expression that contains only rational powers of the variables combined by the four arithmetic operations. An algebraic function is any function f(x) that satisfies an identity

p(x,f(x)) = 0

for all values of x, where p(x,y) is a polynomial in x and y that is NOT independent of y. The content of Abel's famous theorem on the insolvability of the general quintic is that not every algebraic function can be given by an algebraic formula.

Cynergy · **#28** by **Cynergy** » Jul 27, 2010 11:12 pm

Benach wrote:As i said, I can follow you, but hey, I'm a graduated engineer. Only thing I tried to point out is that sometimes it is scary for people to ask questions.

Why?

Benach · **#29** by **Benach** » Jul 28, 2010 12:35 am

Because most people feel dumb. Fortuny I had an excellent math prof at uni and was willing to answer even the silliest questions (Christians asking about "mathematical proofs" of god etc.).

Cynergy · **#30** by **Cynergy** » Jul 28, 2010 1:17 am

Benach wrote:Because most people feel dumb. Fortuny I had an excellent math prof at uni and was willing to answer even the silliest questions (Christians asking about "mathematical proofs" of god etc.).

Well, there are only three kinds of people:

(a) Those who are so clever that being told they are very clever is unremarkable. For example, were I 6' 7" (I'm not), to tell me "You're very tall" would be obvious.

(b) Those who are smart enough to know when they're being flattered.

(c) Those who are dumb enough to believe anything.

Roger Cooke · **#31** by **Roger Cooke** » Jul 28, 2010 1:27 am

Cynergy wrote:
Benach wrote:Because most people feel dumb. Fortuny I had an excellent math prof at uni and was willing to answer even the silliest questions (Christians asking about "mathematical proofs" of god etc.).

Well, there are only three kinds of people:

(a) Those who are so clever that being told they are very clever is unremarkable. For example, were I 6' 7" (I'm not), to tell me "You're very tall" would be obvious.

(b) Those who are smart enough to know when they're being flattered.

(c) Those who are dumb enough to believe anything.

CdesignProponentsist · **#32** by **CdesignProponentsist** » Jul 28, 2010 1:44 am

Roger Cooke wrote:
num1cubfn wrote:
Roger Cooke wrote:
Mononoke wrote:I would start by taking the natural log from both sides

Indeed, and then you would have the equation x log(x) = log(100).

I doubt if the solution to this equation is rational, as the Excel solution appears to imply. I think Excel is simply using limited precision and giving an answer that exaggerates the true precision. I'll run it through Mathematica and see what happens.

(Two minutes later). Mathematica gives 50 decimal places as

3.5972850235404175054976522517822860691355430548866.

Let me know if you want a thousand or a million decimal places. I'm guessing that I can prove fairly easily that x isn't rational. And people who know number theory well can probably invoke one of Vinogradov's theorems to prove that x is transcendental.

Awwww, I thought it was cool that it suddenly dropped to a bunch of zeros.

CURSE YOU EXCEL!!!!!!

Cool, can you please tell me what the millionth digit is?

I finally aborted the evaluation after Mathematica ran for 10 minutes. It would eventually get the answer, but I'm not sure how long it would take. I'll let it run overnight tonight and see what it produces.

Meanwhile, if you want to compute the number yourself, I have found it easy to do. Since this is a convex function, Newton's method is guaranteed to work. I programmed it in Mathematica and had it run 10 steps of the approximation with 1000 digits of accuracy. Since the last two steps were identical, I'm assuming I actually have 1000-place accuracy here. Here's what the input and output looked like:

f[x_]:=(x+Log[100])/(1+Log[x])

Table[N[Nest[f,7/2,k],1000],{k,1,10}]

{3.597879714527434874419322351070673233760002738054671459168194193265202137470008234490837549130587617519958354336843152769674582024289134126366297240420590710425481510271362235671059928940825156156347415477393592735605592680497903208995157477269061456981969612866232060903613749397423272194997022212704451995696293277422154920745064182959534436928918879925354621486342761226470846901298873276879846307573929057083865973020917892092813818262653847048979267473239267321921106136482240169633218672327152178094156832197011014487632696953625135986239195715186083978092751036191188438015530812406382904740841131208826148585359179994947299722580151059179894145040766778848905252010982278805627063044103590641280534041144985198923522345039941085328669258274760420892733816918577019178914144210058654266976131451014303983777973857085972414705421513525356335076448756838046654485937290463730320180286954269306184485612325783503267087010263002159801427044877252613951985218779739412123787965683936843042618280740,
3.597285045094498263223757661882020536815158568443951645929620889874673879520693007258821845742552320356447221211252551620255081991924843794350373134543095737139908302410835032383245223410922965704850076581299209213289573119362118343747839535417063340255733986412168085761865892219430385078158551252710084803442878523561456990539250470123218521505957617948965248921241051711967113602142108378373584597187030655632497449215170042237555946525278265295710079051689330025410186049650555893791117453871506892793925917409783252515279037393790018129229049414672170100219038957135149564187578563585291381191480355922805324484929207939832799835521774504255975635194434946783989833662601469980753348015951122888504427090664622430695376271296244083011218433613087023503391450506822339908529947063830019875359601505625146113346295444949576283487534132071698276888031982126204870046851066928013758531032674966157828641141222109183355159133188417250120522137699195356799032644654646396813418842792059859598057300491,
3.597285023540417533817123822582891478097198507933629058291394533673075764259830332505774726444814016930952076334328052659942603859395862805223923852481207472421038232510522599980671234026592198804959069689774579625633787561469152046877599647617239995511443577326318056881995659979806666044321157132976905245070029213240086534200237119415826179377420946016233473465744479226778267762122844788038265216090397686605762702353136347356527334742937125734151138248151331955542572220294911999934677312924275021653541280623575820827175975916461721806278818844875934134515960450279959524711229341696231745317814053873167632039801787322673820322981800475869056669179542954620246285369451810120750441297766656371385691616526905631337494859365319912453861081242675818108282246437870748802722177360165250003894505421363587468443118294822319562510449624778152525198250178486496842503040219528071816171339000237705157980371216633643692039259335568535729686823421103835696930557646896944512158601684799406765072063236,
3.597285023540417505497652251782286118022885293448071975498356757394646441099209888867706681516367249026755244034574647589375437428509412539588917661525146534089666229958307064287186510773944180137072727937750640080618888756477289809035913107405569113134727463074705343843297908113617261735926618305472039650930848375396575024700856229991441391549692075017978537336652653261166320363050978433700337260717270948821378225581624339880746892223449809585917477640957709941500364700792478931926594410552131954081041453338253993148864678287532188114059269071267252859362088964191036318712781078422378570407429909781250528509619229356083748784952780691608263233374626304495622301442580752880306053751946262361478109984319881901908206106890949831813544145402454088801376134161314820348006347797257083635694148698197482425855853617846279327104889096979395032949379811214394891978854487130088834567199793880491987732891829043085356032408396913663047721682211518061270448478173644484876984180310144611917930163977,
3.597285023540417505497652251782286069135543054886576783720252127972957653242367218909117506138019309787699075175737888060981171759542249685920746259333583642571977612222184426363562408994037086598756787780555280577054872047696242596013673688933149506660361112603686751356857548443528844198878930502277349282502820319751544838065494868735538975801377281703115354295129550504206555421288137065321379648687820387719624291530085482098781309396575220331927612327393643588456904813338356549215905699726362914202592707637900053391624276817452525428717857877575525023648257098333996320243343828968741424410355500384246291967497238810035617978188779827516761848421699766324354393817326746216890579418377264560348085970930773019613554498322217346052540595247406324306257460890541411381047839162251893950316354791027769767839166239560515585527824750997440235577848799034586115974278650623801359769267685457896776036408593799689021049070512575505077198744907879138302170241907403296239383935127408212301774637851,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558986485154257375407589700778316215797508001047976522480317713806344377028505509971869044870199749218918620566342457273648914828988476203838911936114782907546415347888459307637236170570040087886854641667906688531308101054774703302343229145963607299371655512196128601729276853487059227876487450137297356505885673907526716488648504529775513403137496875865038189213401112398982525894639748679809722008261479876445202252265766342087361486383823997321564215331776659788118324324232225397673741152837629197644022508102091929865110973671915127699729214950226446241567946836555579985811833965397732687270614654375288108807942634285848706951345948448048120216665314529575483515916493409537614384160894970494451295072395095125095841108376157946600945379840092140230740598647399655726757759549369148877202397435762850582777652803703491637999127438687972936010356847335680370,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558985191363273649877468862127787139218891217674698017207725879103061299416755295482258129215690094787197199355531366322623109200371547499618906616530070232102394251442860106639068110454636178587217989641576371757673675807442441690827891776005695491402140757000046787995799530828950802713362379899163485131393899063187249445280947027599545384541731846329113406758324466034452084213941027198633011295208673357439308162380960104584397876429540110574743559456830211747100000029851176663991131394068244877627465300460028865474375169754552055569440495860109595624073488106665485302428516577405633396397626788261422868816733453533993168185695811238514933909317914228013608496346367517406992099115987411063383482610339887297275305300111321296094629838891272220595703078968448948635711506991961143260251799986922740847983053735670810046421509516299703732795746545584499346,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558985191363273649877468862127787139218891217674698017207725879103061299416755295482258129215690094787197199355531366322623109200371547499618906616428034008385807002067358650576234775574964970050274465345342270632454828347674409708404509679800320240798894065278709087047361059861995056110447965080499826913987157124362727893766456703905862921434325567525843454765472377973403764733998610090892200786490162542423463964285184479005781629461936305215733609960447437140549885405632194666389946509671971611293314552263234275251312833012378124926183461047926928542352143663613613700485904150829623992360208705642606747780632949868160338892064544668415515103381542720803025180225313763939362281640666313964849850638728851447153815455161463268208630409669828525672851078443291980060426213860323554232925612510339741115888354068796750470862507664920305811744631248451292814,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558985191363273649877468862127787139218891217674698017207725879103061299416755295482258129215690094787197199355531366322623109200371547499618906616428034008385807002067358650576234775574964970050274465345342270632454828347674409708404509679800320240798894065278709087047361059861995056110447965080499826913987157124362727893766456703905862921434325567525843454765472377973403764733998610090892200786490162542423463964285184479005780994811049199676342964730846939445923154311949610873931726459758957529162063410900748125283156546204724223413897144463142859557299496388231948471058105525277509205505423692580039743875977289923725428596682742464674143021285982243531956824641818603041131889613882176335180975587428945364064586157974760762830173877752591853071775558142417801123280911142915720642787624318785587772555407445523105794237669641506440550700455361767037944,
3.597285023540417505497652251782286069135543054886576783720252127972957507555973931818035224037059178330204976748439827655104077645025252558985191363273649877468862127787139218891217674698017207725879103061299416755295482258129215690094787197199355531366322623109200371547499618906616428034008385807002067358650576234775574964970050274465345342270632454828347674409708404509679800320240798894065278709087047361059861995056110447965080499826913987157124362727893766456703905862921434325567525843454765472377973403764733998610090892200786490162542423463964285184479005780994811049199676342964730846939445923154311949610873931726459758957529162063410900748125283156546204724223413897144463142859557299496388231948471058105525277509205505423692580039743875977289923725428596682742464674143021285982243531956824641818603041131889613882176335180975587428945364064586157974760762830173877752591853071775558142417801123280911142915720642787624318785587772555407445523105794237669641506440550700455361767037944}

Oops, looks like it doesn't all display. However, if you quote this post, you'll be able to see what the output was. The thousandth digit is 4 (possibly rounded up, however).

CUT THAT OUT! YOUR GOING TO CRASH THE INTERNET!

Roger Cooke · **#33** by **Roger Cooke** » Jul 28, 2010 1:52 am

CdesignProponentsist wrote:
Roger Cooke wrote:
num1cubfn wrote:
Roger Cooke wrote:
Mononoke wrote:I would start by taking the natural log from both sides

Indeed, and then you would have the equation x log(x) = log(100).

I doubt if the solution to this equation is rational, as the Excel solution appears to imply. I think Excel is simply using limited precision and giving an answer that exaggerates the true precision. I'll run it through Mathematica and see what happens.

(Two minutes later). Mathematica gives 50 decimal places as

3.5972850235404175054976522517822860691355430548866.

Let me know if you want a thousand or a million decimal places. I'm guessing that I can prove fairly easily that x isn't rational. And people who know number theory well can probably invoke one of Vinogradov's theorems to prove that x is transcendental.

Awwww, I thought it was cool that it suddenly dropped to a bunch of zeros.

CURSE YOU EXCEL!!!!!!

Cool, can you please tell me what the millionth digit is?

I finally aborted the evaluation after Mathematica ran for 10 minutes. It would eventually get the answer, but I'm not sure how long it would take. I'll let it run overnight tonight and see what it produces.

Meanwhile, if you want to compute the number yourself, I have found it easy to do. Since this is a convex function, Newton's method is guaranteed to work. I programmed it in Mathematica and had it run 10 steps of the approximation with 1000 digits of accuracy. Since the last two steps were identical, I'm assuming I actually have 1000-place accuracy here. Here's what the input and output looked like:

f[x_]:=(x+Log[100])/(1+Log[x])

Table[N[Nest[f,7/2,k],1000],{k,1,10}]

{3.597879714527434874419322351070673233760002738054671459168194193265202137470008234490837549130587617519958354336843152769674582024289134126366297240420590710425481510271362235671059928940825156156347415477393592735605592680497903208995157477269061456981969612866232060903613749397423272194997022212704451995696293277422154920745064182959534436928918879925354621486342761226470846901298873276879846307573929057083865973020917892092813818262653847048979267473239267321921106136482240169633218672327152178094156832197011014487632696953625135986239195715186083978092751036191188438015530812406382904740841131208826148585359179994947299722580151059179894145040766778848905252010982278805627063044103590641280534041144985198923522345039941085328669258274760420892733816918577019178914144210058654266976131451014303983777973857085972414705421513525356335076448756838046654485937290463730320180286954269306184485612325783503267087010263002159801427044877252613951985218779739412123787965683936843042618280740,
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Oops, looks like it doesn't all display. However, if you quote this post, you'll be able to see what the output was. The thousandth digit is 4 (possibly rounded up, however).

CUT THAT OUT! YOUR GOING TO CRASH THE INTERNET!

Never mind the Internet. I've got all I can do just keeping my home computers from crashing. I just verified the associative law for the group operation on an elliptic curve. If you think the output here was excessive, you ought to see what Mathematica produced before I gave it the "Together" command and caused the whole thing to collapse to {0,0}. Last week, when I was trying to get the same output, I got my laptop stuck on the computation for 18 hours before I managed to abort it.

Tyrennil · **#34** by **Tyrennil** » Jul 30, 2010 2:14 pm

Roger Cooke wrote:
By the way, I had Mathematica run the numerical solution again, asking for 1,000,001 digits (so as to include the 3 ahead of the decimal point). The last two digits were 30, so the millionth digit after the decimal point is 0 (and obviously it couldn't have been rounded up). It took about 15 minutes to run, so I was simply too impatient yesterday.

Couldn't the digits 29 be rounded up to 30 if the digit following 29 is >=5?

Roger Cooke · **#35** by **Roger Cooke** » Jul 31, 2010 11:08 am

Tyrennil wrote:
Roger Cooke wrote:
By the way, I had Mathematica run the numerical solution again, asking for 1,000,001 digits (so as to include the 3 ahead of the decimal point). The last two digits were 30, so the millionth digit after the decimal point is 0 (and obviously it couldn't have been rounded up). It took about 15 minutes to run, so I was simply too impatient yesterday.

Couldn't the digits 29 be rounded up to 30 if the digit following 29 is >=5?

Good point. I stand corrected. Perhaps I'll run it again and get a couple more decimal points, just to be sure. (With luck, they won't all be zeros, although you never know. If you do a million digits of pi, you'll get a string of six 9's in a row near the end.

Pyrion · **#36** by **Pyrion** » Aug 12, 2010 8:09 am

I am quite impressed about all those contributions to my thread. So it is as i suspected, this function can only be approximated, not analytically solved.

And by the way, there are only 10 kinds of people, those who understand binary and those who do not

.

Sityl · **#37** by **Sityl** » Sep 08, 2010 11:02 pm

Cynergy wrote:
Benach wrote:Because most people feel dumb. Fortuny I had an excellent math prof at uni and was willing to answer even the silliest questions (Christians asking about "mathematical proofs" of god etc.).

Well, there are only three kinds of people:

(a) Those who are so clever that being told they are very clever is unremarkable. For example, were I 6' 7" (I'm not), to tell me "You're very tall" would be obvious.

(b) Those who are smart enough to know when they're being flattered.

(c) Those who are dumb enough to believe anything.

You forgot:

(d) Those who are so dumb they don't even realise they're being complimented!

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