Dear Scholar
No, the real value of Pi is an algebraic number
Dear Sarva,
Is it another round?
I spent more than two days to read your work and show you the mistakes in your articles mentioned in your previous question:
Are Different Geometrical Entities Can Be Unified?
We reach a happy end that:
Pi is Really A Transcendental Number
What happened ?!!!
RG should be a serious platform to share knowledge and not to play games.
Dear Sarva,
using the argumentation present in the file you enclosed, EVERY number can be proved to be algebraic.
Dear Sarve, continuing Viera comment let me add:
... and this would contradict the non-countability of the set of reals.
Conclusion: this would be a new theory of reals. But then, please present the full system of new axioms.
Some tips: resign of existence of uncountable sets OR change the definition of real numbers. Obviously, since it would be a different theory, you will need also to introduce a new definition of \pi. Consider also what name would be given to these new numbers including the new \pi (suggested notation: πν - remember please, that this was my original idea:). Honestly - such a theory should be elaborated in many aspects which would be unnecessarily time consuming. Eg. If you want to have this nes set of numbers closed wr. to addition and multiplication according to the same rules as for reals, then you would come to some additive subgroup contained in algebraic numbers. Taking for good, that you wouldn't resign of the any algebraic numbers, one can infere, that the new real numbers and algebraic numbers would coincide. Thus remains only introduce some replacements for such numbers like e and \pi, which are not algebraic. Or resign of their existence. I hope that these remarks will help you in taking the right decision about further developement of disproving transcendence of \pi.
Best regards, Joachim
Dear Sarva,
Few months back you mailed me about "pi". On that time, I told you that "pi" is a transcendental number. Please read this:
https://en.wikipedia.org/wiki/Transcendental_number
This result is very well known undergraduate level results.
Sorry, but your vew text does not explain why the \pi value is equal to (14 – root 2)/4 = 3.1464466…
Obviously, this newly defined \pi is not equal to the ratio of the circumference and the diametere of a circle.
At least no proof has been supplied till now.
Regards, Joachim
.
"Pi is divine"
we had so much fun with Pastafarians ... i can't wait for Pistafarians
.
.
did you know it ?
Pistafarians almost took over Indiana in 1897 :
http://www-personal.umich.edu/~jlawler/aux/pi.html
(The case of Indiana vs. Pi, by M. Brader, an old and excellent usenet post)
.
now, they're storming RG ...
.
Everybody knows that Pi is an integer. The Bible even gives its value, which is 3. You can check in the Book of Kings, 7, 23, which describes a bronze vessel with diameter 10 cubits ("coudées") and perimeter 30 cubits, as well as in the Chronicles 2, 4, 2, which gives the proportions of Solomon's temple.
@Peter In the book of Kings, they used a rope to measure the perimeter with a rope. I guess they did the same with Solomon's temple. This leaves us with 2 hypotheses : - the scribes of the Bible were bad surveyors ("arpenteurs") - God is a bad mathematician
I don't like involving God into inappropriate affairs, since it this unnecessary touching well known Sacrum (of some people, at least).
Best regards, Joachim
PS. To anyone: I would appreciate to not reply to this note in any form.
I agree with Peter's motto, that I don't feel obliged "to take everything other people take seriously, seriously".
@ Peter
Please, be more precise: who is "you" in "yours", who is "they" in "them", and whom "your thinking" (in "I think") is destinated at all?
Let us better decide - each one separately - whether are we continuing hopeless explanations being sure we do not succeed, or are we simply breaking our contribution. No strange comments are appropriate in the current state.
Regards, Joachim
I am declaring to break any further contribution to this thread. JoD
@Stefan Gruner Allow me a few comments concerning the "invasion" of RG by "amateur-hobby-'mathematics' and pseudo-science", or "Pistafarians" as Fabrice Clérot calls them :
1) In view of the context, your statement "In case that PI would not be a transcendental number, that would have been discovered already long time ago" seems to me an unfortunate ... understatement. Pi is transcendental, there is the proof by Lindemann (as well as the akin proof of the transcendence of e by Hermite) that everybody can read and guarantee. Since mathematical science rests on the absolute principle that A and non-A cannot be true at the same time, my position is : never discuss in anyway with any layman who pretends to cast a doubt on an important theorem (a Pistafarian against Hermite or Lindemann) or a fundamental theory (a "quintic solver by radicals" against Galois). Not that I bow in front of authority, but if a theorem is important, it has been checked a hundred times by peers and professionals, and if a theory is important, it has proved its validity in a hundred applications in and outside its original domain. Besides, what a Pistafarian does is not even to point out a flaw here or there in an established (in general not easily accessible) proof. What he wants is to present an "elementary proof" which he would like you to check. Why waste your time if you know the outcome in advance ? Besides, I bet (human nature being what it is) that he would never accept your verdict anyway.
2) There is an intermediary case where my position is less unequivocal. FLT is a perfect illustration. There is a well established proof provided by one of the greatest mathematicians of our time. But it's not within everybody's reach, and anyway, counting all the preliminaries (which start at, say, Phd level), a complete exposing would take thousands of pages. So what should we do when a FLTstafarian presents himself with an "elementary proof" ? He doesn't pretend to disclaim Wiles' proof. What he asks is that you check his own. I've had many times the opportunity, on this very site, to allude to the so called "Krasner trick", based on p-adic considerations, which allows you to get rid in advance of all the "elementary proofs" of FLT which appeal only to operations in a ring which is a UFD (as when doing elementary arithmetic in Z). But this "trick" does not cover all elementary methods, and who knows, as Peter Breuer says, "maybe one of them will be a jackpot winner some day." My empirical conduct is to take a quick look, locate a mathematical flaw (or even, in most cases, a logical fault), and stop there, I mean point it out, but never accept to pursue the discussion - on a different basis, under other assumptions, what do I know ? - before the author acknowledges his error.
3) The most prickly case is that of unsolved conjectures. The Riemann hypothesis is a good example. All we can do here is to go on "feeling". Taking a look at the presented "elementary proof", we can (or not) feel if it contains enough input/machinery to tackle a problem that we know to be very, very deep. But there is no scientific certainty here, only experience and a kind of "art".
4) Let me conclude on your question " Why is ResearchGate providing 'space' for all this amateur , etc." Apart from some kind of "moderation" to prevent polemics to become too hot, I think that the absence of strict scientific rules on RG is meant to not discourage math. fans, whatever their formation or level, on the contrary to encourage the participation of "amateur-hobby-'mathematics' " coming from various horizons. If this is ideed the rule of the game, then a solution against the invasion of "pseudo-science" would be to practice some kind of self control and refrain from answering to "pseudo-questions". More easily said than done ./.
What is the transcendental Number? F.Lindeman prove that, if a1,...,an
are algebraic numbers, and b1,...,bn are nonzero algebraic numbers
then b1ea1 +b2ea2 +...+bnean ≠ 0. But
e ^(iπ)+1=e ^(iπ)+e^(0)=0
since i is an algebraic number then π is not an algebraic number.
The end of the dream that we can construct a square with edge √π.
Dear Scholar
3.1415926... is NOT pi number
Euler's equation is a FUNNY equation
It does NOT accept 3.141... It accepts pi radians 180 degrees
Real pi number is algebraic
Squaring a Circle is DONE now with the TRUE VALUE OF PI
Here is the work
.
well tried Salah !
but, as you see, nothing and no one can compete with "Durga style" mathematics
.
.
i had even weirder reads
http://www.iosrjournals.org/iosr-jm/papers/Vol10-issue1/Version-4/C010141415.pdf
.
@Fabrice Clerot
Oh... Durga is one of the numerous names/forms of Kali, or Parvati, or whatever else, the goddess of preservation, transformation and destruction, and whatever else. She plays a prominent role in R. Zelazny's famous (half Sci Fi) novel "Lord of Light". But after all, her appearence here is not irrelevant, because one of her missions is the destruction of all forms of ignorance (avidyā). I've read Durga's "math" paper in your link, and it suddenly occurred to me that perhaps we should first and forehand define the things we're talking about. So my question to all the followers of this thread is : "What is your definition of a number ? Of an integer ? A rational ? An algebraic ? A real ? " The terms and content of our discussion would appear much more clearly then.
Hi! I have an idea. Pi is not a number. So, the question is it transcedental or integer (as also proposed here) does not exist.
Best!
P.S. By the way, Archimedes' last words were "Nōlī turbāre quadratus meōs!" but he was not translated correctly.
Dear Scholar
Pi is a RATIO of two numbers representing 1. diameter and 2. length of circumference
Further, Pi Constant is NOT NOT NOT NOT NOT NOT NOT NOT essential .
We can find CORRECT AREA & CIRCUMFERENCE using RADIUS ALONE, and WITHOUT PI CONSTANT.
HOW?
Here it is in the attachment
@ Sarva,
Your previous attachment {The area.docx17.12 KB}
Circumference = 6r + (2r -sqrt(2)r)/2
this means that Pi is an algebraic number !!! illusion.
All followers told you, it is a wrong result.
your statement:
Further, Pi Constant is NOT NOT NOT NOT NOT NOT NOT NOT essential .
has no meaning.
To find the circle area and circumference, pi is a must, circumference = 2(pi)r.
All other calculations are approximations.
What if one submits the idea that Pi Constant is NOT (*8) essential to the physicists? I just wonder...
Dear Scholar
What is the purpose of having Pi ?
It was meant for area, circumference etc. of circle.
In square we have square of side and 4* side for area and perimeter.
In triangle 3*side and !/2*altitude*base for perimeter and and area. Here we do not have ANY CONSTANT.
WHY SHOULD WE HAVE A PI CONSTANT FOR CIRCLE?
It is ENOUGH if we know RADIUS as side in square and altitude and base in triangle.
OK
I have a question for Sarva. Since an ellipse is nothing but a kind of "flattened" circle, surely your approach can give us its perimeter ?
Dear Scholar
We can calculate area and circumference without using pi value 3.14... and USING RADIUS only
Hi, to the question "What is the purpose of having Pi ? "
No concrete purpose.
We try to express the fact that we do live in the space. At least, that's where we think we live, that is - we perceive the world in that way.
Our systems to express what we perceive are not perfect, but we try to find out how the world is made and to express it. Pi is an important part of these trials of ours.
Perhaps Pi is not just how many times d enters pi*d. This is a feature we do observe and, being an obvious space feature, it was discouvered first. I wonder, for exemple, what is Pi doing in the Normal (Gaussian) distribution. Could anyone relate it to circles, please? There are mathematicians here.
ToDr. Jan-Martin Wagner Dear Scholar Here is a simple experiment . I did it . You may also REPEAT it. Requirements : a metallic wire, measuring tape Take a wire of length 107 cms. and make it a circle . And measure its diameter. The diameter would be 34.....When Circumference is 107 cms. , its diameter is 34.00....... Pi= 107/34= 3.147......So, it means Cosmic Pi equal to (14-root2)/4=3.14644660941......is 100% TRUE. RSJ REDDY
📷
Come on, the scribes of the Book of Kings 23, 7 did the same experiment and they found 3
@Issam , @Peter
I wonder why you're wasting your time in hopeless discussion. The author's last post clearly shows that he has no precise idea of what a (real) number is
@ Sarva,
Here is an elementary proof that your claim is WRONG.
Assume that pi = p = (14 -- sqrt(2) )/4, then 14 --4p = sqrt(2)
your pi is p = 107/34 ( your previous answer: the ratio of the circumference over the diameter).
Substitute in the equation :
14 -4( 107/34) = sqrt(2)
squaring both sides we obtain (576/289)=2
which is the required contradiction.
I think you need to read more about,
(Rational numbers, irrational numbers, algebraic numbers, transcendental numbers), this will help you to understand the follower's answers.
Dear Scholar
Here is yet another truth on real pi and of its inseparable association with Pythagorean Theorem
.
no one can fight against the "Cosmic pi"
May we all be touched by His Noodly ap.PI.ndages ...
.
@Issam
And you do well. This is not a rational discussion any more.
"No, the real value of Pi is an algebraic number" -- Not at all; see https://www.youtube.com/watch?v=seUU2bZtfgM, starting from 10:00.
S. J. Reddy, a month ago: "WHY SHOULD WE HAVE A PI CONSTANT FOR CIRCLE?"
(This question is taken from the following line of reasoning: "In square we have square of side and 4* side for area and perimeter. In triangle 3*side and !/2*altitude*base for perimeter and and area. Here we do not have ANY CONSTANT. WHY SHOULD WE HAVE A PI CONSTANT FOR CIRCLE? It is ENOUGH if we know RADIUS as side in square and altitude and base in triangle.")
You know the answer already, dear Sarva Jagannadha Reddy: The circle is curved, perfectly, everywhere, but "length" and "area" are only defined for straight and square objects, respectively.
"Length" is just the shortest distance between two points, and in our world that's a straight line.
Therefore, we need a special constant to take into account the round shape of the circle when using straight lines and squares as measures for length and area.
A triangle has straight lines, therefore no special constant is needed for a triangle. A square is the basic defining entity for area, therefore no special constant is needed for a square.
Do you now understand why we need pi for the circle? Do you now understand why all your formulas "just using the radius" (as given in "The area.docx") are fundamentally wrong, therefore?
S. J. Reddy, 2 months ago: "To Dr. Jan-Martin Wagner Dear Scholar Here is a simple experiment . I did it . You may also REPEAT it. Requirements : a metallic wire, measuring tape Take a wire of length 107 cms. and make it a circle . And measure its diameter. The diameter would be 34.....When Circumference is 107 cms. , its diameter is 34.00....... Pi= 107/34= 3.147......So, it means Cosmic Pi equal to (14-root2)/4 = 3.14644660941...... is 100% TRUE. RSJ REDDY"
No: This just means that your measurement wasn't accurate enough.
@Jan-Martin Wagner
Believe my experience, you'll never be able to convince these "Pistafarian" people, be it about the trancendence of Pi, FLT or the impossibity to solve a general quintic polynomial by radicals. The experimental measurement of "Cosmic Pi" by the scribes of the Book of Kings gave Pi=3. Now your contradictor finds 3.14644660941..., hence "100% TRUE". Even the young hero of the movie "Life of Pi" (2012) knew many more decimals, and by heart at that ! Perhaps the ultimate experiment would be to ask a Pistafarian his definition of what a real number is.
@Thong Nguyen Quang Do
Yes, experience is as you described it, and in that respect what I do here seems useless. On the other hand, for me this seemingly useless discussion has brought up the question about highly accurate experimental tests of the numerical value of pi -- and with "highly accurate" I mean to be off well below 0.1 % (because this would be enough to prove Cosmic Pi wrong).
I've started to look for relevant publications, but in a first "quick and dirty" literature search I didn't find any. Therefore I would be happy to receive corresponding information!
I found this document
Les Japonais Alexander J. Yee et Shigeru Kondo explosent leur précédent record. Ils calculent exactement 10 000 000 000 050 décimales du nombre pi, après 371 jours de travail. La machine employée est composée d'un duo de processeurs Xeon X5680 à 3,33 GHz associé à 96 Go de mémoire DDR3, ainsi 24 disques de 2TB.15 mars 2015
Pi, Historique du record de décimales - Math93
https://www.math93.com/index.php/...des.../191-pi-historique-du-record-de-decimales
Sorry, I'm not looking for a numerical calculation or an equation representing a geometrical construction or yet another infinite sum -- I want a real-world hands-on measurement where one has to carefully discuss the magnitude of the error bar to make sure that the final result can be trusted up to the relevant decimal position.
Any "real-world hands-on measurement" would be disputed anyway by the Pistafarians. Instead I suggest you compute somehow the algebraic number given by the "cosmic believer" with high enough accuracy (I have no idea how to proceed, but this is surely feasible) until a discrepancy appears with the decimal expansion of Pi. But why should you or anybody waste time just to refute a lunacy ?
On the one hand, you might be right again, that even such an experiment would not help to convince a strong believer. (However, since S. J. Reddy has made such experiments himself already, I guess he could be impressed by such an experimental result.)
On the other hand, I meant this as a serious scientific question independent of the present discussion with S. J. Reddy: Are there real-world hands-on measurements, with carefully discussed magnitudes of their error bars to make sure that the final result for pi can be trusted up to the relevant decimal position?
So far, I didn't find any relevant publication (just some do-it-yourself stuff with low accuracy: https://www.wired.com/2015/03/area-circle-value-pi/ and https://www.scientificamerican.com/article/circular-reasoning-finding-pi/). Please tell me if you know some. Otherwise this means that there might still be some scientific merits to be earned...
Jan Martin, do you want to play Buffon needles ?
https://en.m.wikipedia.org/wiki/Buffon%27s_needle
This is cool too but in French
http://www.iecl.univ-lorraine.fr/~Gerald.Tenenbaum/PUBLIC/IECN_2003/IECN2003-043-050.pdf
Yes, of course, Buffon's needle could help here -- in principle. But how many times would one have to let drop the needle in order to obtain an accuracy for pi better than 0.1%? And how accurate do we need to fabricate the needle and the stripes in the plane to reach this accuracy?
In the "short-needle case" (where the needle length l is below the stripe width t) one has that pi ≈ 2l/(tPn), with Pn = x / n giving the fraction of x crossings for n drops of the needle. This means that the relative error in such an estimation of pi is the sum of all relative errors of the "ingredients" -- so that, in order to stay below 0.1% for pi, the length of the needle and the stripe width would have to known to such an accuracy, and also Pn would have to be determined to that accuracy.
The "mechanics" should be feasible; for example, using a stripe width of 100 mm, with a tolerance below 0.1 mm (and similarly for the needle). However, I need to think further how to estimate the accuracy of Pn; it's a counting experiment where the number of successes one is asking about (here: x) is of the same order of magnitude as the number of trials (here: n), so I think it's not a Poisson distribution behind it. So, what is it, then?
Again, I'm talking about an accuracy of better than 0.1% because I want to obtain a final result like this: pi ≈ 3.142 ± 0.002, so that 3.146 can be safely excluded from being pi.
Since I understand French more or less well, thanks also for the nice 2003 paper that you provided the reference to.
.
regarding the search for accurate variants of the Buffon needles experiment, you might appreciate
http://www.stat.wmich.edu/wang/688/notes/note14.pdf
.
It was proved in the 19th century by Lindemann that every algebraic power of e is transcendental. This implies, in particular, that pi is transcendental. However, I am not sure about who was the first to prove that pi is transcendental...
Dear Scholar
Lindemann gave a faulty proof that Pi is transcendental
1. Euler's equation e to the power i Pi + 1 = 0 accepts Pi radians 180 degrees and this equation REJECTS Pi constant.
2. How could Lindemann say the rejected Pi constant 3.14 as transcendental ? Do you agree dear ?
------------------------
The idea of transcendence of Pi constant started with the introduction of " infinite series " by James Gregory in 1660.
---------------------------------------
Real Pi is an algebraic number. Here in the attachment you can see a proof for its derivation
Dear Sarva Jagannadha Reddy, you're wrong: Lindemann's proof is correct; but since the mathematics is quite intricate, you probably don't understand it -- and therefore think it's wrong.
Dear
Mathematics is made intricate by us to suit our logic. Frankly speaking ,we have made SCIENCE thus as a SELF-CORRECTING SUBJECT. Newton Laws are being questioned. Einstein Laws are being questioned.
Secondly, Geometry or Cosmometry ( geometry of Cosmos - coined by this Pi student ) is NOT Mathematics or a branch of it. Geometry is an independent subject and a true Science subject like Botany, Chemistry, Physics, Geology, Zoology, etc.
In Euler's equation e to the power i pi + 1 = 0 , substitute 3.14, instead of 180 and see Lindemann is right or wrong in calling 3.14 as a transcendental number.
I personally placed this argument before the local professors. All agreed that Pi radians 180 degrees is NOT same or equal or identical with Pi constant 3.14.
Are the local Professors are wrong ?
So, Lindemann has called polygon's number 3.14159265358 as a transcendental number.
Or he called 180 degrees as a transcendental number.
If we agree this 3.14159265358 of polygon as a transcendental number, we have to necessarily accept that polygon is a transcendental entity.
Are we prepared to accept this blunder ?
Dear Sarva Jagannadha Reddy, please answer these questions:
-- You wrote: "In Euler's equation e to the power i pi + 1 = 0 , substitute 3.14, instead of 180 and see Lindemann is right or wrong in calling 3.14 as a transcendental number." Please show me the exact place in Lindemann's proof where he uses explicitly "180 (degrees)" or "3.14 (radians)".
-- You wrote: "Pi radians 180 degrees is NOT same or equal or identical with Pi constant 3.14." Why do you need to specify "Pi radians 180 degrees"? What do you mean by that? (My problem here is that I don't see why you discussed about this with the local professors, so I try to understand what your thinking is based upon.)
Dear
Euler's equation has relation with this theorem
De Moivre's theorem can be derived from Euler's formula. e^(i * x) = cos(x) + i * sin(x) (often shortened to cis (x)). Euler's identity is a special case of Euler's formula, when x is pi.
Hence here Pi is not Pi constant 3.14 but Pi is Pi Radians 180 degrees in the place of x
Ok
Dear Sarva Jagannadha Reddy, please answer also this question:
-- You wrote: "In Euler's equation e to the power i pi + 1 = 0, substitute 3.14, instead of 180 and see Lindemann is right or wrong in calling 3.14 as a transcendental number." Please show me the exact place in Lindemann's proof where he uses explicitly "180 (degrees)" or "3.14 (radians)".
I would like to remind the followers the answer of prof.Thong
about this thread:
[[[@Stefan Gruner Allow me a few comments concerning the "invasion" of RG by "amateur-hobby-'mathematics' and pseudo-science", or "Pistafarians" as Fabrice Clérot calls them :
1) In view of the context, your statement "In case that PI would not be a transcendental number, that would have been discovered already long time ago" seems to me an unfortunate ... understatement. Pi is transcendental, there is the proof by Lindemann (as well as the akin proof of the transcendence of e by Hermite) that everybody can read and guarantee. Since mathematical science rests on the absolute principle that A and non-A cannot be true at the same time, my position is : never discuss in anyway with any layman who pretends to cast a doubt on an important theorem (a Pistafarian against Hermite or Lindemann) or a fundamental theory (a "quintic solver by radicals" against Galois). Not that I bow in front of authority, but if a theorem is important, it has been checked a hundred times by peers and professionals, and if a theory is important, it has proved its validity in a hundred applications in and outside its original domain. Besides, what a Pistafarian does is not even to point out a flaw here or there in an established (in general not easily accessible) proof. What he wants is to present an "elementary proof" which he would like you to check. Why waste your time if you know the outcome in advance ? Besides, I bet (human nature being what it is) that he would never accept your verdict anyway.
2) There is an intermediary case where my position is less unequivocal. FLT is a perfect illustration. There is a well established proof provided by one of the greatest mathematicians of our time. But it's not within everybody's reach, and anyway, counting all the preliminaries (which start at, say, Phd level), a complete exposing would take thousands of pages. So what should we do when a FLTstafarian presents himself with an "elementary proof" ? He doesn't pretend to disclaim Wiles' proof. What he asks is that you check his own. I've had many times the opportunity, on this very site, to allude to the so called "Krasner trick", based on p-adic considerations, which allows you to get rid in advance of all the "elementary proofs" of FLT which appeal only to operations in a ring which is a UFD (as when doing elementary arithmetic in Z). But this "trick" does not cover all elementary methods, and who knows, as Peter Breuer says, "maybe one of them will be a jackpot winner some day." My empirical conduct is to take a quick look, locate a mathematical flaw (or even, in most cases, a logical fault), and stop there, I mean point it out, but never accept to pursue the discussion - on a different basis, under other assumptions, what do I know ? - before the author acknowledges his error.
3) The most prickly case is that of unsolved conjectures. The Riemann hypothesis is a good example. All we can do here is to go on "feeling". Taking a look at the presented "elementary proof", we can (or not) feel if it contains enough input/machinery to tackle a problem that we know to be very, very deep. But there is no scientific certainty here, only experience and a kind of "art".
4) Let me conclude on your question " Why is ResearchGate providing 'space' for all this amateur , etc." Apart from some kind of "moderation" to prevent polemics to become too hot, I think that the absence of strict scientific rules on RG is meant to not discourage math. fans, whatever their formation or level, on the contrary to encourage the participation of "amateur-hobby-'mathematics' " coming from various horizons. If this is ideed the rule of the game, then a solution against the invasion of "pseudo-science" would be to practice some kind of self control and refrain from answering to "pseudo-questions". More easily said than done ./.]]].
Nothing more to say.
Dear Sarva Jagannadha Reddy, although you have now provided the relevant equations related to the second question I had asked one day ago, you still haven't answered the first one -- here with some boldface markup to make clear what the most important point in that question is:
-- You wrote: "In Euler's equation e to the power i pi + 1 = 0, substitute 3.14, instead of 180 and see Lindemann is right or wrong in calling 3.14 as a transcendental number." Please show me the exact place in Lindemann's proof where he uses (one of) the numerical values "180 (degrees)" or "3.14 (radians)" explicitly.
Since you have just uploaded (again) a short version of the original proof of Lindemann's, it should be easy for you to tell where in this work such an explicit numerical value is used.
It's funny that the document you are posting about Lindemann, besides having flagrant typos (e^ix + 1 = 0 instead of e^iPi + 1 = 0) and being very condensed and not the complete proof, concludes that Pi is transcendental!
Dear Scholar
It is really funny WITHOUT knowing what is Pi number we call it a transcendental number
Here is one paper for the derivation of Pi value
Present number 3.14159265358 is not that of circle. It is polygon number. Are we prepared to accept polygon number as a transcendental number?
@Sarva
It's even more funny that you obtain a formula which you don't bother to check numerically : sqrt 2 = 1,41421..., so (14 - sqrt 2)/4 = 3,14644... > pi = 3,14159...
Many many people have proposed geometric constructions to "square the circle", i.e. to obtain approximate values of pi (not the exact value, contrary to your pretention). Until recently, the record was held by
Until recently, the record was held by Kochanski, who produced the approximation sqrt (40/3 - 2sqrt 3)=3,14153333... See arXiv:1806.02218v1
Dear Scholar
Hippocrates of Chios did Square a Circle 2500 years ago. Here it is. It is unfortunate it was ignored and he was laughed at.
With the algebraic Pi number (14-root2)/4 , this book is written and is for your kind information for the SQUARING OF CIRCLE
"It is really funny WITHOUT knowing what is Pi number we call it a transcendental number" -- No, not at all; in contrast, this is how true mathematics works: to derive conclusions from a minimum of input information, just following basic principles of logical reasoning and making use of previously derived conclusions.
Most importantly, note that it is not correct to say "without knowing what is pi number" when you just mean "without knowing the numerical value of pi" -- because we all know what pi "is", namely the ratio between circumference and diameter of a circle. As such, it is mathematically well defined, and it's not relevant that its numerical value does NOT show up in this definition. (This is an example for what is meant by "minimum of input information".)
Moreover, Lindemann's work is not about just "calling" pi transcendental, but to logically show why it is transcendental -- as a direct consequence of its basic definition (circumference / diameter ratio).
Therefore, since Cosmic Pi is algebraic, it CANNOT be pi.
"Hippocrates of Chios did Square a Circle 2500 years ago." -- Since this would only be possible if pi were algebraic which, however, it isn't, it follows by simple logic that, whatever he did, he did NOT square a circle.
It also follows that if he thought he did square a circle, he made an error somewhere. It also follows that if somebody else thinks a circle was squared 2500 years ago, he is unaware of the present status of mathematical knowledge.
Again: Anybody who supports Cosmic Pi discloses a significant lack of present mathematical knowledge.
Now you are divagating about Hippocrates squaring the circle, etc. This is called "drowning a fish". Sticking to my policy (as recalled by @Issam), I demand that you reply to my remark that (14 - sqrt 2)/4 = 3,14644... > pi = 3,14159... , so your construction is blatantly flawed.
@ Sarva,
Your article published in a PREDATORY JOURNAL.
It is one of the journals listed in the following link:
https://predatoryjournals.com/journals/
We strive to show your wrong results, but it seems
you choose to play games at RG and to repeat the same round every some period!!!
@ Dear Followers,
Do you think to repeat the same answers and the same arguments with the same question make any sense?
Best regards
Hippocrates could not have squared the circle, subject to the straight edge and compas rules for squaring the circle.
The evidence:
1. What one fool can discover another can also discover, and in two and a half thousand years no valid construction to square the circle has been found.
2. To do so would require that $\pi$ be a constructible number, and it isn't as it has been proven to be transcendental.
The circle can be squared but to do so requres relaxing the rules on what constitutes a construction.
Dear Scholar
He did Square the semicircle, full circle and lunes
Here is his work explained using newly discovered ( March 1998) Pi number (14-root2)/4.
@Sarva
So now the new value of Pi is 3,14644... And tomorrow eiPi will no longer equal -1. And there will be darkness at noon... I consider all this as a non answer to my request, meaning that you are ready to issue any gibberish non sense rather than recognizing your mistake.
My dear Sarva Jagannadha Reddy,
pardon me, but may it be the case that you start to suffer from dementia? I just ask because this discussion reminds me of conversations I had with my grandmother during her last years -- since we have been through all this already several times.
So, once more: It's a matter of pure logic that Hippocrates did NOT square anything, and it is clear that your "explanation" is erroneous. However, as long as you keep sidestepping from logical arguments I will not point out your errors explicitly.
Instead, I continue with your nice handwritten notes about Euler's equation: Everything there is correct except for the very last statement that pi = 180°, which is just wrong because pi is not measured in degrees; instead, it is a pure, unit-less number -- also in this equation!
Don't be tempted to think of having to take pi in degrees just because it is used in the argument of sine and cosine. Maybe this is where your above-mentioned local Professors are wrong, too.
We can also try to turn things around and discuss it this way: What do you expect, my dear Sarva Jagannadha Reddy, to obtain as a numerical result when calculating e to the power (i times Cosmic Pi)?
Addendum: Note that calculating the numerical value of the exponential function for any numerical input value (no matter whether it's a real or a complex number) can be obtained from the series expression of the exponential function:
exp(x) = 1 + x + x2/2 + x3/6 + ... + xN/N! + ...
Ok, I have two questions:
1. "Is Tau, the double of PI, a transcendental number?"
2. "Is, in general, the double of a trascendental number also trascendental?"
Trancedental numbers are irrational, so we may prove:
Any non-zero rational multiple of an irrational is irrational.
Proof (trivial)
Suppose z irrational, and for non-zero integers a and b that (a/b)z is rational.
So there exist non-zero integers c and d such that:
(a/b)z=c/d
so:
z=(c/d)(b/a)
But the lefthand side of this is irrational and the right hand side rational - a contradiction, so (a/b)z must be irrational.
Side remark: When Sarva Jagannadha Reddy says, "Real Pi is an ALGEBRAIC number", he has his algebraic home-made value (14-sqrt2)/4 in mind. However, this value has nothing to do with the circle; instead, the circumference / diameter ratio of a circle is transcendental.
@ Sarva,
My kid doesn't like transcendental numbers, and he prefers to use an algebraic number instead. He convinced that your claim is easily understood. So, he decides to send you the following Medal.
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