Let me discuss some aspects from "130th METHOD_27-10-16.docx":
"Are e and pi same to adopt the proof of the transcendence of e to pi?" -- Well, this is not the point. Of course, e and pi are not the same. However, as F. Lindemann has shown, the transcendenceness of e can be used to prove the transcendenceness of pi.
"When a circle is inscribed with the square we see diagonal and circumference together in the square. Diagonal is represented by sqrt2 though we can not measure the exact length of the diagonal with the straightedge." -- The diagonal is a finite straight line. Of course one can measure the length with quite some accuracy; it just depends on the experimental abilities, but there is no fundamental hindering for a highly accurate measurement.
"130th Mathematical Truth on Pi" and "This author very strongly believes that there is no difference between diagonal and circumference. Hence, both entities should be represented by sqrt2." -- This is obviously a contradiction, because "I belive" is not a "mathematical truth".
"Hence, both entities should be represented by sqrt2." -- This is not possible, because pi is transcendental, whereas sqrt2 is not.
"So, 3.1415926… can be called a limit Pi. The real definition is pi = circumference of circle / diameter of the same circle" -- Well, this is not a contradiction, because the "limit value" of pi is obtained from the "real definition".
"In the case of limit pi, 3.1415926… as pi value is obtained from the following: pi = perimeter of polygon (=3.1415926...) / diameter of circle (1). So, the limit pi is not pure, then what it is? It is a hybrid of polygon and circle." -- No, because the diameter of the circle is also a diameter of the polygon (namely, the largest). But htat's completely irrelevant, because the only relevant thing is the proper definition of pi (circumference / diameter).
"Even if circumference is accepted as transcendental, how can we divide it by diameter, which is non-transcendental ? i.e. pi = transcendental circumference / non-transcendental diameter" -- So what? Dividing by 1 doesn't change anything, so it's clear that the result is transcendental.
"What we are actually doing is, we divide perimeter of polygon by the diameter of circle, but the definition of pi is, dividing the circumference of the circle by the diameter of the circle itself." -- Of course! The diameter is the only well-known object here, but the circle's perimeter is a problem. There is no direct measurement prescription for the length of a curved line; "length" is defined for straight lines only (distance between two points). Therefore, the circle is approximated by regular polygons, which come arbitrarily close to the circle. The larger the number of edges in the polygon, the smaller the error bercomes; so, the perimeters of polygon and circle differ by an amount that can be made arbitrarily small.
"So, 3.1415926… can be obtained both by geometrically, with sqrt3, and the same number, without any change, by adopting “infinite series” without sqrt3." -- No, 3.1415926… cannot be obtained geometrically, with sqrt3, because only an approximative value can. And of course, because that approximative value is obtained geometrically from a polygon, it is algebraic. However, this approximative value isn't pi itself, so there is no contradiction; the true value of pi is obtained only in the limit of a series of polygons with arbitrarily many edges. Due to this construction using only algebraic numbers, it is not per se clear from this approach whether pi is algebraic or transcendental. But this is irrelevant, because the proof that pi (and here I mean the "limit value" of pi!) is indeed transcendental doesn't use the polynomial approximation.
"So, pi constant (3.14) = pi radians (180°)." -- No, not at all! A given angle has different numerical values when measured in degree or in radians. However, pi has always the same value, no matter whether it represents radians or not.
"The individual areas of square, triangle and circle are aligned ingeniously that there exists, a certain order maintained by the three. This is possible only when all the three are of one category either algebraic or transcendental." -- Fiddle-faddle! "Magnitude" is a completely different category and has nothing to do with "transcendenceness". Euler's number e = 2.71828... is transcendent, and the algebraic numbers 2 and 3 are smaller and larger than e, respectively -- but that doesn't make e algebraic!
"As all the three are overlapping with each other, this indicates all the three are algebraic and no possibility of any one, especially circle, to be a transcendental entity." --- Again, just Fiddle-faddle (for the reason just explained)!
"So, this equation, (Circle – Triangle) + (Square – Circle) = Square / 2 = a² / 2, tells us that all the three: square, circle and triangle are algebraic." -- Fiddle-faddle! The left-hand side computes likes this: (Circle – Triangle) + (Square – Circle) = Circle – Triangle + Square – Circle = Circle – Circle + Square – Triangle = Square – Triangle. So, there is no circle involved in this equation; it doesn't contribute anything; any other arbitrary value could have been added and subtracted simultaneously, and this would make no difference. This is nothing but eyewash.
"If the impossibility of measuring direct by straightedge (even after circumference is straightened) of the circumference is the reason; the same difficulty exists in measuring the exact length of diagonal of a square." -- No, not at all! The difficulty only lies in the curvature; a straight line can be measured without difficulties.
"But circle and diagonal are finite entities having terminal ends, and having thus finite magnitude, to be represented by a finite number sqrt2 of diagonal to circumference also." -- So what? Also pi is a number with a finite value. It's just that sqrt2 is algebraic, whereas pi is not.
Dear Scholar
As long as your mind is FIXED that pi is a transcendental number you look a shadow is true rather the the original one
"As long as your mind is FIXED that pi is a transcendental number you look a shadow is true rather the the original one." -- As long as you don't understand the constraints on pi given by both an inscribed and a circumscribed polygon (and I'm not talking about the limit), you also don't know what you're fighting against. (Say, do you know the story of Don Quichotte?)
Again: "I belive" is not a "mathematical truth"!
Dear Scholar
Your 3.141....is a polygon number and you support it is pi of the circle.
I challenge you , show me second geometrical method
"I challenge you, show me second geometrical method." -- Sorry: a second geometrical method for what, please? Since pi is a transcendental number, there is no other way to introduce pi than by stating that pi = circumference / diameter. What else do you need, and why?
Dear Scholar
Any truth must have the support from many angles.
Validity & reliability is a must.
2. 3.141....of polygon is transcendental .
3. It is NOT Pi number.
4.Hence Pi is NOT a transcendental number
5. If you insist I must accept 3.141....of polygon as PI OF THE CIRCLE I INSIST show me at least a 2nd geometrical method.
6.One geometrical method is INSUFFICIENT
"show me at least a 2nd geometrical method." -- Sorry, I still don't get it: What shall that second method do, precisely?
Dear Scholar
For that reason I comfortable argue Exhaustion Method is NOT NOT NOT enough to say 3.141.... of polygon to ATTRIBUTE it as Pi of the circle
Thank you, dear Sarva Jagannadha Reddy, for this explanation; now I understand better. But I need one more explanation, please: What do you mean by "Exhaustion Method" -- using only inscribed polygons, or using both inscribed and circumscribed polygons?
Dear Scholar
The Exhaustion Method (EM ) is that of Eudoxus of Cnidus of 410 BC.
Archimedes of Syracuse of 240 BC had made much more scientific.
With the DOUBLING of sides successively of both inscribed and Circumscribed polygon in and about circle THE GAP BETWEEN CIRCLE AND BOTH TYPES OF POLYGONS EXHAUSTS . Hence this method is called EM.
Theoretically it looks right
But polygon is angular in nature and circle is a curve
It means EXHAUSTION never takes place.
So, do I understand it right that you are speaking of a geometrical method which allows to derive the exact value of pi?
And do I understand it right that you disagree with the polygon exhaustion method to derive the exact value of pi because exhaustion is never complete, and because a polygon always remains a polygon and never becomes a curved line (as for the circle)?
There is a reason why I have emphasized the condition "to derive the exact value of pi": Within our discussion I can accept your disagreement with the polygon approach to derive the exact value of pi; in fact, within our discussion I don't care about the exact true value for pi at all. This is so because I only care about the Cosmic Pi value.
Now, the point here is that my question about this specific Cosmic Pi value is whether it can be correct or whether it cannot be correct -- and this "can or cannot" is important, since I'm not interested in the question whether Cosmic Pi is correct or not.
And this is so because the question about can/cannot makes life much easier: If it is found that Cosmic Pi cannot be correct, I don't need to deal with the question anymore whether it is correct or not.
You see, my dear Sarva Jagannadha Reddy, I'm not here to defend "standard pi" = 3.1415..., but to put Cosmic Pi to the test. Therefore, there is no need for a second geometrical method as you requested.
Dear Scholar
With your approach you will never understand the truth about Cosmic Pi
"With your approach you will never understand the truth about Cosmic Pi" -- So, what do you think is my approach, please? And why is that approach of mine a hindrance to understand Cosmic Pi?
Let's look at it differently: Why should you care about my testing approach? You don't have to be afraid of any such test if Cosmic Pi is the truth, don't you? If Cosmic Pi is the truth it would pass any test.
Dear Scholar
Cosmic Pi is NOT NOT NOT mine
184+ methods have tested the presence of Cosmic Pi.
You can DISPROVE IT ( But do not go for Circumscribed polygon)
A doctor can test a patient using Stethoscope putting it on thorax ,dorsal & ventral. and not on thigh.
Choosing circumscribed polygon is like putting stethoscope on thigh.
"You can DISPROVE IT ( But do not go for Circumscribed polygon)" -- Why not?
"Choosing circumscribed polygon is like putting stethoscope on thigh." -- Why?
Dear Scholar
I AGREE for your satisfaction that present pi number 3.141.... is RIGHT RIGHT RIGHT
OK
Have a peace of mind
"184+ methods have tested the presence of Cosmic Pi." -- No, these aren't tests for pi, because none of them is truely related to a circle. Instead, they are geometrical constructions that, interestingly, all involve the value (14 – sqrt2)/4.
This means that, between all these geometrical constructions, there is a connection which has nothing to do with a circle. So, the real truth about 3.1464... still has to be discovered.
Dear Scholar
You are mistaken
The presence of Cosmic Pi is tested this way.
Dear Sarva Jagannadha Reddy, without looking for the true connection between all these geometrical constructions you will never understand the real and full truth about (14 – sqrt2)/4.
Dear Scholar
How is it possible without link Cosmic Pi exists?
"How is it possible without link Cosmic Pi exists?" -- On the one hand, there exist many numbers in the universe that are close to 3.14 but don't have anything to do with the circle. On the other hand, the work of Gernot Hoffmann ("Circle a Square"; http://docs-hoffmann.de/circsqua22042005.pdf) shows that, nevertheless, (14 – sqrt2)/4 "is a good approximation and delivers the best solution so far".
However, "Cosmic Pi" is just an approximation. In fact, there seems to exist an algebra/calculus-based proof that not only is it impossible to represent pi by a fraction u/v of integers u, v (i.e., that pi is irrational), but it cannot even be represented by an expression of the type [u ± sqrt(w)]/v with integers u, v, w; have a look at the end (around 17'30") of the following recent Mathologer video: https://www.youtube.com/watch?v=jGZtVl4XfGo
Dear Scholar
Prof.Gernot Hofmann gave his own proof
"Prof.Gernot Hofmann gave his own proof" -- Sorry: proof of what, please?
Dear Scholar
He gave proof for EXACT CIRCLING A SQUARE WITH COSMIC PI
No, he didn't, because his construction is not exact. And this is not a surprise: As I told you already, he wrote so himself ("Neither for ‘circling a square’ nor for ‘squaring a circle’ exists an accurate solution by ruler and compass.").
Gernot Hoffmann just used a geometrical construction of his own, but he clearly stated that the true radius of the circle (of exact equal area as the square) is different from the one obtained using "Cosmic Pi", resulting in the small mismatch of Square-Area/Circle-Area = 1.001545, which is equivalent to an accuracy of 0.15% for the approximation using "Cosmic Pi".
"How is it possible without link [to the circle that] Cosmic Pi exists?" -- Dear Sarva Jagannadha Reddy, without looking for the true connection between all your geometrical constructions you will never understand the real and full truth about (14 – sqrt2)/4.
That "Cosmic Pi" is not linked to the circle follows from an algebra/calculus-based proof that not only pi is irrational, but it cannot even be represented by an expression of the type [u ± sqrt(w)]/v with integers u, v, w; have a look at the end (around 17'30") of the following recent Mathologer video: https://www.youtube.com/watch?v=jGZtVl4XfGo
Dear Scholar
Here we BASICALLY differ
You are conditioned by your University education . They ( that education) spontaneously enter your mind and agitate you which ultimately confuses you.
In my case I am a NON- MATHEMATICIAN and hence no class room education influenced me.
This is the reason I could study Cosmic Pi from 1998 non-stop , day and night. And I have a CLEAR picture of Cosmic Pi
If I were a student of Mathematics I WOULD NOT HAVE THOUGHT OF PI CONSTANT. This was the answer I gave in the past when many mathematics professors said seeing Cosmic Pi that I would have done great work in other fields of mathematics if I were a mathematician.
"Here we BASICALLY differ" -- Which is just a fact; it's neither good nor bad.
"You are conditioned by your University education. They (that education) spontaneously enter your mind and agitate you which ultimately confuses you." -- Here, I honestly disagree! Yes, I passed the university successfully (diploma and Ph.D. in physics), but that doesn't mean that I'm "conditioned" in that way; how can you say that without having looked into my mind? Yes, somtimes pre-knowledge acts as a hindrance, but since I'm aware of that effect I've less and less fallen for it.
"In my case I am a NON-MATHEMATICIAN and hence no class room education influenced me." -- This is OK; it just means we have to be patient with each other.
"This is the reason I could study Cosmic Pi from 1998 non-stop, day and night. And I have a CLEAR picture of Cosmic Pi." -- Unfortunately, your picture, although it is clear, will most likely be limited to the part of the matter which is accessible to you as a non-mathematician. However, I think I'm able to see more in it, since at least partially, I'm a mathematician. Moreover, I guess that, unfortunately, your picture of "standard pi" is not as clear as it is for Cosmic Pi -- what do you think?
"If I were a student of Mathematics I WOULD NOT HAVE THOUGHT OF PI CONSTANT." -- Well, but now you have to study mathematics, at least a little bit, because you stepped out from your private room into the mathematical world and started to tell big stories about pi. And of course you're not the only person in the world who has thought deeply about pi!
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