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3.14 » History and Philosophy of Pi
Pi (
Pi or 3.1415) = 3.1415926535...  Dictionary.com Definition
currently the value of Pi is known to 6.4 billion places
Pi 3.14 math history and philosophy

Probably no symbol in mathematics has evoked as much mystery,
romanticism, misconception and human interest as the number Pi (
Pi or 3.1415)

Socrates  Socrates, Plato, Aristotle  Plato  Socrates, Plato, Aristotle  Aristotle  Socrates, Plato, Aristotle  Pi 3.14  Socrates, Plato, Aristotle  Atlantis  Socrates, Plato, Aristotle  Fascinating Facts
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OTGPi (noun) ... plural = pis.

Mathematics.  A transcendental number, approximately 3.14159, represented by the symbol Pi or 3.1415, that expresses the ratio of the circumference to the diameter of a circle and appears as a constant in many mathematical expressions.

The 16th letter of the Greek alphabet.

:  BEYOND INFINITY

Deep in the nature of man is the will to go further than any human has ever been before.  This quest is symbolized by the Greek letter Pi or 3.1415, which evokes infinity.  Humans are still in pursuit of the end of its innumerable string of decimals...

Book:  The Joy of Pi by David Blatner: $14.40 from Amazon.comNo number (3.14...) has captured the attention and imagination of number fanatics and nerds throughout the ages as much as the ratio of a circle's circumference to its diameter - a.k.a. Pi.  With incisive historical insight and a refreshing sense of humor, this page brings us the story of Pi Pi or 3.1415 and humankind's fascination with it, from Archimedes to da Vinci to the modern day Chudnovsky brothers, who holed up in their Manhattan apartment with a homemade supercomputer churning out digits of pi into the billions.

Breezy narratives tell the history of Pi Pi or 3.1415 and the quirky stories of those obsessed with it; sidebars recount fascinating Pi trivia; dozens of snippets and factoids reveal Pi's many fascinating facets of Mother Nature's numeric perfection combining chaos and order into apparent provable scientific perfection.

Pi 3.14 math history and philosophy

Interesting Pi Facts

bulletYou can determine your hat size by measuring the circumference of your head, then divide by Pi Pi or 3.1415, and round off to the nearest one-eighth inch.
bulletThe height of an elephant (from foot to shoulder) = 2 x Pi x the diameter of its foot.
bulletIt is more accurate to say that a circle has an infinite number of corners than it is to say it has no corners.
bulletOne of the more accurate fractions for pi is 104348/33215.  It is accurate to 0.00000001056%.
bulletThe Babylonians, in 2000 B.C.E., were the first people known to find a value for Pi.
bulletThe Bible uses the value of 3 for Pi.  This verse comes from 1 Kings 7:23:   "And he made a molten sea, ten cubits from brim to brim:  it was round all about, and its height was five cubits:  and a line of thirty cubits did compass about it."
bulletIf you were to type one billion decimals of Pi, they would stretch from New York City to the middle of Kansas.
bulletPeople once thought that trying to square the circle was an illness called Morbus Cyclometricus.
bulletTo calculate the circumference of the known universe, you would only have to use 39 decimals of Pi and be off by one proton.
bulletHalf of the circumference of a circle with a diameter of 2 is Pi.  The area inside the circle is also Pi Pi or 3.1415.
bulletThe most accurate version of Pi was by Dr. Kanada of the University of Tokyo calculated the value of Pi to 206,158,430,000 places in September 1999, surpassing the previous record by more than 150 billion digits.
bulletThe most inaccurate version of Pi In 1897, the General Assembly of Indiana enacted Bill No. 246, stating that Pi was just plain 4.
bulletMemorizing Pi On February 18, 1995, Hiroyuki Goto of Tokyo, Japan recited Pi Pi or 3.1415 to 42,195 places at the NHK Broadcasting Center, Tokyo.
bulletPi and Atmospheric Pressure ... Jonathan Bradshaw points out that standard atmospheric pressure is defined to be P= 0.101325 MPa (this is a human-defined value, which is approximately the average pressure at sea level.)  Curiously, if you take the square root of this number and then divide 1 by the result (the reciprocal of the value), you get 3.14153.

Pi 3.14 math history and philosophy

Fermat's Enigma : The Epic Quest to Solve the World's Greatest Mathematical Problem $9.60 at Amazon.comPi and the Length of Rivers
From Fermat's Enigma, by Simon Singh

FYI .. i read this book and it is friggin' awesome!!!

"Professor Hans-Henrik Stolum, an earth scientist at Cambridge University has calculated the ratio between the actual length of rivers from source to mouth and their direct length as the crow flies.  Although the ratio varies from river to river, the average value is slightly greater than 3, that is to say that the actual length is roughly three times greater than the direct distance. In fact the ratio is approximately 3.14, which is close to the value of the number Pi Pi or 3.1415.

The ratio of Pi Pi or 3.1415 is most commonly found for rivers flowing across very gently sloping planes, such as those found in Brazil or the Siberian tundra."

Pi 3.14 math history and philosophy

Raphael: The School of AthensThe Story of Pi
by Lazarus Mudehwe

Undoubtedly, Pi Pi or 3.1415 is one of the most famous and most remarkable numbers you have ever met.  The number, which is the ratio of circumference of a circle to its diameter, has a long story about its value.

Even nowadays supercomputers are used to try and find its decimal expansion to as many places as possible.  For Pi is one of those numbers that cannot be evaluated exactly as a decimal --- it is in that class of numbers called irrationals.

The hunt for Pi began in Egypt and in Babylon about two thousand years before Christ.  The Egyptians obtained the value (4/3)^4 and the Babylonians the value 3 1/8 for Pi Pi or 3.1415.  About the same time, the Indians used the square root of 10 for Pi.  These approximations to Pi had an error only as from the second decimal place.

	(4/3)^4	   =	3,160493827...
	3 1/8	   =	3.125
	root 10	   =	3,16227766...
	Pi	   =	3,1415926535...

Philosophy of Pi 3.14The next indication of the value of Pi occurs in the Bible.  It is found in 1 Kings chapter 7 verse 23, where using the Authorized Version, it is written "... and he made a molten sea, ten cubits from one brim to the other: it was round about ... and a line of thirty cubits did compass it round about."  Thus their value of Pi was approximately 3.  Even though this is not as accurate as values obtained by the Egyptians, Babylonians and Indians, it was good enough for measurements needed at that time.

Jewish rabbinical tradition asserts that there is a much more accurate approximation for Pi Pi or 3.1415 hidden in the original Hebrew text of the said verse and 2 Chronicles 4:2.  In English, the word 'round' is used in both verses.  But in the original Hebrew, the words meaning 'round' are different. Now, in Hebrew, etters of the alphabet represent numbers.  Thus the two words represent two numbers. And - wait for this - the ratio of the two numbers represents a very accurate continued fraction representation of Pi! Question is, is that a coincidence or ...

Another major step towards a more accurate value of Pi Pi or 3.1415 was taken when the great Archimedes put his mind to the problem about 250 years before Christ.  He developed a method (using inscribed and circumscribed 6-, 12-, 48-, 96-gons) for calculating better and better approximations to the value of Pi, and found that 3 10/71 < Pi < 3 10/70.  

Today we often use the latter value 22/7 for work which does not require great accuracy.  We use it so often that some people think it is the exact value of Pi!

As time went on other people were able come up with better approximations for Pi. About 150 AD, Ptolemy of Alexandria (Egypt) gave its value as 377/120 and in about 500 AD the Chinese Tsu Ch'ung-Chi gave the value as 355/113.  These are correct to 3 and 6 decimal places respectively.

	377/120	       =	3,14166667...
	22/7	       =	3,142857143...
	355/113	       =	3,14159292...
	Pi	       = 	3,1415926535...

It took a long time to prove that it was futile to search for an exact value of Pi, ie to show that it was irrational.  This was proved by Lambert in 1761.  In 1882, Lindemann proved that Pi Pi or 3.1415 was more than irrational --- it was also transcendental --- that is, it is not the solution of any polynomial equation with integral coefficients.  This has a number of consequences:
bulletIt is not possible to square a circle. In other words, it is not possible to draw (with straight edge, compass and pencil only) a square exactly equal in area to a given circle.  This problem was set by the Greeks two thousand years ago and was only put to rest with Lindemann's discovery.
bulletIt's not possible to represent Pi Pi or 3.1415 as an exact expression in surds, like root2, root7 or root5+root3/root7, etc.

History of Pi - amazon.com book $9.56From that time on interest in the value of Pi has centered on finding the value to as many places as possible and on finding expressions for Pi and its approximations, such as these found by the Indian mathematician Ramanujan:

	(1 + (root3)/5)*7/3	 	=	3.14162371...
	(81 + (19^2)/22)^(1/4)   	=	3.141592653...
	63(17+15root5)/25(7+15root5) 	=	3.141592654...
	Pi		 	=	3.141592654...

The last approximation is so good (9dp) that my ancient Casio calculator tells me it's the same as Pi! (Sadly, many people would believe my calculator).

Finding info on the web is one of the easiest tasks in existence.  Steve Berlin has a nice article, and this site offers software that can be used to get Pi to plenty decimal places.  Want to change the value of Pi Pi or 3.1415?  Sorry, the voting is over, but the results are here.  I could go on and on, but instead I'll just leave you with The Albany Pi Club which has several links, including the brilliant Uselessness of Pi page and the recently started Joy of Pi page.

Pi 3.14 math history and philosophy

Ancient Pi (Pi):  Knowers of the Universe
by Charles William Johnson

Any practical attempt to divide the diameter of a circle into its own circumference can only meet with failure.  Such a procedure is entirely theoretical in nature.  Dividing unlikes, a straight line (the diameter of a circle) into a curved line (the circumference of a circle) can only be met with frustration.  

The kind of frustration that is portrayed throughout history in humankind's attempt to measure the incommensurable.  No matter how hard one may try, even with the assistance of contemporary electronic computers, bending either the straight line or the curved line, alters the nature of the problem and yields an impossibility.  As soon as one of the lines is bent the results are tainted.  

Then, there is the question of the very thickness of the lines being measured in length.  Whether one measures the inner part of the curved line of the circumference or the outer edge makes a great deal of difference; especially, when one is attempting to achieve an exactness in the concept of Pi () to hundreds or even thousands of decimal places.

If we realize that the measurement of the ratio between the diameter and the circumference of a circle is entirely theoretical and speculative, then we may also realize that the result shall always represent an approximation.  In fact, the very fact that Pi is always expressed in terms of an unending fraction (with mathematicians searching it to the nth number of decimal places), should cause us to accept the idea that Pi can only be an approximation. (As Lambert illustrated in 1767, " Pi or 3.1415 is not a rational number, i.e., it cannot be expressed as a ratio of two integers"; Beckmann, p.100.)

Once we realize that Pi Pi or 3.1415 represents a fractional expression in numbers, it were as though either nature itself were wrong, or the numbers must surely be able to be manipulated to render whole numbers.  The ancients sought to work with whole numbers.  However, once we realize that the ancient reckoning system may have been based upon the concept of a floating decimal place, then we should understand that all numbers, in fact, may be visualized as whole numbers. The cut-off point becomes one of arbitrary choice at times.  

What is Pi? With regard to the concept of Pi Pi or 3.1415, contemporary mathematicians have not decided to accept that arbitrary cut-off point, and continue to search for the unending decimal expression of Pi.  

At one time, not too long ago, Pi was simply represented to be 3.1416 Pi or 3.1415; and, in a practical sense, it served all purposes of constructing things out of matter and energy.  

Today, the unending expression of Pi to hundreds of thousands of decimal places serves no practical purpose that we know of, at least, other than that of an unending contest to discover the ultimate expression of Pi.  One has only to admire the relation of the diameter of any circle to its circumference to note that particular expression.

Throughout history, the expression of Pi has taken on many variations. Petr Beckmann (Cfr., A History of (Pi), Golem, 1971), offers an exemplary analysis of the concept throughout history. The Babylonians 3 1/8; the Egyptians 4(8/9) ²; Siddhantas, 3.1416; Brahmagupta, 3.162277; Chinese, 3.1724; Liu Hui, 3.141024 < < 3.142704; Liu Hui, 3.14159; Tsu Chung-Chih, 3.1415926 < < 3.1415927; Archimedes, 3.14084 < < 3.142858 (3 1/7); Heron, 3.1738; Ptolemy, 3.14167; Fibonacci, = 864:275 = 3.141818; Vičte, 3.141592635 < < 3.1415926537; and, finally in the computer language of FORTAN: 3.14159265358979324. 

Again, citing Beckmann (p.101):  "There is no practical or scientific value in knowing more than the 17 decimal places used in the foregoing, already somewhat artificial, application".

Nonetheless, in 1844, Johann Martin Zacharias Dase calculated Pi or 3.1415 to 200 decimal places, with the first zero appearing at the 32nd decimal place ---meaning, possibly that the exercise should have ended there. It has not; just as Pi is an unending fraction, so is the human practice of finding the number of unending decimal places in Pi.

Decimal hunting games aside, the practical uses of knowing Pi (the ratio of the diameter of a circle to its circumference) even as an approximation has infinite applications in astronomy. And, the ancients were on the whole astronomers; knowers of the universe.  This ratio becomes significant in calculating the movements of the planets and the stars; in computing their coming and going in the sky.  Once more, since we are dealing with movement, the movement of the planetary bodies and the stars, we are always speaking about approximations; even in and especially so in astronomy.  Therefore, the approximations to Pi Pi or 3.1415 serve a purpose in knowing the approximate movements of the planets.  Such are the problems concerning the measurement of moving bodies. As soon as they have been measured, they have already moved from that measurement.

When we observe the measurements offered by Tsu Chung-Chih given above, it becomes obvious that ancient approximations were at times far ahead of latter day computations.  And, then there is the problem that one may obtain Pi to the nth decimal place, but such decimal expressions are beyond the human capacity to measure or even observe matter-energy to such a minute degree.

Smells like Pi to MeIn our analyses, we cannot cite any specific ancient documents for the computation of Pi Pi or 3.1415 among the ancients.  Yet, the historically significant numbers that do exist within the ancient reckoning systems may reveal some partial aspects of the computations themselves.  No matter which contemporary studies we examine, Pi is always given in relation to the number ca. 3.1-something, as a guidepost. Yet, it may be the case that the ancients conceived of Pi in relation to the number of divisions that made up the circle; the number of degrees or segments contained therein.

The concept of Pi refers to the constant ratio of the diameter: circumference of any circle; irrespective of the number of degrees contained within that circle.  Historically, the Babylonians came to use the number 360 for the divisional segments within a circle, and we have employed that same number ever since.  The abstracted universal circle, then, would have a constant diameter of one (1.0), and the length of its circumference would be Pi (Pi or 3.1415) of that: 3.1-something (whichever one might choose). Hence, diameter is 1.0 in length; while, circumference is 3.141592654 (for example) in length.

Now, if we consider the circumference to be divided into 360 degrees (or segments; angular divisions with lines cutting through the center of the circle as we know them), then using the contemporary figure for Pi (3.141592654), the length of the circumference could be 360 units, while the length of the diameter would be 114.591559 (i.e., 360 /Pi or 3.1415).

Now, let us suppose that the circle is divided into 260 degrees (something that we are unaccustomed to considering, in fact). If we employ the same length of the diameter of the previous example (114.591559), then the relational figure for Pi for a 260-degree circle would be: 2.268928028.  With that something very intriguing develops.  Within ancient Nineveh, there exists an historically significant cited as 2268.  

One could imagine that the 2268 fractal number may relate to the concept of proportion (i.e., Pi Pi or 3.1415) regarding a 260-division circle.  The number 260 is relevant because during ancient times there existed in various cultures a calendar based on a 260c day-count.  Furthermore, the Great Cycle of the sun, known as Precession, also involves a fractal of 260 (i.e., 26000 years). Now, were we to consider the Nineveh number for representing Pi Pi or 3.1415 on a 260-degree circle, then the constant value for the diameter would then be 114.638448 (i.e., 260 / 2.268).

Throughout history, an inexact representation of Pi has always been cited as that of 3 1/7 (or, 3.142857); a reciprocal of seven number.  However, when we consider that the length of the diameter of a 360-degree circle yields a number that approximates a reciprocal of seven number (114.591559), we can consider the possibility of employing 114.285714 in its place.

The use of the reciprocal of seven number (114.284714) for the length of the 360 and 260 circle would offer the following values for Pi, respectively:
.

360 / 114.285714 = 3.150000008 (Pi proportion for 360c circle)
260 / 114.285714 = 2.275000006 (Pi proportion for 260c circle)

Note that the 3.15 number offers a mediatio/duplatio series based on the 63c, which was significant in ancient reckoning systems:  315, 630, 1260, 2520, 5040, 10080, 20160, 40320, 80640, 161280, 322560, 645120,1290240, 2580480 (a Precession number/fractal); and, 63, 126, 189, 252, 315, 378, 441, 504, 567 (kemi), 630, 693 (Sothic), 756 (Giza), 819 (k'awil; maya), 882, 945, 1008, 1071, 1134 (Nineveh, 2 x 1134 = 2268), 1197, etc.

Note that the 2275 fractal number is relevant for the computational series within the ancient reckoning system of the 364c day-count:  2275, 4550, 9100, 18200, 36400, etc. Also, note that the difference between the Nineveh 2268c and the Pi-like number 2275 is seven (2275 - 2268 = 7); which could be easily translated from one series to the other by remainder math based on multiples of seven.

Many of the distinctive historically significant numbers of the ancient reckoning system reflect a relationship based on the reciprocal of seven.  Consider the maya long count period number of 1872000, which has received so much speculation regarding its beginning and ending date.  Also, consider the period called the k'awil of the maya cited as consisting of 819c days.

Now, notice the number that obtains from the division resulting from half of the long count period figure by the k'awil: 936 / 819 = 1.142857143.  The same figure obtains regarding the constant length of a diameter of a circle based on a Pi-like number in relation to the reciprocal of seven as explained earlier.

Other relationships obtain regarding similar historically significant numbers from other systems.  The Great Pyramid entails the number 756c as its baseline.  Also, there exists the 432c number/fractal associated with the Consecration.  If we double the 432 figure and divide by the 756c, the same result obtains: 864 / 756 = 1.142857143.

Consider: 360 x .864 = 311.04 (31104 being an historically significant number for China and Mesoamerica).

The significance of seven and its reciprocal becomes obvious throughout the historically significant numbers/fractals.  Even the obvious relationship, of the 364c day-count of ancient Mesoamerica, which was employed for computations, reveals a direct basis of seven: 364 / 7 = 52. Immediately, one will recognize the 52c that is so well-known in ancient Mesoamerica as the calendar round (52 years times 365 days = 18980 days; and 52 years times 360 days = 18720! days).  And, the ancient kemi appear to have employed a 54c in its place: 7 times 54 = 378 (2 x 378 = 756; or, 7 x 108 = 756).  No matter where one turns, the number seven and its reciprocal make their appearance. The reasoning behind this procedure may be rather obvious, although we have not discerned it previously.

The number 1.142857143 concerns the ratio 8/7ths.  The Aztec Calendar appears to be based upon a spatial division that reflects the logic of 7:8 or 8:7, depending upon the rings and segments to be considered (Cfr., Earth/matriX No.88). If one were attempting to consider the diameter of the Solar System, or the Universe, knowing that these events consist of imaginary circles (ellipses), then the use of the unit 1.0 for the length of their respective diameters would not be of much value.  And, furthermore, if the ancients had employed the contemporary (and possibly past) concept of Pi (based on a close approximation to 3.141592654, give or take a fraction), then the numbers would have been unmanageable and not very attractive.

The apparent relational aspects of the many different historical numbers found in the many distinctive ancient reckoning systems suggest a common origin and reasoning. If the length of the diameter of the solar system or the Universe were assigned a value consisting of the reciprocal of seven (i.e., 1.142857143), then this would be the next best thing to working with whole numbers for computing the time cycles of the movement of the planetary bodies and the stars. 

Furthermore, knowing the actual measurement of Pi Pi or 3.1415 (the exact proportion of the diameter: circumference ratio) could be compensated with remainder math adjustments quite easily. Consider the following computations:

 

1.142857 x 819 = 935.999883 (936) (maya long count fractal)
1.285714 x 819 = 1052.999766 (1053)
1.428571 x 819 = 1169.999649 (1170) (Venus sidereal count)
1.571428 x 819 = 1286.999532 (1287)
1.714285 x 819 = 1403.999415 (1404) (kemi count; 351c)
1.857142 x 819 = 1520.000298 (1521) (39²)
1.142857 x 315 = 359.999955 (360) (360c; kemi; maya)
1.285714 x 315 = 404.99991 (405) (1296000c; kemi)
1.428571 x 315 = 449.999865 (450) (maya long count; 9 base system)
1.571428 x 315 = 494.99982 (495) (99c lunar count)
1.714285 x 315 = 539.999775 (540) (kemi count)
1.857142 x 315 = 584.99973 (585) (Venus synodic count)

One of the most interesting relationships of this nature concerns the 2268c Nineveh count:
.

1.142857 x 2268 = 2591.999676 (2592) (Platonic Year, 25920 years)

Scholars consider the figure of 3 1/7ths to have been an erroneous computation for Pi.  Yet, we have never really known how the ancients computed their mathematics.  The few documents that remain (such as the Rhind document of the ancient kemi) concern everyday matters; not the mathematics and geometry of the study of the Universe.  

By employing the reciprocal of seven in the computations, which is what an initial analysis of the historically significant numbers reveals, the ancients may have been seeking an easier method for arriving at their knowledge of the Universe than what is offered by the precise unending fractional expression of Pi Pi or 3.1415, the proportion of the diameter to the circumference of a circle.  This may be further understood when we realize that the comings and goings of the planetary bodies and the stars throughout the Universe do not travel on perfect Pi-like circles.

The ancients may have employed distinct constant fractals/numbers for adjustments in their computations: the length of the diameter may have been based on 114.2857, 114.591559, 114.638448; etc; the distance of the circumference ay have been related to the 260c, 360c, 378c, 936c, etc.; and, the Pi ratio (proportion) of the diameter: circumference may have been 2.268, 3.15, 3.1416, 3.142857, 819, etc. 

The distinctive historically significant numbers reflect different aspects of the computations and their corresponding adjustments.  From this dynamic perspective, the historically significant numbers may be communicating to us a much more precise knowledge of astronomy and mathematical and geometrical computations than we have been willing to concede to the ancients.

Pi 3.14 math history and philosophy

Useless Facts about Pi

Euler's Relation

This proof is, of course, due to Euler. 

exp(z)=sum(0->infinity)z^n/n!
z=iy then exp(iy)=sum(0->infinity)(iy)^n/n!
=sum(0->infinity)[(iy)^(2n)/(2n)!+(iy)^(2n+1)/(2n+1)!]
=sum(0->infinity)(-1)^n[y^(2n)/(2n)!+iy^(2n+1)/(2n+1)!]
which, because they have identical power series, is
cos(y)+i*sin(y)
setting y=pi we have
exp(i*pi)=cos(pi)+isin(pi)=-1

Irrationality

This is Niven's proof. 

Suppose pi is rational; then there exists a,b natural numbers pi = a/b.
Let f be in Q[x] f=x^n(ax-b)^n/n! (pi = a/b, remember?).
Suppose we define g in Q[x] as sum(0->n)(-1)^n(d/dx)^(2n)f.
It should be obvious that d^2g/(dx)^2 is f-g.
Getting trickier, we see that
(d/dx)[(dg/dx)sin(x)-g*cos(x)] is [d^2g/(dx)^2+g]sin(x) = f*sin(x).
Integrating f*sin(x) from zero to pi with respect to x yields g(0)+g(pi).

At zero and pi, the first n derivatives of f vanish.  But then the coefficients become integers by an easy result on products of n consecutive integers being divisible by n!. So our integral is an integer.  But the maximum of f on (0,pi) is (a*pi)^n/n! and the max of sin(x) is 1.  Both f and sin are bounded by zero from below.  Thus the integral is bounded strictly between zero and an arbitratily small number, resulting in a contradiction. Therefore pi cannot be rational.

Transcendence

Niven again. This is considerably easier than Lindemann's  Suppose pi be algebraic.  Then, so would be i*pi. Let S be a set containing all the conjugates of i*pi in its minimal polynomial over Q.  By Euler's relation, the product(t in S)[exp(t)+1]=0.  Let 1 + sum(q in some set T)exp(q) be equal to that product by expanding it out.  T is then just the set of all the sums over the subsets of S.

There is a polynomial the set of whose roots is T, and it is in Q[x]. It is obviously product(q in T)(x-q). It has rational coefficients because a permutation of the roots merely causes permutation of the various sums of the roots which comprise T, making our candidate polynomial symmetric, and thus in Q[x] (it's fixed under those automorphisms, so it must be).  We can easily make a polynomial in Z[x] out of this, and I call it g.  (Divide by the lead coef to get the old one back.) In the spirit of the first proof, let's define h = a^s*x^(p-1)*g^p/(p-1)! for some p, with a the leading coef of g, and

s = degree of g * p - 1. Again like the first proof we define
f = sum(0->s+p)(d/dx)^n h
Since deg f = s+p, (d/dx)[exp(-x)f]=-exp(-x)h
Now integrating from zero to z, we have exp(-z)f(z)-f(0) = I, or
f(z)-exp(z)f(0) = exp(z)I.
Letting z assume values in T and summing we have
sum(q in T)[f(q)-exp(q)f(0)] = sum(q in T)f(q) + f(0)
recalling one of the aforementioned products.

The sum of the f(q) is an integer, as all derivatives in the defining sum of order less than p vanish, and the remaining terms have the product of enough integers from differentiation to make them integral, canceling the (p-1)!.

Furthermore, the expression is symmetric under permutation of the elements of T (or for that matter, S) so it is in Z.
f(0) is obviously in Z, for until the terms in the sum defining it have been differentiated enough times, they vanish, and once they no longer vanish, they have been differentiated enough to cancel out their denominator. That would make our integral integral. (Nice pun, eh?) We have 

sum(q in T)[exp(q)integral(0 to q)exp(-z)f(z)dz]
supposing B bounds |a*z*g| in the disk |z| less than max(q in T)|q| and C be the greatest bound on the |exp(q-z)g| (q in T) in that same disk, the terms beneath the integral signs in the sum are all bounded by a^(m-1)*C*B^(p-1)/(p-1)!, which may be made arbitrarily small for p sufficiently large. Thus the integral in question cannot be an integer, we have a contradiction, and thus pi cannot be algebraic. Therefore it is transcendental.
(psst! think about what might happen if p were prime)

Quadrature of the circle with ruler and compass

This is a consequence of the transcendence of pi with primarily historical importance.  It is known as one of the three Greek problems.  The other two were doubling the cube and trisecting the angle (for arbitrary angles).  They are disposed of below as kind of a package deal.

Straightedge and compass constructions amount to solving at best quadratic polynomials.  Reduction of order establishes this for the intersection of two circles thus drawn, and the other cases are trivial.  Since such constructions amount to solving polynomials over Q, and pi Pi or 3.1415 is transcendental, pi cannot be constructed.  A square with area equal to a circle requires the construction of the length sqrt(pi), also an impossibility given the above.

Delving into Galois theory (not even very deeply), one may determine further the nature of constructible numbers. In particular, one finds that the only constructible nth roots of unity must have n divisible only by two and Fermat primes. This has obvious implications regarding the construction of regular polygons.  The preceding observations imply that all constructible numbers must have degree a power of two, and the degree of a minimal polynomial for a root of unity is phi(n) where phi is the number of relatively prime natural numbers less than n, so one might simply observe this from the properties of phi. Namely,

phi(n)/n = prod(p|n)(1-1/p) (in Q!)so that
phi[2^k*prod(p|n,p!=2)p] = 2^(k-1)prod(p|n,p!=2)(p-1)

And, of course, in the case of Fermat primes, p-1 is a power of two.  One notes that in doubling the cube and trisecting arbitrary angles, that with the exception of a small number of particular angles, one is required to solve cubic equations, or equivalently, construct numbers with degree three over Q.

Mother Nature and Pi 3.14The Content of n-Spheres

Finding the content of an n-sphere is by definition equivalent to integrating the characteristic function of the underlying set, that is, the function which is one within the n-sphere and zero outside it.  This leads us to consider all radially symmetric functions.  Enough talk, let's get on with the math.
int((R+)^n)f(sum(1->n)x_k^a_k)prod(1->n)x_k^(b_k-1)dx
has a redundant set of parameters, but its form is useful.
=prod(1->n)(1/a_k)int((R+)^n)f(sum(1->n)y_k)prod(1->n)y_k^(b_k/a_k-1)dy
via the obvious y_k = x_k^a_k
=prod(1->n)(1/a_k)int((R+)^n)
	f(z_1)(z_1-sum(2->n)z_k)^(b_1/a_1-1)prod(2->n)z_k^(b_k/a_k-1)dz
by z1 = sum(1->n)y_k, z_k = y_k otherwise.
=prod(1->n)(1/a_k)(int(R+)f(w)w^((sum(1->n)b_k/a_k)-1)dw)
   * (int((R+)^(n-1))(1-sum(1->n-1)t_k)^(b_1/a_1-1)
	* prod(1->n-1)t_k^(b_(k+1)/a_(k+1)-1)dt)
which follows directly from the transformation w = z_1, w*t_k = z_(k+1) Letting f = exp, it becomes apparent that
int((R+)^(n-1))(1-sum(1->n-1)t_k)^(b_1/a_1-1)
   * prod(1->n-1)t_k^(b_(k+1)/a_(k+1)-1)dt
= (prod(1->n)gamma(b_k/a_k))/gamma(sum(1->n)b_k/a_k)
The case of the n-sphere is given by a_k = 2, b_k = 1, f = the characteristic function of (0,r^2) where r is the radius. The reduced integral is
int(R+)f(x)x^(n/2-1)dx = 2r^n/n
The constant in this instance evaluates to sqrt(pi)^n/gamma(n/2) Accounting for the restriction of the integration to (R+)^n, the content of an n-sphere is pi^(n/2)r^n/gamma(1+n/2). By differentiation one obtains the expression for the content of the surface: 2pi^(n/2)r^(n-1)/gamma(n/2)

Thus Pi or 3.1415's properties extend to n-dimensional geometry as well.

Pi 3.14 math history and philosophy

A History of Pi

A little known verse of the Bible reads
And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I Kings 7, 23)
The same verse can be found in II Chronicles 4, 2. It occurs in a list of specifications for the great temple of Solomon, built around 950 BC and its interest here is that it gives Pi or 3.1415 = 3. Not a very accurate value of course and not even very accurate in its day, for the Egyptian and Mesopotamian values of 25/8 = 3.125 and sqrt10 = 3.162 have been traced to much earlier dates:  though in defense of Solomon's craftsmen it should be noted that the item being described seems to have been a very large brass casting, where a high degree of geometrical precision is neither possible nor necessary. 

There are some interpretations of this which lead to a much better value.

The fact that the ratio of the circumference to the diameter of a circle is constant has been known for so long that it is quite untraceable.  The earliest values of Pi or 3.1415 including the 'Biblical' value of 3, were almost certainly found by measurement. In the Egyptian Rhind Papyrus, which is dated about 1650 BC, there is good evidence for 4(8/9)2 = 3.16 as a value for Pi or 3.1415.

ArchimedesThe first theoretical calculation seems to have been carried out by Archimedes of Syracuse (287-212 BC). He obtained the approximation

223/71 < pi < 22/7.
Before giving an indication of his proof, notice that very considerable sophistication involved in the use of inequalities here.

Archimedes knew, what so many people to this day do not, that pi does not equal 22/7, and made no claim to have discovered the exact value. If we take his best estimate as the average of his two bounds we obtain 3.1418, an error of about 0.0002.Pi or 3.1415

Here is Archimedes' argument.

Consider a circle of radius 1, in which we inscribe a regular polygon of 3 cross 2n-1 sides, with semiperimeter bn, and ascribe a regular polygon of 3 cross 2n-1 sides, with semiperimeter an.

The diagram for the case n = 2 is on the right.

The effect of this procedure is to define an increasing sequence

b1, b2, b3, ...
and a decreasing sequence
a1, a2, a3, ...
such that both sequences have limit Pi or 3.1415.

Using trigonometrical notation, we see that the two semiperimeters are given by

an = K tan(pi/K), bn = K sin(pi/K),
where K = 3 cross 2n-1. Equally, we have
an+1 = 2K tan(pi/2K), bn+1 = 2K sin(pi/2K),
and it is not a difficult exercise in trigonometry to show that
(1) . . . (1/an + 1/bn) = 2/an+1

(2) . . . an+1bn = (bn+1)2.

Archimedes, starting from a1 = 3 tan(Pi or 3.1415/3) = 3sqrt3 and b1 = 3 sin(Pi or 3.1415/3) = 3sqrt3/2, calculated a2 using (1), then b2 using (2), then a3 using (1), then b3 using (2), and so on until he had calculated a6 and b6. His conclusion was that
b6 < pi < a6.
It is important to realize that the use of trigonometry here is unhistorical:  Archimedes did not have the advantage of an algebraic and trigonometrical notation and had to derive (1) and (2) by purely geometrical means. Moreover he did not even have the advantage of our decimal notation for numbers, so that the calculation of a6 and b6 from (1) and (2) was by no means a trivial task.  So it was a pretty stupendous feat both of imagination and of calculation and the wonder is not that he stopped with polygons of 96 sides, but that he went so far.

Pi 3.14 math history and philosophy

Archimedes' Constant

Much more scholarly expositions concerning are available, for example, Beckmann's book and Borwein & Borwein's book. Our treatment of this, the most famous of the transcendental constants, is necessarily incomplete.

The area enclosed by a circle of radius 1 is =3.1415926535... and its circumference is 2. How is it that the same mysterious appears in both formulas? We give the answer here along with a little story. Of course, appears in higher dimensional analogs (spherical volume and surface area) as well.

History of Pi - amazon.com book $9.56In the 3rd century B.C., Archimedes considered inscribed and circumscribed regular polygons of 96 sides and deduced that

The following recursion



(often called the Borchardt-Pfaff algorithm) essentially gives Archimedes' estimate on the fourth iteration. The Mathcad PLUS 6.0 file wayman.mcd discusses this procedure further.  Click here if you have 6.0 and don't know how to view web-based Mathcad files.

Another connection between geometry and arises in Buffon's needle problem. Suppose a needle of length 1 is thrown at random on a plane marked by parallel lines of distance 1 apart. What is the probability that the needle will land in a position which crosses a line? The answer is .

Here is a completely different probabilistic interpretation of . Suppose two integers are chosen at random. What is the probability that they are coprime, i.e., have no common factor exceeding 1? The answer is .

Archimedes' constant was proved to be irrational by Lambert in 1761 and transcendental by Lindemann in 1882. The first truly attractive formula for computing decimal digits of was found by Machin

The advantage of Machin's formula is that the second term converges very rapidly and the first is nice for decimal arithmetic. Using this, Machin became the first individual to correctly compute 100 digits of .

We skip over many years of history and mention only one recent algorithm. The Borwein quartically convergent algorithm is related to Ramanujan's work on elliptic integrals:




then decreases monotonically to 1/ and

I believe that this is the basis for Kanada's record-breaking evaluation of to over 200 billion digits.  Quintically convergent algorithms and, more generally, pth-order iterative algorithms for p>4, were discussed by Borwein & Borwein.

The End

The End
The End

Here are the first 10,000 Digits of Pi:

3.
1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959 5082953311 6861727855 8890750983 8175463746 4939319255 0604009277 0167113900 9848824012 8583616035 6370766010 4710181942 9555961989 4676783744 9448255379 7747268471 0404753464 6208046684 2590694912 9331367702 8989152104 7521620569 6602405803 8150193511 2533824300 3558764024 7496473263 9141992726 0426992279 6782354781 6360093417 2164121992 4586315030 2861829745 5570674983 8505494588 5869269956 9092721079 7509302955 3211653449 8720275596 0236480665 4991198818 3479775356 6369807426 5425278625 5181841757 4672890977 7727938000 8164706001 6145249192 1732172147 7235014144 1973568548 1613611573 5255213347 5741849468 4385233239 0739414333 4547762416 8625189835 6948556209 9219222184 2725502542 5688767179 0494601653 4668049886 2723279178 6085784383 8279679766 8145410095 3883786360 9506800642 2512520511 7392984896 0841284886 2694560424 1965285022 2106611863 0674427862 2039194945 0471237137 8696095636 4371917287 4677646575 7396241389 0865832645 9958133904 7802759009 9465764078 9512694683 9835259570 9825822620 5224894077 2671947826 8482601476 9909026401 3639443745 5305068203 4962524517 4939965143 1429809190 6592509372 2169646151 5709858387 4105978859 5977297549 8930161753 9284681382 6868386894 2774155991 8559252459 5395943104 9972524680 8459872736 4469584865 3836736222 6260991246 0805124388 4390451244 1365497627 8079771569 1435997700 1296160894 4169486855 5848406353 4220722258 2848864815 8456028506 0168427394 5226746767 8895252138 5225499546 6672782398 6456596116 3548862305 7745649803 5593634568 1743241125 1507606947 9451096596 0940252288 7971089314 5669136867 2287489405 6010150330 8617928680 9208747609 1782493858 9009714909 6759852613 6554978189 3129784821 6829989487 2265880485 7564014270 4775551323 7964145152 3746234364 5428584447 9526586782 1051141354 7357395231 1342716610 2135969536 2314429524 8493718711 0145765403 5902799344 0374200731 0578539062 1983874478 0847848968 3321445713 8687519435 0643021845 3191048481 0053706146 8067491927 8191197939 9520614196 6342875444 0643745123 7181921799 9839101591 9561814675 1426912397 4894090718 6494231961 5679452080 9514655022 5231603881 9301420937 6213785595 6638937787 0830390697 9207734672 2182562599 6615014215 0306803844 7734549202 6054146659 2520149744 2850732518 6660021324 3408819071 0486331734 6496514539 0579626856 1005508106 6587969981 6357473638 4052571459 1028970641 4011097120 6280439039 7595156771 5770042033 7869936007 2305587631 7635942187 3125147120 5329281918 2618612586 7321579198 4148488291 6447060957 5270695722 0917567116 7229109816 9091528017 3506712748 5832228718 3520935396 5725121083 5791513698 8209144421 0067510334 6711031412 6711136990 8658516398 3150197016 5151168517 1437657618 3515565088 4909989859 9823873455 2833163550 7647918535 8932261854 8963213293 3089857064 2046752590 7091548141 6549859461 6371802709 8199430992 4488957571 2828905923 2332609729 9712084433 5732654893 8239119325 9746366730 5836041428 1388303203 8249037589 8524374417 0291327656 1809377344 4030707469 2112019130 2033038019 7621101100 4492932151 6084244485 9637669838 9522868478 3123552658 2131449576 8572624334 4189303968 6426243410 7732269780 2807318915 4411010446 8232527162 0105265227 2111660396 6655730925 4711055785 3763466820 6531098965 2691862056 4769312570 5863566201 8558100729 3606598764 8611791045 3348850346 1136576867 5324944166 8039626579 7877185560 8455296541 2665408530 6143444318 5867697514 5661406800 7002378776 5913440171 2749470420 5622305389 9456131407 1127000407 8547332699 3908145466 4645880797 2708266830 6343285878 5698305235 8089330657 5740679545 7163775254 2021149557 6158140025 0126228594 1302164715 5097925923 0990796547 3761255176 5675135751 7829666454 7791745011 2996148903 0463994713 2962107340 4375189573 5961458901 9389713111 7904297828 5647503203 1986915140 2870808599 0480109412 1472213179 4764777262 2414254854 5403321571 8530614228 8137585043 0633217518 2979866223 7172159160 7716692547 4873898665 4949450114 6540628433 6639379003 9769265672 1463853067 3609657120 9180763832 7166416274 8888007869 2560290228 4721040317 2118608204 1900042296 6171196377 9213375751 1495950156 6049631862 9472654736 4252308177 0367515906 7350235072 8354056704 0386743513 6222247715 8915049530 9844489333 0963408780 7693259939 7805419341 4473774418 4263129860 8099888687 4132604721 5695162396 5864573021 6315981931 9516735381 2974167729 4786724229 2465436680 0980676928 2382806899 6400482435 4037014163 1496589794 0924323789 6907069779 4223625082 2168895738 3798623001 5937764716 5122893578 6015881617 5578297352 3344604281 5126272037 3431465319 7777416031 9906655418 7639792933 4419521541 3418994854 4473456738 3162499341 9131814809 2777710386 3877343177 2075456545 3220777092 1201905166 0962804909 2636019759 8828161332 3166636528 6193266863 3606273567 6303544776 2803504507 7723554710 5859548702 7908143562 4014517180 6246436267 9456127531 8134078330 3362542327 8394497538 2437205835 3114771199 2606381334 6776879695 9703098339 1307710987 0408591337 4641442822 7726346594 7047458784 7787201927 7152807317 6790770715 7213444730 6057007334 9243693113 8350493163 1284042512 1925651798 0694113528 0131470130 4781643788 5185290928 5452011658 3934196562 1349143415 9562586586 5570552690 4965209858 0338507224 2648293972 8584783163 0577775606 8887644624 8246857926 0395352773 4803048029 0058760758 2510474709 1643961362 6760449256 2742042083 2085661190 6254543372 1315359584 5068772460 2901618766 7952406163 4252257719 5429162991 9306455377 9914037340 4328752628 8896399587 9475729174 6426357455 2540790914 5135711136 9410911939 3251910760 2082520261 8798531887 7058429725 9167781314 9699009019 2116971737 2784768472 6860849003 3770242429 1651300500 5168323364 3503895170 2989392233 4517220138 1280696501 1784408745 1960121228 5993716231 3017114448 4640903890 6449544400 6198690754 8516026327 5052983491 8740786680 8818338510 2283345085 0486082503 9302133219 7155184306 3545500766 8282949304 1377655279 3975175461 3953984683 3936383047 4611996653 8581538420 5685338621 8672523340 2830871123 2827892125 0771262946 3229563989 8989358211 6745627010 2183564622 0134967151 8819097303 8119800497 3407239610 3685406643 1939509790 1906996395 5245300545 0580685501 9567302292 1913933918 5680344903 9820595510 0226353536 1920419947 4553859381 0234395544 9597783779 0237421617 2711172364 3435439478 2218185286 2408514006 6604433258 8856986705 4315470696 5747458550 3323233421 0730154594 0516553790 6866273337 9958511562 5784322988 2737231989 8757141595 7811196358 3300594087 3068121602 8764962867 4460477464 9159950549 7374256269 0104903778 1986835938 1465741268 0492564879 8556145372 3478673303 9046883834 3634655379 4986419270 5638729317 4872332083 7601123029 9113679386 2708943879 9362016295 1541337142 4892830722 0126901475 4668476535 7616477379 4675200490 7571555278 1965362132 3926406160 1363581559 0742202020 3187277605 2772190055 6148425551 8792530343 5139844253 2234157623 3610642506 3904975008 6562710953 5919465897 5141310348 2276930624 7435363256 9160781547 8181152843 6679570611 0861533150 4452127473 9245449454 2368288606 1340841486 3776700961 2071512491 4043027253 8607648236 3414334623 5189757664 5216413767 9690314950 1910857598 4423919862 9164219399 4907236234 6468441173 9403265918 4044378051 3338945257 4239950829 6591228508 5558215725 0310712570 1266830240 2929525220 1187267675 6220415420 5161841634 8475651699 9811614101 0029960783 8690929160 3028840026 9104140792 8862150784 2451670908 7000699282 1206604183 7180653556 7252532567 5328612910 4248776182 5829765157 9598470356 2226293486 0034158722 9805349896 5022629174 8788202734 2092222453 3985626476 6914905562 8425039127 5771028402 7998066365 8254889264 8802545661 0172967026 6407655904 2909945681 5065265305 3718294127 0336931378 5178609040 7086671149 6558343434 7693385781 7113864558 7367812301 4587687126 6034891390 9562009939 3610310291 6161528813 8437909904 2317473363 9480457593 1493140529 7634757481 1935670911 0137751721 0080315590 2485309066 9203767192 2033229094 3346768514 2214477379 3937517034 4366199104 0337511173 5471918550 4644902636 5512816228 8244625759 1633303910 7225383742 1821408835 0865739177 1509682887 4782656995 9957449066 1758344137 5223970968 3408005355 9849175417 3818839994 4697486762 6551658276 5848358845 3142775687 9002909517 0283529716 3445621296 4043523117 6006651012 4120065975 5851276178 5838292041 9748442360 8007193045 7618932349 2292796501 9875187212 7267507981 2554709589 0455635792 1221033346 6974992356 3025494780 2490114195 2123828153 0911407907 3860251522 7429958180 7247162591 6685451333 1239480494 7079119153 2673430282 4418604142 6363954800 0448002670 4962482017 9289647669 7583183271 3142517029 6923488962 7668440323 2609275249 6035799646 9256504936 8183609003 2380929345 9588970695 3653494060 3402166544 3755890045 6328822505 4525564056 4482465151 8754711962 1844396582 5337543885 6909411303 1509526179 3780029741 2076651479 3942590298 9695946995 5657612186 5619673378 6236256125 2163208628 6922210327 4889218654 3648022967 8070576561 5144632046 9279068212 0738837781 4233562823 6089632080 6822246801 2248261177 1858963814 0918390367 3672220888 3215137556 0037279839 4004152970 0287830766 7094447456 0134556417 2543709069 7939612257 1429894671 5435784687 8861444581 2314593571 9849225284 7160504922 1242470141 2147805734 5510500801 9086996033 0276347870 8108175450 1193071412 2339086639 3833952942 5786905076 4310063835 1983438934 1596131854 3475464955 6978103829 3097164651 4384070070 7360411237 3599843452 2516105070 2705623526 6012764848 3084076118 3013052793 2054274628 6540360367 4532865105 7065874882 2569815793 6789766974 2205750596 8344086973 5020141020 6723585020 0724522563 2651341055 9240190274 2162484391 4035998953 5394590944 0704691209 1409387001 2645600162 3742880210 9276457931 0657922955 2498872758 4610126483 6999892256 9596881592 0560010165 5256375678

 


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