From Wikipedia, the free encyclopedia
List of numbers – Irrational and suspected irrational numbers
γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δ_{S} – α – e – π – δ 


Number system 
Evaluation of π 
Binary 
11.00100100001111110110...^{[1]} 
Decimal 
3.14159265358979323846264338327950288... 
Hexadecimal 
3.243F6A8885A308D31319...^{[2]} 
Rational approximations 
3, ^{22}⁄_{7}, ^{333}⁄_{106}, ^{355}⁄_{113}, ^{103993}/_{33102}, ...^{[3]}
(listed in order of increasing accuracy)

Continued fraction 
[3; 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1, 1, ... ]^{[4]}

Trigonometry 
π radians = 180 degrees 
.^ This almost mystical ratio between a circle’s diameter and circumference is an irrational number whose quest has persisted since the time of Pythagoras and the ancient Greeks – and whose mystery still captivates mathematicians today. Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^ I find it amazing that with our advanced mathmatics, we can't even find something as simple as the exact ratio of a circumfrence of a circle and its diameter. Michael Naylor, WWU 28 January 2010 1:39 UTC www.ac.wwu.edu [Source type: Original source]
It is approximately equal to 3.141593 in the usual decimal notation (see the table for its representation in some other bases). The constant is also known as Archimedes Constant, although this name is rather uncommon in modern, western, Englishspeaking contexts.
.^ Thessaloniki is considered as one of the most important cities of Greece (in a time that didn’t exist officially a Greek state) and its population is consisted of Turks, Greeks and Jews . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[5]}
π is an
.^ It's what you would write in a love letter: "My love for you is like piMy love for you is endless; my love for you is irrational; my love for you cannot be shown in fraction form." Michael Naylor, WWU 28 January 2010 1:39 UTC www.ac.wwu.edu [Source type: Original source]
Consequently, its
decimal representation never ends or repeats. It is also a
transcendental number, which implies, among other things, that no finite sequence of algebraic operations on integers (powers, roots, sums, etc.) can be equal to its value; proving this was a late achievement in mathematical history and a significant result of 19th century German mathematics.
.^ They may choose to believe there is no God even when the knowledge of Gods existence is built into all that is around them as stated in Romans. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ I'm tempted to let Diamond's "25/8" argument stand in the section "NonGenerous Interpretation," since it is a reminder of how useless the counterclaim "But there were more accurate values of pi known in antiquity" actually is. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ But I don't think the failure of the writer to break into a formal mathematical description of pi in this passage is evidence for the nonexistence of God or even an error by the writer. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
The Greek letter π, often spelled out
pi in text, was adopted for the number from the Greek word for
perimeter "περίμετρος", first by
William Jones in 1707, and popularized by
Leonhard Euler in 1737.
^{[6]}
Fundamentals
Lowercase π is used to symbolize the constant
The letter π
Main article:
pi (letter)
Circumference = π × diameter
.^ Conversely, could a “team science” application with a single PI receive criticism (and a worse score) for not using the Multiple PI model? Multiple Principal Investigators FAQ 28 January 2010 1:39 UTC grants.nih.gov [Source type: FILTERED WITH BAYES]
^ This might be a valid alternative spelling, a misspelling based on the Greek variant of the name, or a purposeful alteration that will make the letters of his name to sum to the same total as Antiochus Epiphanes. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ Unfortunately for themselves, the Greeks had been the first to make use of this name, with its classical associations, and to give it a wide extension to the north in interests of Hellenic expansion. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[citation needed]} It is not capitalized (Π) even at the beginning of a sentence. When referring to this constant, the symbol π is always pronounced
/ˈpaɪ/,
^{[citation needed]} "pie" in
English, which is the conventional English pronunciation of the Greek letter.
.^ GREEK CAPITAL LETTER PI . Letter Database 11 September 2009 20:49 UTC www.eki.ee [Source type: Academic]
^ GREEK SMALL LETTER PI . Letter Database 11 September 2009 20:49 UTC www.eki.ee [Source type: Academic]
.^ In other words they use the above quote to support their theory that most Greeks have Albanian ancestors. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ This might be a valid alternative spelling, a misspelling based on the Greek variant of the name, or a purposeful alteration that will make the letters of his name to sum to the same total as Antiochus Epiphanes. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ On the text of the book, the words “Macedonian population” is in the FYROM Slavic language, but on the picture and above the heads of the people to the right, circled in black, the word “Bulgarians” in Greek is equally clear. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[7]} π is
Unicode character U+03C0 ("
Greek small letter pi").
^{[8]}
Geometric definition
Area of the circle = π × area of the shaded square
The ratio
^{C}/
_{d} is constant, regardless of a circle's size.
.^ This almost mystical ratio between a circle’s diameter and circumference is an irrational number whose quest has persisted since the time of Pythagoras and the ancient Greeks – and whose mystery still captivates mathematicians today. Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^ I find it amazing that with our advanced mathmatics, we can't even find something as simple as the exact ratio of a circumfrence of a circle and its diameter. Michael Naylor, WWU 28 January 2010 1:39 UTC www.ac.wwu.edu [Source type: Original source]
Alternatively π can be also defined as the ratio of a circle's
area (A) to the area of a square whose side is equal to the
radius:
^{[7]}^{[9]}
.^ However, seeing as how they are all dependent on Mark, and Mark is an allegorical fiction, the fact that they all copy his claims doesn't really amount to much does it? SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
This can be considered a problem when π occurs in areas of mathematics that otherwise do not involve geometry. For this reason, mathematicians often prefer to define π without reference to geometry, instead selecting one of its
analytic properties as a definition.
.^ The startup package and other institutional support should be comparable to that given to other recently hired faculty into tenuretrack or equivalent positions. PA07297: NIH Pathway to Independence (PI) Award (K99/R00) 28 January 2010 1:39 UTC grants.nih.gov [Source type: Academic]
Irrationality and transcendence
π is an
irrational number, meaning that it cannot be written as the
ratio of two integers. The belief in the irrationality of π is mentioned by
Muhammad ibn Mūsā alKhwārizmī^{[11]} in the 9th century.
Maimonides also mentions with certainty the irrationality of π in the 12th century.
^{[12]} This was proved in 1768 by
Johann Heinrich Lambert.
^{[13]} .^ The Bulgarian propagandists found what they required in the word “Macedonia” a name with no more definite significance than Wessex or Languedoc. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ Its website states that in the beginning of the 20th century three vilayets existed in Macedonia, the one of Bitola, Thessaloniki and Skopje.” 130 The fact is that there was no vilayet of Skopje. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ In reality, no significant population affiliated with their ethnic/national identity until the 20th century. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
One of those, due to
Ivan Niven, is widely known.
^{[14]}^{[15]} A somewhat earlier similar proof is by
Mary Cartwright.
^{[16]}
π is also a
transcendental number, meaning that there is no
polynomial with
rational coefficients for which π is a
root.
^{[17]} This was proved by
Ferdinand von Lindemann in 1882. An important consequence of the transcendence of π is the fact that it is not
constructible.
.^ Now because the "prophecy" up to that point needed to be accurate to be convincing, modern historians use the "prophecies of Merlin" as historical material ... SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ More to the point, the reasoning "X does not exist because X is a vicious son of a bitch" doesn't follow at all. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ All of this occurred within an area whose population numbered only around two million in 1990. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[18]} .^ Anyway, I just wanted to thank you guys for all the time and effort you put into these threads to help people like me. TomTom v8 Software and Maps [Archive]  Page 2  GSMForum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]
Decimal representation
 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510
 See the links below and those at sequence A000796 in OEIS for more digits.
.^ There are only a couple of agencies statewide that can effectively handle such investigations because there is no investigation that is more difficult than surveillance.
^ The Bulgarian propagandists found what they required in the word “Macedonia” a name with no more definite significance than Wessex or Languedoc. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ With an official 35% unemployment rate (the grey market is estimated to more than 20% of the GDP) and about a third of its population below the poverty line, it is heavily dependent upon foreign investment for its economy to function. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
.^ It's relevant because you've dismissed less signficiant biblical contradictions and errors as not being "true" contradictions, so I provided you with an example of a "true" contradiction. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ I like the idea of categorizing errors but since I think an error can fall under more than one characterization I think it is a better idea not to divide errors into separate type sections. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^{[21]}^{[22]}
Because π is an
irrational number, its decimal representation does not
repeat, and therefore does not terminate.
.^ Pi has fascinated mathematicians and philosophers for centuries. Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^ Anyway, I just wanted to thank you guys for all the time and effort you put into these threads to help people like me. TomTom v8 Software and Maps [Archive]  Page 2  GSMForum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]
^{[23]} Despite much analytical work, and
supercomputer calculations that have determined over 1
trillion digits of the decimal representation of π, no simple
base10 pattern in the digits has ever been found.
^{[24]} Digits of the decimal representation of π are available on many web pages, and there is
software for calculating the decimal representation of π to billions of digits on any
personal computer.
Estimating π
π can be empirically estimated by drawing a large circle, then measuring its diameter and circumference and dividing the circumference by the diameter. Another geometrybased approach, attributed to
Archimedes,
^{[25]} is to calculate the
perimeter,
P_{n} , of a
regular polygon with
n sides
circumscribed around a circle with diameter
d. Then
That is, the more sides the polygon has, the closer the approximation approaches π. Archimedes determined the accuracy of this approach by comparing the perimeter of the circumscribed polygon with the perimeter of a regular polygon with the same number of sides
inscribed inside the circle. Using a polygon with 96 sides, he computed the fractional range:
3+^{10}⁄_{71} < π <
3+^{1}⁄_{7}.
^{[26]}
π can also be calculated using purely mathematical methods. Most formulae used for calculating the value of π have desirable mathematical properties, but are difficult to understand without a background in
trigonometry and
calculus. However, some are quite simple, such as this form of the
GregoryLeibniz series:
^{[27]}
While that series is easy to write and calculate, it is not immediately obvious why it yields π. In addition, this series converges so slowly that nearly 300 terms are needed to calculate π correctly to 2 decimal places.^{[28]} However, by computing this series in a somewhat more clever way by taking the midpoints of partial sums, it can be made to converge much faster. Let
and then define
then computing
.^ [Elisabeth Barker, "Macedonia, its place in Balkan power politics", (originally published in 1950 by the Royal Institute of International Affairs), p.10] . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ In that same time period, the problem of mixed populations was “solved” by a series of forced relocations. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
This computation is an example of the
van Wijngaarden transformation.
^{[29]}
History
The
Great Pyramid of Giza is estimated to have originally been 280
cubits in height by 440 cubits in length at each side. The ratio of 440/280 is approximately equal to π/2.
The earliest evidenced conscious use of an accurate approximation for the length of a circumference with respect to its radius is of 3+1/7 in the designs of the
Old Kingdom pyramids in Egypt. The
Great Pyramid at Giza, constructed c.25502500 BC, was built with a perimeter of 1760
cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π.
.^ This is certainly old enough for such physical activity to be very stressful, while still being young enough for the feat to be reasonably performed. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^{[32]}^{[33]} The same
apotropaic proportions were used earlier at the Pyramid of
Meidum c.2600 BC. This application is archaeologically evidenced, whereas textual evidence does not survive from this early period.
The early history of π from textual sources roughly parallels the development of mathematics as a whole.^{[34]} Some authors divide progress into three periods: the ancient period during which π was studied geometrically, the classical era following the development of calculus in Europe around the 17th century, and the age of digital computers.^{[35]}
Antiquity
.^ This almost mystical ratio between a circle’s diameter and circumference is an irrational number whose quest has persisted since the time of Pythagoras and the ancient Greeks – and whose mystery still captivates mathematicians today. Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^ That was only slightly more than the rhetorical equivalent of a grunt. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ And all the while, I am working a full day, my soninlaw works more than 60 hours a week, and everyone in my family works and pays for our education. Things I find annoying... [Archive]  Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]
.^ The three last had alliances both with Athens and Philip but their proMacedonian activity of 344/3 BC showed they were leaning towards Philip. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ The external evidence for dating the book of Revelation around 96 AD is far too weak for one to allow it to influence, much less dominate, one's theological perspective on the book of Revelation. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^{[7]} The Indian text
Shatapatha Brahmana gives π as 339/108 ≈ 3.139.
Archimedes' Pi approximation
Archimedes (287–212 BC) was the first to estimate π rigorously. He realized that its magnitude can be bounded from below and above by inscribing circles in
regular polygons and calculating the outer and inner polygons' respective perimeters:
^{[36]}
By using the equivalent of 96sided polygons, he proved that 3^{10}/_{71} < π < 3^{1}/_{7}.^{[36]} The average of these values is about 3.14185.
Around AD 265, the
Wei Kingdom mathematician
Liu Hui provided a simple and rigorous
iterative algorithm to calculate π to any degree of accuracy. He himself carried through the calculation to a 3072gon and obtained an approximate value for π of 3.1416.
^{[38]} Later, Liu Hui invented a
quick method of calculating π and obtained an approximate value of 3.14 with only a 96gon,
^{[38]} by taking advantage of the fact that the difference in area of successive polygons forms a geometric series with a factor of 4.
Around 480, the Chinese mathematician
Zu Chongzhi demonstrated that π ≈ 355/113, and showed that 3.1415926 < π < 3.1415927
^{[38]} using Liu Hui's algorithm applied to a 12288gon.
.^ Therefore it remained Greek after 550 years, most of them under Ottoman rule, . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ This would be especially difficult, since most of the apostles were known to have been martyred many years earlier. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
Second Millennium AD
Until the
second millennium AD, estimations of π were accurate to fewer than 10 decimal digits. The next major advances in the study of π came with the development of
infinite series and subsequently with the discovery of
calculus, which permit the estimation of π to any desired accuracy by considering sufficiently many terms of a relevant series. Around 1400,
Madhava of Sangamagrama found the first known such series:
This is now known as the
Madhava–Leibniz series^{[39]}^{[40]} or GregoryLeibniz series since it was rediscovered by
James Gregory and
Gottfried Leibniz in the 17th century.
.^ With an official 35% unemployment rate (the grey market is estimated to more than 20% of the GDP) and about a third of its population below the poverty line, it is heavily dependent upon foreign investment for its economy to function. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
However, by transforming the series into
Madhava was able to estimate π as 3.14159265359, which is correct to 11 decimal places. The record was beaten in 1424 by the
Persian mathematician,
Jamshīd alKāshī, who gave an estimate π that is correct to 16 decimal digits.
.^ First, it will jeopardise the sustainability of the European edifice since flexibility leads to the application of double standards and second it will certainly lead to veto situations in defence of national interests. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ Unfortunately for themselves, the Greeks had been the first to make use of this name, with its classical associations, and to give it a wide extension to the north in interests of Hellenic expansion. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ B] Unfortunately for themselves, the Greeks had been the first to make use of this name, with its classical associations, and to give it a wide extension to the north in interests of Hellenic expansion. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
He was so proud of the calculation, which required the greater part of his life, that he had the digits engraved into his tombstone.
^{[41]}
.^ This is another indication that perhaps all three of these books were written around the same time, during the mid to late 60's. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^ The ethnicities were clearly labelled as Serbian, Bulgarian, Greek, Albanian and Turkish or Muslim (as in the Turkish censuses of around the same time). Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ In that same time period, the problem of mixed populations was “solved” by a series of forced relocations. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
The first such representation was the
Viète's formula,
by
John Wallis in 1655.
Isaac Newton himself derived a series for π and calculated 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time."
^{[42]}
In 1706
John Machin was the first to compute 100 decimals of π, using the formula
with
.^ In the Middle Ages and into the nineteenth century, the term ´Macedonian´ was used ENTIRELY in reference to a geographical region. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ In the Middle Ages and into the nineteenth century, the term ‘Macedonian’ was used ENTIRELY in reference to a geographical region . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ A VRF consists of an IP routing table, a derived forwarding table, a set of interfaces that use the forwarding table, and a set of rules and routing protocols that determine what goes into the forwarding table. IPSec VPN Accounting  Cisco Systems 11 September 2009 20:49 UTC www.cisco.com [Source type: Reference]
A remarkable record was set by the calculating prodigy
Zacharias Dase, who in 1844 employed a Machinlike formula to calculate 200 decimals of π in his head at the behest of
Gauss.
.^ Note: This excerpt clearly states that their definition of “Macedonia” was only established in the 19th century . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ In their first year and a half in the Primera División, only two Arsenal players were shown the red card: no other team was treated so mildly. Gnagarforum 28 January 2010 1:39 UTC aik.pi.se [Source type: Academic]
^ The point is that less than 20 years ago their own scholars were conceding that their population was undifferentiated from the Bulgarians prior to the middle of the 19th century . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
(To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas.
.^ It was then that I noticed the same thing and it MAKES ME NUTS. Also, they look around for the monitor, like they haven't been staring at themselves in it on every commercial break. Things I find annoying... [Archive]  Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]
)
Theoretical advances in the 18th century led to insights about π's nature that could not be achieved through numerical calculation alone.
Johann Heinrich Lambert proved the irrationality of π in 1761, and
AdrienMarie Legendre also proved in 1794 π
^{2} to be irrational. When
Leonhard Euler in 1735 solved the famous
Basel problem – finding the exact value of
William Jones' book
A New Introduction to Mathematics from 1706 is said to be the first use of the
Greek letter π for this constant, but the notation became particularly popular after
Leonhard Euler adopted it in 1737.
^{[43]} He wrote:
There are various other ways of finding the Lengths or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for instance, in the Circle, the Diameter is to the Circumference as 1 to (16/5 − 4/239) − 1/3(16/5^{3} − 4/239^{3}) + ... = 3.14159... = π^{[7]}
Computation in the computer age
.^ It has so many different forms and “national” variations in space and time that is often argued whether they can all be accommodated under one roof”. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ As such the differences between this dialect and Bulgarian were emphasized and the similarities (although many) were purposely ignored. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ Pi Variations: Posters & Tshirts of That Very Irrational number, Pi Pi Posters and Pi T shirts featuring Pi: The first 5,000 digits Tshirt & poster . Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
.^ Rating: 0.0/ 10 (0 votes cast) Quite many Slavomacedonians migrated to USA in the begining of 20th century. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ We might as well add Finlay to the vast number of sources that failed to record any “ethnic Macedonians” prior to the 20th century! Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ VN:F [1.6.4_902] Rating: 0 (from 0 votes) Tags: 19th century , 20th century , bulgarians , delchev , foreign , gruev , macedonian bulgarians , macedonian greeks , new york times , newspapers 1 Comment » . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
John von Neumann et al. used
ENIAC to compute 2037 digits of π in 1949, a calculation that took 70 hours.
^{[44]}^{[45]} Additional thousands of decimal places were obtained in the following decades, with the milliondigit milestone passed in 1973. Progress was not only due to faster hardware, but also new algorithms. One of the most significant developments was the discovery of the
fast Fourier transform (FFT) in the 1960s, which allows computers to perform arithmetic on extremely large numbers quickly.
.^ Rating: 0.0/ 10 (0 votes cast) Quite many Slavomacedonians migrated to USA in the begining of 20th century. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ Its website states that in the beginning of the 20th century three vilayets existed in Macedonia, the one of Bitola, Thessaloniki and Skopje.” 130 The fact is that there was no vilayet of Skopje. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[46]} One of his formulas is the series,
which deliver 14 digits per term.
^{[46]} .^ Dr Adequate 19th June 2006, 01:47 AM :confused: J is the one who is supposed to tend to use YHWH as the name for God, as opposed to E, who uses Elohim. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ J is the one who is supposed to tend to use YHWH as the name for God, as opposed to E, who uses Elohim. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
.^ Is there a step required before I attempt to install it? TomTom v8 Software and Maps [Archive]  Page 2  GSMForum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]
A breakthrough was made in 1975, when
Richard Brent and
Eugene Salamin independently discovered the
Brent–Salamin algorithm, which uses only arithmetic to double the number of correct digits at each step.
^{[47]} The algorithm consists of setting
and iterating
until a_{n} and b_{n} are close enough. Then the estimate for π is given by
Using this scheme, 25 iterations suffice to reach 45 million correct decimals. A similar algorithm that quadruples the accuracy in each step has been found by
Jonathan and
Peter Borwein.
^{[48]} .^ Being unable to win people by peaceful propaganda, the Comitadjis do not hesitate to make use of the most atrocious methods. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[49]} This beats the previous record of 2,576,980,370,000 decimals, set by Daisuke Takahashi on the T2KTsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.
^{[50]}
.^ The first one was #45, "How did Simon Peter learn that Jesus was the Christ?" SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^{[51]} The formula,
is remarkable because it allows extracting any individual
hexadecimal or
binary digit of π without calculating all the preceding ones.
^{[51]} Between 1998 and 2000, the
distributed computing project
PiHex used a modification of the BBP formula due to
Fabrice Bellard to compute the
quadrillionth (1,000,000,000,000,000:th) bit of π, which turned out to be 0.
^{[52]}
If a formula of the form
were found where
.^ The Greeks, of all recent immigrants, are the most fond of athletic exercise, and many of them find their way to the best gymnasiums in the city where they reside. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^ It would seem that there were only three other people in the world, and he just killed one of them. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ Thessaloniki is considered as one of the most important cities of Greece (in a time that didn’t exist officially a Greek state) and its population is consisted of Turks, Greeks and Jews . Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
.^ Ironically, these are the only three times Balaam is named in the New Testament, all in the space of only about 14 pages in my Bible. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^ It has so many different forms and “national” variations in space and time that is often argued whether they can all be accommodated under one roof”. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
The previous formula (found by Plouffe for π with
b=2 and
c=4, but also found for log(9/10) and for a few other irrational constants), implies that π is a
SC* number.
and others of form,
In the previous formula, if k is of the form 4m + 3, then the formula has the particularly simple form,
for some rational number
.^ Many highly respected Bible teachers have stated it as though it were proven fact, offering no evidence in defense. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
Pi and continued fraction
The sequence of partial denominators of the simple
continued fraction of π does not show any obvious pattern:
^{[4]}
or
Memorizing digits
Main article:
Piphilology
Recent decades have seen a surge in the record for number of digits memorized.
.^ Well, obviously it's useful to at least six people picto2. TomTom v8 Software and Maps [Archive]  Page 2  GSMForum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]
^ We might as well add Finlay to the vast number of sources that failed to record any “ethnic Macedonians” prior to the 20th century! Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
In 2006,
Akira Haraguchi, a retired Japanese engineer, claimed to have recited 100,000 decimal places.
^{[55]} This, however, has yet to be verified by
Guinness World Records.
.^ However, if the book of Revelation was written in 67 AD, then the chase would have occurred in 69 AD, when John was 60 years old. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^ In the Balkans, if you want to be recognized as a Nation, you need to have history 3000 years old. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
^{[56]} It took him 24 hours and 4 minutes to recite to the 67,890th decimal place of π without an error.
^{[57]}
.^ Mom's Apple Pi 20" x 30" poster, glossy . Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^ World's Largest PI Poster  1 Million Digits of Pi . Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^ The Pi Diner and the PIOMatic: Hundreds of Millions of Digits of Pi on Tap . Pi T shirts Pi Posters  Pi Tshirt Pi Tshirt Pi Poster Pi Mugs Pi Tshirts Pi Tshirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]
^{[58]} .^ However, to the credit of the mapmakers it did not make claims that the region of FYROM has some unredeemed territories elsewhere in the Balkans. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
.^ If the real intention of the author(s) had been to provide an etiological myth to explain the presence of salt columns, the narrative easily could have been composed in such a way as to account explicitly for why there are so many. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ Put another way, if a particular passage can be validly interpreted such that there is no contradiction, then it can't necessarily be considered a contradiction. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ While later the author informs us ”In the district of Uskub , where there are some Servianspeaking refugees and people speaking a Bulgarian dialect containing many Servian words “. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
.^ I love how Homer was by the drink machine and tackled James Bond as he tried to escape. Things I find annoying... [Archive]  Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]
^ SecondHand Hearsay: If the apostle John were to have written us a letter stating that he received the Revelation in 96 AD, that would be first hand information. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
The
Cadaeic Cadenza contains the first 3835 digits of π in this manner.
^{[61]} Piems are related to the entire field of humorous yet serious study that involves the use of
mnemonic techniques to remember the digits of π, known as
piphilology.
.^ There is not a shred of evidence that a language other than Greek was spoken in ancient Macedonia and in countries conquered by Alexander the Great. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
However, this method proves inefficient for large memorizations of π. Other methods include remembering patterns in the numbers and the
method of loci.
^{[62]}^{[63]}
Advanced properties
Numerical approximations
.^ The term Nephilim was used for dramatic effect, as the term "Huns" was used to indicate Germans during the World Wars, centuries after there were no longer true Huns. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ June 2006, 05:51 PM In Hebrew there's no distinct term for ancestor; "father" would be used. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ In Hebrew there's no distinct term for ancestor; "father" would be used. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^{[17]} Formulas for calculating π using elementary arithmetic typically include
series or
summation notation (such as "..."), which indicates that the formula is really a formula for an infinite sequence of approximations to π.
^{[64]} The more terms included in a calculation, the closer to π the result will get.
Consequently, numerical calculations must use
approximations of π.
.^ It should be noted that although the name Balaam is not used in reference to the church of Thyatira, examination of Revelation 2:14 to 2:20 shows that they have many of the same heretical characteristics (fornication and eating things sacrificed to idols) as Pergamos. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^ As such the differences between this dialect and Bulgarian were emphasized and the similarities (although many) were purposely ignored. Bulgarians  History Of Macedonia 15 September 2009 6:11 UTC historyofmacedonia.com [Source type: FILTERED WITH BAYES]
The approximations
^{22}/
_{7} and
^{355}/
_{113}, with 3 and 7 significant figures respectively, are obtained from the simple
continued fraction expansion of π. The approximation
^{355}⁄_{113} (3.1415929...) is the best one that may be expressed with a threedigit or fourdigit
numerator and denominator; the next good approximation
^{103993}/
_{33102} (3.14159265301...) requires much bigger numbers, due to the large number 292 in the continued fraction expansion.
^{[3]}
The earliest numerical approximation of π is almost certainly the value
3.
^{[36]} In cases where little precision is required, it may be an acceptable substitute. That 3 is an underestimate follows from the fact that it is the ratio of the
perimeter of an
inscribed regular hexagon to the
diameter of the
circle.
Open questions
.^ I'm SO anal about EVERYTHING ELSE that I type, that I just give myself a pass on that one. Things I find annoying... [Archive]  Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]
^ Certainly, if God normally reacted that way, most civilized nations would be visited by pestilence every five or ten years; moreover, when God tells Moses to take a census (in the Book of Numbers, chapter 1 [3]) neither Moses nor the tribes of Israel are punished for it. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
^ (Lot's wife) The fact that it doesn't function very well as an etiological myth raises a question in my mind as to whether it was really intended as one. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
Use in mathematics and science
π is ubiquitous in mathematics, appearing even in places that lack an obvious connection to the circles of Euclidean geometry.^{[69]}
Geometry and trigonometry
For any circle with radius
r and diameter
d = 2
r, the circumference is π
d and the area is π
r^{2}. Further, π appears in formulas for areas and volumes of many other geometrical shapes based on circles, such as
ellipses,
spheres,
cones, and
tori.
^{[70]} Accordingly, π appears in
definite integrals that describe circumference, area or volume of shapes generated by circles. In the basic case, half the area of the
unit disk is given by:
^{[71]} and
From the unitcircle definition of the
trigonometric functions also follows that the sine and cosine have period 2π. That is, for all
x and integers
n, sin(
x) = sin(
x + 2π
n) and cos(
x) = cos(
x + 2π
n). Because sin(0) = 0, sin(2π
n) = 0 for all integers
n. Also, the angle measure of 180° is equal to π radians. In other words, 1° = (π/180) radians.
In modern mathematics, π is often
defined using trigonometric functions, for example as the smallest positive
x for which sin
x = 0, to avoid unnecessary dependence on the subtleties of Euclidean geometry and integration. Equivalently, π can be defined using the
inverse trigonometric functions, for example as π = 2 arccos(0) or π = 4 arctan(1). Expanding inverse trigonometric functions as
power series is the easiest way to derive infinite series for π.
Complex numbers and calculus
Euler's formula depicted on the complex plane. Increasing the angle
φ to
π radians (180°) yields Euler's identity.
where
i is the
imaginary unit satisfying
i^{2} = −1 and
e ≈ 2.71828 is
Euler's number. This formula implies that imaginary powers of
e describe rotations on the
unit circle in the complex plane; these rotations have a period of 360° = 2π. In particular, the 180° rotation
φ = π results in the remarkable
Euler's identity
Euler's identity is famous for linking several basic mathematical concepts in one concise, elegant expression.
A consequence is that the
gamma function of a halfinteger is a rational multiple of √π.
Physics
.^ If the answer to that is "No," then why is the part of the article that says as much described as giving merely a "generous interpretation" of the text rather than an accurate one? SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
Using units such as
Planck units can sometimes eliminate π from formulae.






Probability and statistics
Note that since
for any probability density function
f(
x), the above formulas can be used to produce other integral formulas for π.
^{[80]}
Buffon's needle problem is sometimes quoted as an empirical approximation of π in "popular mathematics" works. Consider dropping a needle of length
L repeatedly on a surface containing parallel lines drawn
S units apart (with
S >
L). If the needle is dropped
n times and
x of those times it comes to rest crossing a line (
x > 0), then one may approximate π using the
Monte Carlo method:
^{[81]}^{[82]}^{[83]}^{[84]} Though this result is mathematically impeccable, it cannot be used to determine more than very few digits of π
by experiment. Reliably getting just three digits (including the initial "3") right requires millions of throws,
^{[81]} and the number of throws grows
exponentially with the number of digits desired. Furthermore, any error in the measurement of the lengths
L and
S will transfer directly to an error in the approximated π. For example, a difference of a single
atom in the length of a 10centimeter needle would show up around the 9th digit of the result.
.^ There is much less to the web site than it appears since many articles are only stubs. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
Geomorphology and chaos theory
Under ideal conditions (uniform gentle slope on an homogeneously erodible substrate), the ratio between the actual length of a river and its straightline from source to mouth length tends to approach π.
^{[85]} Albert Einstein was the first to suggest that rivers have a tendency towards an ever more loopy path because the slightest curve will lead to faster currents on the outer side, which in turn will result in more erosion and a sharper bend. The sharper the bend, the faster the currents on the outer edge, the more the erosion, the more the river will twist and so on. However, increasing loopiness will result in rivers doubling back on themselves and effectively shortcircuiting, creating an
oxbow lake.
.^ This article is primarily directed toward those readers who are well versed in or at least familiar with differences between these two programs. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^ Now let us examine the similarities between these two passages of Revelation 3:1518 and Zephaniah 1:718. The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]
^{[86]}
In popular culture
Probably because of the simplicity of its definition, the concept of pi and, especially its decimal expression, have become entrenched in popular culture to a degree far greater than almost any other mathematical construct.^{[87]} It is, perhaps, the most common ground between mathematicians and nonmathematicians.^{[88]} Reports on the latest, mostprecise calculation of π (and related stunts) are common news items.^{[89]}^{[90]}^{[91]}
On November 7, 2005,
alternative musician
Kate Bush released the album,
Aerial. The album contains the song "π" whose lyrics consist principally of Bush singing the digits of π to music, beginning with "3.14"
^{[94]}
.^ The Gospel of Mark, however, gives many reasons to consider it a purely fictitious story, in which characters are fabricated and play out roles in a narrative. SkepticWiki And The Bible [Archive]  JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]
This part of the story was left out of the
film adaptation of the novel.
See also
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 ^ See Alfred S. Posamentier and Ingmar Lehmann, Pi: A Biography of the World's Most Mysterious Number. Prometheus Books, 2004. ISBN 9781591022008.
 ^ E.g., MSNBC, Man recites pi from memory to 83,431 places July 3, 2005; Matt Schudel, Obituaries: "John W. Wrench, Jr.: Mathematician Had a Taste for Pi" The Washington Post, March 25, 2009, p. B5.
 ^ The Big Question: How close have we come to knowing the precise value of pi? The Independent, 8 January 2010
 ^ Pi, a mathematical story that would take 49,000 years to tell
 ^ Pi Day activities.
 ^ MIT, E to the U.
 ^ Blatner, David (20080314). "UK  Magazine  3.14 and the rest". BBC News. http://news.bbc.co.uk/1/hi/magazine/7296224.stm. Retrieved 20100102.
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