# Pi: Wikis

### Did you know ...

More interesting facts on Pi

# Encyclopedia

When a circle's diameter is 1 unit, its circumference is pi units
Number system Evaluation of π List of numbers – Irrational and suspected irrational numbers γ – ζ(3) – √2 – √3 – √5 – φ – ρ – δS – α – e – π – δ 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] (This continued fraction is not periodic. Shown in linear notation) Trigonometry π radians = 180 degrees
.π (sometimes written pi) is a mathematical constant whose value is the ratio of any circle's circumference to its diameter in Euclidean space; this is the same value as the ratio of a circle's area to the square of its radius.^ 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 T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 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, English-speaking contexts. .Many formulae from mathematics, science, and engineering involve π, which is one of the most important mathematical and physical constants.^ 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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

[5]
π is an .irrational number, which means that its value cannot be expressed exactly as a fraction m/n, where m and n are integers.^ It's what you would write in a love letter: "My love for you is like pi--My 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. .Throughout the history of mathematics, there has been much effort to determine π more accurately and to understand its nature; fascination with the number has even carried over into non-mathematical culture.^ They may choose to believe there is no God even when the knowledge of Gods 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 "Non-Generous 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 non-existence 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

Lower-case π is used to symbolize the constant

### The letter π

Circumference = π × diameter
.The name of the Greek letter π is pi, and this spelling is commonly used in typographical contexts when the Greek letter is not available or its usage could be problematic.^ 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 history-of-macedonia.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. .In Greek, the name of this letter is pronounced [pi].^ 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]

.The constant is named "π" because "π" is the first letter of the Greek words περιφέρεια (periphery) and περίμετρος (perimeter), probably referring to its use in the formula to find the circumference, or perimeter, of a circle.^ 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 history-of-macedonia.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 history-of-macedonia.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
In Euclidean plane geometry, π is defined as the ratio of a circle's circumference to its diameter:[7]
$\pi = \frac{C}{d}.$
The ratio C/d is constant, regardless of a circle's size. .For example, if a circle has twice the diameter d of another circle it will also have twice the circumference C, preserving the ratio C/d.^ 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 T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 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]
$\pi = \frac{A}{r^2}.$
.These definitions depend on results of Euclidean geometry, such as the fact that all circles are similar.^ 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. .A common choice is to define π as twice the smallest positive x for which cos(x) = 0.[10] The formulas below illustrate other (equivalent) definitions.^ The start-up package and other institutional support should be comparable to that given to other recently hired faculty into tenure-track or equivalent positions.
• PA-07-297: NIH Pathway to Independence (PI) Award (K99/R00) 28 January 2010 1:39 UTC grants.nih.gov [Source type: Academic]

### Irrationality and transcendence

Squaring the circle: This was a problem proposed by the ancient geometers. In 1882, it was proven that π is transcendental, and consequently this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.
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 al-Khwārizmī[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] .In the 20th century, proofs were found that require no prerequisite knowledge beyond integral calculus.^ 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 history-of-macedonia.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 history-of-macedonia.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 history-of-macedonia.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. .Because the coordinates of all points that can be constructed with compass and straightedge are constructible numbers, it is impossible to square the circle: that is, it is impossible to construct, using compass and straightedge alone, a square whose area is equal to the area of a given circle.^ 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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

[18] .This is historically significant, for squaring a circle is one of the easily understood elementary geometry problems left to us from antiquity; many amateurs in modern times have attempted to solve each of these problems, and their efforts are sometimes ingenious, but in this case, doomed to failure: a fact not always understood by the amateur involved.^ 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 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

### Decimal representation

The decimal representation of π truncated to 50 decimal places is:[19]
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.
.While the decimal representation of π has been computed to more than a trillion (1012) digits,[20] elementary applications, such as estimating the circumference of a circle, will rarely require more than a dozen decimal places.^ 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 history-of-macedonia.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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

.For example, the decimal representation of π truncated to 11 decimal places is good enough to estimate the circumference of any circle that fits inside the earth with an error of less than one millimetre, and the decimal representation of π truncated to 39 decimal places is sufficient to estimate the circumference of any circle that fits in the observable universe with precision comparable to the radius of a hydrogen atom.^ 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. .This sequence of non-repeating digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing ever more of these digits and investigating π's properties.^ Pi has fascinated mathematicians and philosophers for centuries.
• Pi T shirts Pi Posters - Pi Tshirt Pi T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 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 - GSM-Forum 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 base-10 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 geometry-based approach, attributed to Archimedes,[25] is to calculate the perimeter, Pn , of a regular polygon with n sides circumscribed around a circle with diameter d. Then
$\pi = \lim_{n o \infty}\frac{P_{n}}{d}.$
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: 31071 < π < 317.[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 Gregory-Leibniz series:[27]
$\pi = 4\sum^\infty_{k=0} \frac{(-1)^k}{2k+1} = \frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}\cdots.\!$
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
$\pi_{0,1} = \frac{4}{1},\ \pi_{0,2} =\frac{4}{1}-\frac{4}{3},\ \pi_{0,3} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5},\ \pi_{0,4} =\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}, \ldots\!$
and then define
$\pi_{i,j} = \frac{\pi_{i-1,j}+\pi_{i-1,j+1}}{2} ext{ for all }i,j\ge 1$
then computing .π10,10 will take similar computation time to computing 150 terms of the original series in a brute-force manner, and $\pi_{10,10}=3.141592653\ldots$, correct to 9 decimal places.^ [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 history-of-macedonia.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 history-of-macedonia.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.
Archimedes used the method of exhaustion to approximate the value of π.
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.2550-2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits; the ratio 1760/280 ≈ 2π. .Egyptologists such as Professors Flinders Petrie[30] and I.E.S Edwards[31] have shown that these circular proportions were deliberately chosen for symbolic reasons by the Old Kingdom scribes and architects.^ 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

.That the ratio of the circumference to the diameter of a circle is the same for all circles, and that it is slightly more than 3, was known to Ancient Egyptian, Babylonian, Indian and Greek geometers.^ 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 T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 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 son-in-law 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 earliest known textually evidenced approximations date from around 1900 BC; they are 25/8 (Babylonia) and 256/81 (Egypt), both within 1% of the true value.^ The three last had alliances both with Athens and Philip but their pro-Macedonian activity of 344/3 BC showed they were leaning towards Philip.
• Bulgarians | History Of Macedonia 15 September 2009 6:11 UTC history-of-macedonia.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
Liu Hui's π algorithm
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 96-sided polygons, he proved that   310/71 < π < 31/7.[36] The average of these values is about 3.14185.
Ptolemy, in his Almagest, gives a value of 3.1416, which he may have obtained from Apollonius of Perga.[37]
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 3072-gon 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 96-gon,[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 12288-gon. .This value would remain the most accurate approximation of π available for the next 900 years.^ Therefore it remained Greek after 550 years, most of them under Ottoman rule, .
• Bulgarians | History Of Macedonia 15 September 2009 6:11 UTC history-of-macedonia.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]

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:
${\pi} = 4\sum^\infty_{k=0} \frac{(-1)^k}{2k+1} = \frac{4}{1} - \frac{4}{3} + \frac{4}{5} - \frac{4}{7} + \cdots\!$
This is now known as the Madhava–Leibniz series[39][40] or Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. .Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate.^ 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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

However, by transforming the series into
\begin{align} \pi &= \sqrt{12}\sum^\infty_{k=0} \frac{(-3)^{-k}}{2k+1} \ &= \sqrt{12}\sum^\infty_{k=0} \frac{(-\frac{1}{3})^k}{2k+1} \ &= \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right), \end{align}
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 al-Kāshī, who gave an estimate π that is correct to 16 decimal digits.
.The first major European contribution since Archimedes was made by the German mathematician Ludolph van Ceulen (1540–1610), who used a geometric method to give an estimate of π that is correct to 35 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 history-of-macedonia.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 history-of-macedonia.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 history-of-macedonia.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]
.Around the same time, the methods of calculus and determination of infinite series and products for geometrical quantities began to emerge in Europe.^ 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 history-of-macedonia.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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

The first such representation was the Viète's formula,
$\frac2\pi = \frac{\sqrt2}2 \cdot \frac{\sqrt{2+\sqrt2}}2 \cdot \frac{\sqrt{2+\sqrt{2+\sqrt2}}}2 \cdot \cdots\!$
found by François Viète in 1593. Another famous result is Wallis' product,
$\frac{\pi}{2} = \prod^\infty_{k=1} \frac{(2k)^2}{(2k)^2-1} = \frac{2}{1} \cdot \frac{2}{3} \cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5} \cdot \frac{6}{7} \cdot \frac{8}{7} \cdot \frac{8}{9} \cdots\ = \frac{4}{3} \cdot \frac{16}{15} \cdot \frac{36}{35} \cdot \frac{64}{63} \cdots\!$
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
$\frac{\pi}{4} = 4 \, \arctan \frac{1}{5} - \arctan \frac{1}{239}\!$
with
$\arctan \, x = \sum^\infty_{k=0} \frac{(-1)^k x^{2k+1}}{2k+1} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots\!$
.Formulas of this type, now known as Machin-like formulas, were used to set several successive records and remained the best known method for calculating π well into the age of computers.^ 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 history-of-macedonia.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 history-of-macedonia.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 Machin-like formula to calculate 200 decimals of π in his head at the behest of Gauss. .The best value at the end of the 19th century was due to William Shanks, who took 15 years to calculate π with 707 digits, although due to a mistake only the first 527 were correct.^ 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 history-of-macedonia.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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

(To avoid such errors, modern record calculations of any kind are often performed twice, with two different formulas. .If the results are the same, they are likely to be correct.^ 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 Adrien-Marie Legendre also proved in 1794 π2 to be irrational. When Leonhard Euler in 1735 solved the famous Basel problem – finding the exact value of
$\sum^\infty_{k=1} \frac{1}{k^2} = \frac{1}{1^2} + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + \cdots\!$
which is π2/6, he established a deep connection between π and the prime numbers. Both Legendre and Euler speculated that π might be transcendental, which was finally proved in 1882 by Ferdinand von Lindemann.
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(16/53 − 4/2393) + ... = 3.14159... = π[7]

### Computation in the computer age

.Although practically a physicist needs only 39 digits of Pi to make a circle the size of the observable universe accurate to one atom of hydrogen, the number itself as a mathematical curiosity has created many challenges in different fields.^ 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 history-of-macedonia.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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

^ Pi Variations: Posters & T-shirts of That Very Irrational number, Pi Pi Posters and Pi T shirts featuring Pi: The first 5,000 digits T-shirt & poster .
• Pi T shirts Pi Posters - Pi Tshirt Pi T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]

.The advent of digital computers in the 20th century led to an increased rate of new π calculation records.^ 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 history-of-macedonia.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 history-of-macedonia.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 history-of-macedonia.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 million-digit 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.
.In the beginning of the 20th century, the Indian mathematician Srinivasa Ramanujan found many new formulas for π, some remarkable for their elegance, mathematical depth and rapid convergence.^ 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 history-of-macedonia.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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

[46] One of his formulas is the series,
$\frac{1}{\pi} = \frac{2 \sqrt 2}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$
and the related one found by the Chudnovsky brothers in 1987,
$\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!$
which deliver 14 digits per term.[46] .The Chudnovskys used this formula to set several π computing records in the end of the 1980s, including the first calculation of over one billion (1,011,196,691) decimals in 1989. It remains the formula of choice for π calculating software that runs on personal computers, as opposed to the supercomputers used to set modern records.^ 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]

.Whereas series typically increase the accuracy with a fixed amount for each added term, there exist iterative algorithms that multiply the number of correct digits at each step, with the downside that each step generally requires an expensive calculation.^ Is there a step required before I attempt to install it?
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 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
$a_0 = 1 \quad \quad \quad b_0 = \frac{1}{\sqrt 2} \quad \quad \quad t_0 = \frac{1}{4} \quad \quad \quad p_0 = 1\!$
and iterating
$a_{n+1} = \frac{a_n+b_n}{2} \quad \quad \quad b_{n+1} = \sqrt{a_n b_n}\!$
$t_{n+1} = t_n - p_n (a_n-a_{n+1})^2 \quad \quad \quad p_{n+1} = 2 p_n\!$
until an and bn are close enough. Then the estimate for π is given by
$\pi \approx \frac{(a_n + b_n)^2}{4 t_n}.\!$
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] .The methods have been used by Yasumasa Kanada and team to set most of the π calculation records since 1980, up to a calculation of 206,158,430,000 decimals of π in 1999. As of January 2010, the record is almost 2.7 trillion digits.^ 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 history-of-macedonia.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 T2K-Tsukuba System, a supercomputer at the University of Tsukuba northeast of Tokyo.[50]
.An important recent development was the Bailey–Borwein–Plouffe formula (BBP formula), discovered by Simon Plouffe and named after the authors of the paper in which the formula was first published, David H. Bailey, Peter Borwein, and Simon Plouffe.^ 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,
$\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right),$
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
$\pi = \sum_{k=0}^\infty \frac{1}{b^{ck}} \frac{p(k)}{q(k)},$
were found where .b and c are positive integers and p and q are polynomials with fixed degree and integer coefficients (as in the BPP formula above), this would be one the most efficient ways of computing any digit of π at any position in base bc without computing all the preceding digits in that base, in a time just depending on the size of the integer k and on the fixed degree of the polynomials.^ 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 history-of-macedonia.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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

.Plouffe also describes such formulas as the interesting ones for computing numbers of class SC*, in a logarithmically polynomial space and almost linear time, depending only on the size (order of magnitude) of the integer k, and requiring modest computing resources.^ 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 history-of-macedonia.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.
In 2006, Simon Plouffe, using the integer relation algorithm PSLQ, found a series of formulas.[53] Let q = eπ (Gelfond's constant), then
$\frac{\pi}{24} = \sum_{n=1}^\infty \frac{1}{n} \left(\frac{3}{q^n-1} - \frac{4}{q^{2n}-1} + \frac{1}{q^{4n}-1}\right)$
$\frac{\pi^3}{180} = \sum_{n=1}^\infty \frac{1}{n^3} \left(\frac{4}{q^n-1} - \frac{5}{q^{2n}-1} + \frac{1}{q^{4n}-1}\right)$
and others of form,
$\pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{a}{q^n-1} + \frac{b}{q^{2n}-1} + \frac{c}{q^{4n}-1}\right)$
where k is an odd number, and abc are rational numbers.
In the previous formula, if k is of the form 4m + 3, then the formula has the particularly simple form,
$p\pi^k = \sum_{n=1}^\infty \frac{1}{n^k} \left(\frac{2^{k-1}}{q^n-1} - \frac{2^{k-1}+1}{q^{2n}-1} + \frac{1}{q^{4n}-1}\right)$
for some rational number .p where the denominator is a highly factorable number, though no rigorous proof has yet been given.^ 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]
$\pi=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,1,84,\cdots]$
or
$\pi=3+ extstyle \frac{1}{7+ extstyle \frac{1}{15+ extstyle \frac{1}{1+ extstyle \frac{1}{292+ extstyle \frac{1}{1+ extstyle \frac{1}{1+ extstyle \frac{1}{1+ extstyle \frac{1}{2+ extstyle \frac{1}{1+ extstyle \frac{1}{3+ extstyle \frac{1}{1+ extstyle \frac{1}{14+ extstyle \frac{1}{2+ extstyle \frac{1}{1+ extstyle \frac{1}{1+ extstyle \frac{1}{2+ extstyle \frac{1}{2+ extstyle \frac{1}{2+ extstyle \frac{1}{2+ extstyle \frac{1}{1+ extstyle \frac{1}{84+\cdots}}}}}}}}}}}}}}}}}}}}}$
However, there are generalized continued fractions for π with a perfectly regular structure, such as:[54]
$\pi= extstyle \frac{4}{1+ extstyle \frac{1^2}{2+ extstyle \frac{3^2}{2+ extstyle \frac{5^2}{2+ extstyle \frac{7^2}{2+ extstyle \frac{9^2}{2+ extstyle \frac{11^2}{2+\cdots}}}}}}}= 3+ extstyle \frac{1^2}{6+ extstyle \frac{3^2}{6+ extstyle \frac{5^2}{6+ extstyle \frac{7^2}{6+ extstyle \frac{9^2}{6+ extstyle \frac{11^2}{6+\cdots}}}}}}= extstyle \frac{4}{1+ extstyle \frac{1^2}{3+ extstyle \frac{2^2}{5+ extstyle \frac{3^2}{7+ extstyle \frac{4^2}{9+ extstyle \frac{5^2}{11+\cdots}}}}}}$

### Memorizing digits

Recent decades have seen a surge in the record for number of digits memorized.
.Well before computers were used in calculating π, memorizing a record number of digits had become an obsession for some people.^ Well, obviously it's useful to at least six people picto2.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 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 history-of-macedonia.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. .The Guinness-recognized record for remembered digits of π is 67,890 digits, held by Lu Chao, a 24-year-old graduate student from China.^ 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 history-of-macedonia.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]
.On June, 17th, 2009 Andriy Slyusarchuk, a Ukrainian neurosurgeon, medical doctor and professor claimed to have memorized 30 million digits of pi, which were printed in 20 volumes of text.^ Mom's Apple Pi 20" x 30" poster, glossy .
• Pi T shirts Pi Posters - Pi Tshirt Pi T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 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 T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]

^ The Pi Diner and the PI-O-Matic: Hundreds of Millions of Digits of Pi on Tap .
• Pi T shirts Pi Posters - Pi Tshirt Pi T-shirt Pi Poster Pi Mugs Pi Tshirts Pi T-shirts 28 January 2010 1:39 UTC mathematicianspictures.com [Source type: Academic]

[58] .Although he did not recite the entire 30 million digits that he claims to have memorized, some media claim that he was able to recite ten randomly selected sequences from the printed text of the 30 million digits.^ 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 history-of-macedonia.com [Source type: FILTERED WITH BAYES]

.There are many ways to memorize π, including the use of "piems", which are poems that represent π in a way such that the length of each word (in letters) represents a digit.^ 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 Servian-speaking refugees and people speaking a Bulgarian dialect containing many Servian words “.
• Bulgarians | History Of Macedonia 15 September 2009 6:11 UTC history-of-macedonia.com [Source type: FILTERED WITH BAYES]

.Here is an example of a piem, originally devised by Sir James Jeans: How I need (or: want) a drink, alcoholic in nature (or: of course), after the heavy lectures (or: chapters) involving quantum mechanics.[59][60] Notice how the first word has 3 letters, the second word has 1, the third has 4, the fourth has 1, the fifth has 5, and so on.^ 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]

^ Second-Hand 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. .In other languages there are similar methods of memorization.^ 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 history-of-macedonia.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]

### Numerical approximations

.Due to the transcendental nature of π, there are no closed form expressions for the number in terms of algebraic numbers and functions.^ 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. .For many purposes, 3.14 or 22/7 is close enough, although engineers often use 3.1416 (5 significant figures) or 3.14159 (6 significant figures) for more precision.^ 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 history-of-macedonia.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 355113 (3.1415929...) is the best one that may be expressed with a three-digit or four-digit 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

.The most pressing open question about π is whether it is a normal number—whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely "randomly", and that this is true in every integer base, not just base 10.[65] Current knowledge on this point is very weak; e.g., it is not even known which of the digits 0,...,9 occur infinitely often in the decimal expansion of π,[66] although it is clear that at least two such digits must occur infinitely often, since otherwise π would be rational, which it is not.^ 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]

Bailey and Crandall showed in 2000 that the existence of the above mentioned Bailey-Borwein-Plouffe formula and similar formulas imply that the normality in base 2 of π and various other constants can be reduced to a plausible conjecture of chaos theory.[67]
It is also unknown whether π and e are algebraically independent, although Yuri Nesterenko proved the algebraic independence of {π, eπ, Γ(1/4)} in 1996.[68]

## 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 = 2r, the circumference is πd and the area is πr2. 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] $\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}$ and
$\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\,dx = \pi$
gives half the circumference of the unit circle.[70] More complicated shapes can be integrated as solids of revolution.[72]
From the unit-circle 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 + 2πn) and cos(x) = cos(x + 2πn). Because sin(0) = 0, sin(2πn) = 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.
A complex number z can be expressed in polar coordinates as follows:
$z = r\cdot(\cos\varphi + i\sin\varphi)$
The frequent appearance of π in complex analysis can be related to the behavior of the exponential function of a complex variable, described by Euler's formula
$e^{i\varphi} = \cos \varphi + i\sin \varphi \!$
where i is the imaginary unit satisfying i2 = −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
$e^{i \pi} = -1.\!$
$e^{i \pi} + 1 = 0.\!$
Euler's identity is famous for linking several basic mathematical concepts in one concise, elegant expression.
There are n different n-th roots of unity
$e^{2 \pi i k/n} \qquad (k = 0, 1, 2, \dots, n - 1).$
$\int_{-\infty}^{\infty}e^{-x^2}dx=\sqrt{\pi}.$
A consequence is that the gamma function of a half-integer is a rational multiple of √π.

### Physics

.Although not a physical constant, π appears routinely in equations describing fundamental principles of the Universe, due in no small part to its relationship to the nature of the circle and, correspondingly, spherical coordinate systems.^ 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.
$\Lambda = {{8\pi G} \over {3c^2}} \rho$
$\Delta x\, \Delta p \ge \frac{h}{4\pi}$
$R_{ik} - {g_{ik} R \over 2} + \Lambda g_{ik} = {8 \pi G \over c^4} T_{ik}$
$F = \frac{\left|q_1q_2\right|}{4 \pi \varepsilon_0 r^2}$
$\mu_0 = 4 \pi \cdot 10^{-7}\,\mathrm{N/A^2}\,$
$\frac{P^2}{a^3}={(2\pi)^2 \over G (M+m)}$

### Probability and statistics

In probability and statistics, there are many distributions whose formulas contain π, including:
$f(x) = {1 \over \sigma\sqrt{2\pi} }\,e^{-(x-\mu )^2/(2\sigma^2)}$
$f(x) = \frac{1}{\pi (1 + x^2)}.$
Note that since $\int_{-\infty}^{\infty} f(x)\,dx = 1$ 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] $\pi \approx \frac{2nL}{xS}.$ 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 10-centimeter needle would show up around the 9th digit of the result. .In practice, uncertainties in determining whether the needle actually crosses a line when it appears to exactly touch it will limit the attainable accuracy to much less than 9 digits.^ 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 straight-line 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 short-circuiting, creating an ox-bow lake. .The balance between these two opposing factors leads to an average ratio of π between the actual length and the direct distance between source and mouth.^ 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 3:15-18 and Zephaniah 1:7-18.
• The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]

[86]

## In popular culture

A whimsical "Pi plate"
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 non-mathematicians.[88] Reports on the latest, most-precise calculation of π (and related stunts) are common news items.[89][90][91]
Pi Day (March 14, from 3.14) is observed in many schools.[92] At least one cheer at the Massachusetts Institute of Technology includes "3.14159!"[93]
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]
.In Carl Sagan's novel Contact, pi played a key role in the story and suggested that there was a message buried deep within the digits of pi placed there by whoever created the universe.^ 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.

## References

1. ^ Alexander D. Poularikas (1999). The handbook of formulas and tables for signal processing. CRC Press. p. 9.8. ISBN 9780849385797.
2. ^
3. ^ a b Gourdon, Xavier; Pascal Sebah. "Collection of approximations for π". Numbers, constants and computation. Retrieved 2007-11-08.
4. ^ a b A001203: Continued fraction for Pi, On-Line Encyclopedia of Integer Sequences
5. ^ Howard Whitley Eves (1969). An Introduction to the History of Mathematics. Holt, Rinehart & Winston.
6. ^ Comanor, Milton; Ralph P. Boas (1976). "Pi". in William D. Halsey. Collier's Encyclopedia. 19. New York: Macmillan Educational Corporation. pp. 21–22.
7. ^ a b c d e "About Pi". Ask Dr. Math FAQ. Retrieved 2007-10-29.
8. ^ "Characters Ordered by Unicode". W3C. Retrieved 2007-10-25.
9. ^ Richmond, Bettina (1999-01-12). "Area of a Circle". Western Kentucky University. Retrieved 2007-11-04.
10. ^ Rudin, Walter (1976) [1953]. Principles of Mathematical Analysis (3e ed.). McGraw-Hill. p. 183. ISBN 0-07-054235-X.
11. ^ Glimpses in the history of a great number: Pi in Arabic mathematics by Mustafa Mawaldi
12. ^ Commentary to Mishneh Torah, beginning of Eruvin
13. ^ Lambert, Johann Heinrich (1761), "Mémoire sur quelques propriétés remarquables des quantités transcendentes circulaires et logarithmiques", Histoire de l'Académie, (Berlin) XVII: 265–322, 1768
14. ^
15. ^ Richter, Helmut (1999-07-28). "Pi Is Irrational". Leibniz Rechenzentrum. Retrieved 2007-11-04.
16. ^ Jeffreys, Harold (1973). Scientific Inference (3rd ed.). Cambridge University Press.
17. ^ a b Mayer, Steve. "The Transcendence of π". Retrieved 2007-11-04.
18. ^ "Squaring the Circle". cut-the-knot. Retrieved 2007-11-04.
19. ^ A000796: Decimal expansion of Pi, On-Line Encyclopedia of Integer Sequences
20. ^ "Current publicized world record of pi". Retrieved 2007-10-14.
21. ^ Young, Robert M. (1992). Excursions in Calculus. Washington: Mathematical Association of America (MAA). p. 417. ISBN 0883853175.
22. ^
23. ^ Weisstein, Eric W., "Pi Digits" from MathWorld.
24. ^ Boutin, Chad (2005-04-26). "Pi seems a good random number generator - but not always the best". Purdue University. Retrieved 2007-11-04.
25. ^ Groleau, Rick (09-2003). "Infinite Secrets: Approximating Pi". NOVA. Retrieved 2007-11-04.
26. ^ Beckmann, Petr (1989). A History of Pi. Barnes & Noble Publishing. ISBN 0880294183.
27. ^ Eymard, Pierre; Jean-Pierre Lafon (02 2004). "2.6". The Number π. Stephen S. Wilson (translator). American Mathematical Society. p. 53. ISBN 0821832468. Retrieved 2007-11-04.
28. ^ Lampret, Spanish, Vito (2006). "Even from Gregory-Leibniz series π could be computed: an example of how convergence of series can be accelerated" (PDF). Lecturas Mathematicas 27: 21–25. Retrieved 2007-11-04.
29. ^ A. van Wijngaarden, in: Cursus: Wetenschappelijk Rekenen B, Process Analyse, Stichting Mathematisch Centrum, (Amsterdam, 1965) pp. 51–60.
30. ^ Petrie, W.M.F. 1940 Wisdom of the Egyptians: pp 27
31. ^ Edwards. I.E.S. 1979: The Pyramids of Egypt: pp269
32. ^ Jackson and Stamp (2002) Pyramid: Beyond Imagination. pp153
33. ^ Lightbody, D.I. 2008. Egyptian Tomb Architecture: The Archaeological Facts of Pharaonic Circular Symbolism. British Archaeological Reports. pp36
34. ^ Beckmann, Petr (1976). A History of π. St. Martin's Griffin. ISBN 0-312-38185-9.
35. ^ "Archimedes' constant π". Retrieved 2007-11-04.
36. ^ a b c O'Connor, J J; E F Robertson (2001-08). "A history of Pi". Retrieved 2007-10-30.
37. ^ C. Boyer, A History of Mathematics, Wiley, p. 168.
38. ^ a b c C. Boyer, A History of Mathematics, Wiley, p. 202.
39. ^ George E. Andrews, Richard Askey, Ranjan Roy (1999), Special Functions, Cambridge University Press, p. 58, ISBN 0521789885
40. ^ Gupta, R. C. (1992), "On the remainder term in the Madhava-Leibniz's series", Ganita Bharati 14 (1-4): 68–71
41. ^ Charles Hutton (1811). Mathematical Tables; Containing the Common, Hyperbolic, and Logistic Logarithms.... London: Rivington. p. 13.
42. ^ The New York Times: Even Mathematicians Can Get Carried Away
43. ^ "About: William Jones". Famous Welsh. Retrieved 2007-10-27.
44. ^ "An {ENIAC} Determination of pi and e to more than 2000 Decimal Places", Mathematical Tables and Other Aids to Computation, 4 (29), pp. 11–15. (January,1950)
45. ^ "Statistical Treatment of Values of First 2,000 Decimal Digits of e and of pi Calculated on the ENIAC", Mathematical Tables and Other Aids to Computation, 4 (30), pp. 109–111. (April,1950)
46. ^ a b
47. ^ Brent, Richard (1975), Traub, J F, ed., "Multiple-precision zero-finding methods and the complexity of elementary function evaluation", Analytic Computational Complexity (New York: Academic Press): 151–176, retrieved 2007-09-08
48. ^ Borwein, Jonathan M; Borwein, Peter; Berggren, Lennart (2004). Pi: A Source Book. Springer. ISBN 0387205713.
49. ^ "Pi calculated to 'record number' of digits". bbc.co.uk. Retrieved 2010-01-06.
50. ^ Pi-obsessed Japanese reach 2.5 trillion digits 2009-08-20
51. ^ a b Bailey, David H.; Borwein, Peter B.; and Plouffe, Simon (April 1997). "On the Rapid Computation of Various Polylogarithmic Constants" (PDF). Mathematics of Computation 66 (218): 903–913. doi:10.1090/S0025-5718-97-00856-9.
52. ^
53. ^
54. ^ Lange, L. J. (May 1999), "An Elegant Continued Fraction for π", The American Mathematical Monthly 106 (5): 456–458
55. ^ Otake, Tomoko (2006-12-17). "How can anyone remember 100,000 numbers?". The Japan Times. Retrieved 2007-10-27.
56. ^ "Pi World Ranking List". Retrieved 2007-10-27.
57. ^ "Chinese student breaks Guiness record by reciting 67,890 digits of pi". News Guangdong. 2006-11-28. Retrieved 2007-10-27.
58. ^ Профессор Андрей Слюсарчук установил мировой рекорд по возможностям человеческой памяти http://www.mk.ru/health/303812.html?phrase_id=1446233
59. ^ Weisstein, Eric W. "Pi Wordplay." From MathWorld—A Wolfram Web Resource. Retrieved on 2009-03-12.
60. ^ Borwein, Jonathan M (2005-09-25). "The Life of Pi: From Archimedes to Eniac and Beyond" (PDF). Dalhousie University Computer Science. Retrieved 2007-10-29.
62. ^ Liu, Yicong (2004-05-19). "Oh my, memorizing so many digits of pi.". Silver Chips Online. Retrieved 2007-11-04.
63. ^ Raz A, Packard MG, Alexander GM, Buhle JT, Zhu H, Yu S, Peterson BS. (2009). A slice of pi: An exploratory neuroimaging study of digit encoding and retrieval in a superior memorist. Neurocase. 6:1-12. doi:10.1080/13554790902776896 PMID 19585350
64. ^ Weisstein, Eric W (2007-09-27). "Pi Formulas". MathWorld. Retrieved 2007-11-10.
65. ^ Weisstein, Eric W (2005-12-22). "Normal Number". MathWorld. Retrieved 2007-11-10.
66. ^
67. ^ Peterson, Ivars (2001-09-01). "Pi à la Mode: Mathematicians tackle the seeming randomness of pi's digits". Science News Online. Retrieved 2007-11-10.
68. ^ Nesterenko, Yuri V (1996). "Modular Functions and Transcendence Problems". Comptes rendus de l'Académie des sciences Série 1 322 (10): 909–914.
69. ^ "Japanese breaks pi memory record". BBC News. 2005-07-02. Retrieved 2007-10-30.
70. ^ a b
71. ^ Weisstein, Eric W (2006-01-28). "Unit Disk Integral". MathWorld. Retrieved 2007-11-08.
72. ^ Weisstein, Eric W (2006-05-04). "Solid of Revolution". MathWorld. Retrieved 2007-11-08.
73. ^ Miller, Cole. "The Cosmological Constant" (PDF). University of Maryland. Retrieved 2007-11-08.
74. ^ Imamura, James M (2005-08-17). "Heisenberg Uncertainty Principle". University of Oregon. Retrieved 2007-11-09.
75. ^ Einstein, Albert (1916). "The Foundation of the General Theory of Relativity" (PDF). Annalen der Physik. Retrieved 2007-11-09.
76. ^ Nave, C. Rod (2005-06-28). "Coulomb's Constant". HyperPhysics. Georgia State University. Retrieved 2007-11-09.
77. ^ "Magnetic constant". NIST. 2006 CODATA recommended values. Retrieved 2007-11-09.
78. ^ Weisstein, Eric W (2004-10-07). "Gaussian Integral". MathWorld. Retrieved 2007-11-08.
79. ^ Weisstein, Eric W (2005-10-11). "Cauchy Distribution". MathWorld. Retrieved 2007-11-08.
80. ^ Weisstein, Eric W (2003-07-02). "Probability Function". MathWorld. Retrieved 2007-11-08.
81. ^ a b Weisstein, Eric W (2005-12-12). "Buffon's Needle Problem". MathWorld. Retrieved 2007-11-10.
82. ^ Bogomolny, Alex (2001-08). "Math Surprises: An Example". cut-the-knot. Retrieved 2007-10-28.
83. ^ Ramaley, J. F. (October 1969). "Buffon's Noodle Problem". The American Mathematical Monthly 76 (8): 916–918. doi:10.2307/2317945.
84. ^ "The Monte Carlo algorithm/method". datastructures. 2007-01-09. Retrieved 2007-11-07.
85. ^ Hans-Henrik Stølum (1996-03-22). "River Meandering as a Self-Organization Process". Science 271 (5256): 1710–1713. doi:10.1126/science.271.5256.1710.
86. ^ Kaleem Omar (2002-12-10). "Poetic Licence: Pi calculated to 1.24 trillion digits". Daily Times. Retrieved 2009-12-11.
87. ^ See, e.g, Lennart Berggren, Jonathan M. Borwein, and Peter B. Borwein (eds.), Pi: A Source Book. Springer, 1999 (2nd ed.). ISBN 978-0-387-98946-4.
88. ^ See Alfred S. Posamentier and Ingmar Lehmann, Pi: A Biography of the World's Most Mysterious Number. Prometheus Books, 2004. ISBN 978-1-59102-200-8.
89. ^ 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.
90. ^ The Big Question: How close have we come to knowing the precise value of pi? The Independent, 8 January 2010
91. ^ Pi, a mathematical story that would take 49,000 years to tell
92. ^ Pi Day activities.
93. ^ MIT, E to the U.
94. ^ Blatner, David (2008-03-14). "UK | Magazine | 3.14 and the rest". BBC News. Retrieved 2010-01-02.

# Quotes

Up to date as of January 14, 2010
(Redirected to Pi (film) article)

### From Wikiquote

Pi is a 1998 film about a paranoid mathematician who searches for a key number that will unlock the universal patterns found in nature.
Directed by Darren Aronofsky and written by Darren Aronofsky, Sean Gullette, and Eric Watson.
faith in chaos

## Maximillian Cohen

• Something's going on. It has to do with that number. .There's an answer in that number.
• You see the simplicity of the circle.^ You should see 3 options Select number 1 which says Run it for Navcore 8.010 and higher.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

.You see the maddening complexity of the endless string of numbers, 3.14 off into infinity.
• 11:15, restate my assumptions: 1. Mathematics is the language of nature.^ WarEagleAU 11-15-2008, 03:27 Alright, let me give yall some info on my problem.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

^ BuMobiles 11-18-2008, 15:20 Thank you very much.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

^ BuMobiles 11-18-2008, 15:40 BU, as a Moderator, do you think you could find out why I am unable to post attachments, please.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

2. Everything around us can be represented and understood through numbers. 3. If you graph these numbers, patterns emerge. .Therefore: There are patterns everywhere in nature.
• 9:13, Personal note: When I was a little kid my mother told me not to stare into the sun.^ However, as far as I can conclude from my research into this topic, very little external evidence exists that would warrant any bold statement as to the date of the writing.
• The Seven Churches of Revelation 15 September 2009 4:42 UTC www.matthewmcgee.org [Source type: Original source]

^ There was petty persecution; Bulgarian Christians crossed from Macedonia into Bulgaria proper and told their tales of woe .
• Bulgarians | History Of Macedonia 15 September 2009 6:11 UTC history-of-macedonia.com [Source type: FILTERED WITH BAYES]

So once when I was six, I did. At first the brightness was overwhelming, but I had seen that before. I kept looking, forcing myself not to blink, and then the brightness began to dissolve. My pupils shrunk to pinholes and everything came into focus and for a moment I understood. The doctors didn't know if my eyes would ever heal. I was terrified, alone in that darkness. Slowly daylight crept in through the bandages, and I could see, but something else had changed inside of me. .That day I had my first headache.
• I'm trying to understand our world.^ And all the while, I am working a full day, my son-in-law 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]

.I don't deal with petty materialists like you.
• 12:53-Restate my assumptions: Mathematics is the language of nature.^ But he didn't do that and it looks like a ridiculous stretch (IMHO of course) to interpret what looks like normal everyday descriptive language as a formal mathematical statement.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

Everything around us can be represented by numbers. .If one graphs the system of any numbers, patterns emerge, therefore there are patterns everywhere in nature.
• 12:50, press return...
• Failed treatments to date: Beta blockers, calcium channel blockers, adrenalin injections, high dose ibuprofen, steroids, Trager Mentastics, violent exercise, cafergot suppositories, caffeine, acupuncture, marijuana, Percodan, Midrine, Tenormin, Sansert, homeopathics.^ When i put them on my system the keygen found them but when i went to switch map there was onlt one map listed instead of three maps and the one listed was a 725 map i kept on the system.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

^ Also a calcium channel blocker named Verapamil helps as a preventive for some.
• Things I find annoying... [Archive] - Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]

• 10:15, personal note: It's fair to say I'm stepping out on a limb, but I am on the edge and that's where it happens.
• My new hypothesis: If we're built from Spirals while living in a giant Spiral, then is it possible that everything we put our hands to is infused with the Spiral?
• If the number's there I'll find it!

## Sol Robeson

.
• There will be no order, only chaos.
• Have you met Archimedes?^ 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 only relevance of bringing in other examples of contradictions would be if somebody asserted that there were no Biblical contradictions, which no one here has done.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

^ Bri 5th July 2006, 06:44 PM There is no inconsistency if you use the translation of Genesis 21:31 from the Hebrew version that I have.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

.The one with the black spots, you see?^ Not inside any other folder) Start with one voice only to see you have it right.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

^ See if you have more than one .tlv file per voice.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

You remember Archimedes of Syracuse, eh? The king asks Archimedes to determine if a present he's received is actually solid gold. Unsolved problem at the time. It tortures the great Greek mathematician for weeks - insomnia haunts him and he twists and turns in his bed for nights on end. Finally, his equally exhausted wife - she's forced to share a bed with this genius - convinces him to take a bath to relax. While he's entering the tub, Archimedes notices the bath water rise. Displacement, a way to determine volume, and that's a way to determine density - weight over volume. And thus, Archimedes solves the problem. He screams "Eureka" and he is so overwhelmed he runs dripping naked through the streets to the king's palace to report his discovery.
• Number? How many digits was it? .100? 1,000? 216?
• Hold on.^ It should be noted that Mr. Georgievski holds a Bulgarian passport as do more than 100,000 citizens of the FYROM. 125 .
• Bulgarians | History Of Macedonia 15 September 2009 6:11 UTC history-of-macedonia.com [Source type: FILTERED WITH BAYES]

You have to slow down. You're losing it. You have to take a breath. Listen to yourself. .You're connecting a computer bug I had with a computer bug you might have had and some religious hogwash.^ You might connect your 910 to your puter and look at your 910 files with Windows Explorer.
• TomTom v8 Software and Maps [Archive] - Page 2 - GSM-Forum 11 September 2009 20:49 UTC forum.gsmhosting.com [Source type: FILTERED WITH BAYES]

You want to find the number 216 in the world, you will be able to find it everywhere. 216 steps from a mere street corner to your front door. 216 seconds you spend riding on the elevator. .When your mind becomes obsessed with anything, you will filter everything else out and find that thing everywhere.
• The Ancient Japanese considered the Go board to be a microcosm of the universe.^ You deserve to know the truth and have peace of mind knowing for certain whether or not your significant other is engaging in a relationship outside of yours.

^ He told me that (even though it's almost impossible to plan for it) you should try to put your forearm out in front of you...
• Things I find annoying... [Archive] - Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]

^ Canadians!!!:D "Crocs shoes are becoming more and more popular everywhere you go.
• Things I find annoying... [Archive] - Nick DiPaolo Forum 15 September 2009 6:11 UTC www.nickdip.com [Source type: FILTERED WITH BAYES]

Although when it is empty it appears to be simple and ordered, in fact, the possibilities of gameplay are endless. .They say that no two Go games have ever been alike.^ And the text doesn't say how long they were there beyond the two nights necessary for the incest to happen.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

^ Hilmi: They say they are Greeks when no coercion, no constraint, is brought to bear on them.
• Bulgarians | History Of Macedonia 15 September 2009 6:11 UTC history-of-macedonia.com [Source type: FILTERED WITH BAYES]

Just like snowflakes. So, the Go board actually represents an extremely complex and chaotic universe.
• That is the truth of our world, Max. It can't be easily summed up with math.
• As soon as you discard scientific rigor, you're no longer a mathematician, you're a numerologist.

## Lenny Meyer

• Each letter's a number. Like the Hebrew A, Alef is 1. B, Bet is 2. You understand? But look at this. The numbers are inter-related. Like take the Hebrew word for father, 'Ab' - Alef Bet... 1, 2 equals 3. Alright? Hebrew word for mother, 'em' - Alef Mem... 1, 40 equals 41. Sum of 3 and 41... 44. Alright? Now, Hebrew word for child, alright, mother... father... child, 'Yeled' - that's 10, 30, and 4... 44.
• The Torah is just a long string of numbers. Some say that it's a code sent to us from God.
• You gave it to those Wall Street bastards?

## Marcy Dawson

• It's survival of the fittest, Max, and we've got the fucking gun.

## Dialogue

Sol Robeson: This is insanity, Max.
Maximillian Cohen: Or maybe it's genius.

Rabbi Cohen: Who do you think you are? You are only a vessel from our god. You are carrying a delivery that was meant for us.
Maximillian Cohen: It was given to me.

Maximillian Cohen: Studying the pattern made Euclid conscious of itself. I had to... Before it died it spit out the number. That consciousness is the number?
Sol Robeson: No, Max. It's only a nasty bug.
Maximillian Cohen: It's more than that, Sol.
Sol Robeson: No, it's not. It's a dead end. There's nothing there.
Maximillian Cohen: It's a door, Sol. It's a door.
Sol Robeson: A door at the front of a cliff. .You're driving yourself over the edge.^ You're changing the meaning of the text to suit yourself.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

^ March 2007, 09:36 PM You're changing the meaning of the text to suit yourself.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

^ ChristineR 23rd March 2007, 08:01 AM You're changing the meaning of the text to suit yourself.
• SkepticWiki And The Bible [Archive] - JREF Forum 11 September 2009 20:49 UTC forums.randi.org [Source type: Original source]

# Study guide

Up to date as of January 14, 2010

### From Wikiversity

.Pi (π) is found by dividing the circumference by the diameter.^ Pi is the ratio of the circumference of a circle to its diameter, not the other way around.
• NPR: A Challenge for Pi Day: Write Your Own Pi-ku 28 January 2010 1:39 UTC www.npr.org [Source type: General]

^ "Strictly speaking, pi represents the constant ratio of a circle's diameter to its circumference..."
• NPR: A Challenge for Pi Day: Write Your Own Pi-ku 28 January 2010 1:39 UTC www.npr.org [Source type: General]

^ Circumference by Diameter of the fruit I eat apple PI .
• NPR: A Challenge for Pi Day: Write Your Own Pi-ku 28 January 2010 1:39 UTC www.npr.org [Source type: General]

C/D = π. .The first one hundred digits of pi are the following.^ Here are the first few digits: pi =3.
• We are in the Digits of Pi and Live Forever 28 January 2010 1:39 UTC sprott.physics.wisc.edu [Source type: Original source]

^ However, the (base 16 I assume) digits of pi are calculable in sequence via one of several algorithms.
• We are in the Digits of Pi and Live Forever 28 January 2010 1:39 UTC sprott.physics.wisc.edu [Source type: Original source]

^ The sequence isn't simply Pi itself but any encoded sequence in the sequence itself as in every third digit followed by the 42nd and divided by the first 5.
• We are in the Digits of Pi and Live Forever 28 January 2010 1:39 UTC sprott.physics.wisc.edu [Source type: Original source]

π = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679
.We use pi in calculating circles.^ If you use pi = 3.14 in your 2-pi-r-calculation, you'll be off by millions of lightyears.
• Michael Naylor, WWU 28 January 2010 1:39 UTC www.ac.wwu.edu [Source type: Original source]

# Wiktionary

Up to date as of January 15, 2010

## German

German Wikipedia has an article on:
Pi
Wikipedia de

### Noun

Pi n.
1. (mathematics) pi, an important constant
2. pi (greek letter)

#### Synonyms

• Kreiszahl f. (2)

# Simple English

Pi, or π, is the mathematical constant equal to a circle's circumference divided by its diameter.

## Contents

Pi may also refer to:

## Letter Pi (π)

]]

• Pi, the Greek letter
• Lowercase π:
• π (film), a 1998 film directed by Darren Aronofsky.
• $\pi\left(x\right)\,\!$ (pi function), the prime-counting function
• Pi bond
• Osmotic pressure
• A song from the Kate Bush album Aerial
• Shorthand for pion
• A men's fragrance from Givenchy
• The symbol for political life used in Boethius' Consolation of Philosophy
• Uppercase Π:
• The product symbol
• $\Pi\left(x\right)\,\!$ (Pi function), the Gamma function when offset to coincide with the factorial
• $\pi\left(x\right)\,\!$, the reciprocal of the Pi function

## Pi as a word

• Pi (instrument)
• Pi (.hack), a fictional character from the multimedia franchise .hack//G.U.
• Pi Magazine
• Pi Recordings, a record label
• "Pi", a song from the Hard 'n Phirm album Horses and Grasses, which the song ends at 3:14
• Pi, a member of Team Robo in Sega Soccer Slam
• Pi, a fictional character in the novel Life of Pi
• Pi County, county in Sichuan, China

# Citable sentences

Up to date as of December 01, 2010

Here are sentences from other pages on Pi, which are similar to those in the above article.