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(Pi): Facts and Figures
 Edited by The Starman





Pi () to 100 decimal places is:

      3. 14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679


A  Circle  of   (Pi)

is the ratio of a circle's circumference to its diameter (π = c / d) which also means
  that the circumference of a circle is Pi times its diameter (c = πd) or twice Pi times its radius (c = 2πr). If we make the diameter 1 unit, then its  
circumference will equal π units. The area (A) inside a circle is Pi times the radius squared or A = π r2 (see links below for proof *). If a circle has a diameter of 2 units, then its area will equal π units.
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*
For various proofs that A = π r2, try these pages: I'm still researching this!.

is the 16th letter of the Greek alphabet (it also denoted the number 80 in ancient Greece). Note that the pronunciation of this letter in Greek is like the English word 'Pea' (the same way they say the name of the letter 'P') or perhaps like the p and i in the word 'Pit.' But it's NEVER pronounced like the English word 'Pie' in Greece! To create that type of sound, Greek could use the diphthong 'ai' (it means two vowels together; like the 'oi' in the English word 'oil'); so, 'pai' in Greek would sound like 'pie' in English.

So, how did this Greek letter become the Mathematical symbol it's known for today?
In the mid to late 1600's, some mathematicians were using (a lowercase Pi divided by a lowercase Greek Delta) to note the perimeter divided by the diameter of a circle; 3.14159... . The symbol, π, was used in this manner because it's the first letter of the Greek words, perifereia (periphery) and perimetroV (or perimeter; which could also mean the circumference of a circle). Many geometrical terms in English come from Greek, including of course, geometry itself, gewmetria (gew = earth + metria = measure).

But William Jones is often cited as being the first author to use the Greek letter π for this constant in his 1706 work, A New Introduction to Mathematics.


If you bring everything up one dimension to get a 3D value for Pi. The ratio of a sphere's surface area to the area of the circle seen if you cut the sphere in half is exactly 4.

The volume of a sphere is 4/3*r3 and its surface area is 4/*r2.

The circle is the shape with the least perimeter length to area ratio (for a given shape area). Centuries ago mathematicians were also philosophers. They considered the circle to be the 'perfect' shape because of this. The sphere is the 3D shape with the least surface area to volume ratio (for a given volume)





Fractions approaching Pi
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Fraction Decimal Approximation Percent Deviation from Pi
25 / 8   3.125 (exactly)   -0.52816 %
22 / 7   3.142857142857   +0.04025 %
333 / 106   3.14150943396   -0.00264896 %
355 / 113   3.14159292035   +0.00000849 %
104348 / 33215   3.14159265392   +0.00000001 %
837393900 / 266550757   3.14159265358980   +2.2 x 10^(-13) %

The following lists the repeating decimal portion (underlined) of fractions related to those approaching the value of Pi:


    15/106 = 0.141509433962264

    16/113 = 0.14159292035398230088495575221238938053097345132743
               36283185840707964601769911504424778761061946902654
               867256637168

4703/33215 = 0.14159265392142104470871594

The Babylonians were the first to use (25 / 8)

The fraction (22 / 7) is probably the most famous approximation for Pi (about 3.142857). It's actually composed of 3 + (1 / 7) and the fractional part (1/7) is famous in its own right as being the first fraction to produce a 'repeating decimal' number with more than a single digit! (1/7 = 0.142857 142857 142857 ... etc.)

(355 / 113) which is about 3.141592920354 and only 0.00000849% larger than Pi.





If one were to find the circumference of a circle the size of the known universe, requiring that the circumference be accurate to within the radius of one proton only 39 decimal places of Pi would be necessary.





Some Facts about Pi
(in Chronological Order)

The Babylonians are credited as having recorded the first value for Pi (around 2000 BCE) and they used (25 / 8).

The earliest known reference to Pi is on a Middle Kingdom papyrus scroll, written around 1650 BC by Ahmes the scribe.

In around 200 BC Archimedes found that Pi was between (223/71) and (22/7). His error was no more than 0.008227 %. He did this by approximating a circle as a 96 sided polygon.

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Ludolph Van Ceulen (1540 - 1610) spent most of his life working out Pi to 35 decimal places. Pi is sometimes known as Ludolph's Constant

Another name for Pi in Germany is 'die Ludolphsche Zahl' after Ludolph van Ceulen, the German mathematician who devoted his life to calculating 35 decimals of pi.

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In 1706 William Jones first gave the Greek letter π its current mathematical definition.

The first person to use the Greek letter Pi was Welshman William Jones in 1706. He used it as an abbreviation for the periphery of a circle with unit diameter. Euler adopted the symbol and it quickly became a standard notation.

A rapidly converging formula for calculation of Pi found by Machin in 1706 was pi/4 = 4 * arctan (1/5) - arctan (1/239).

 

Basic facts on π (including Irrationality and Transcendence):
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In 1768 Johann Lambert proved Pi is irrational.

Pi is irrational. An irrational number is a number that cannot be expressed in the form (a / b) where a and b are integers.

In 1882 Ferdinand Lindemann, proved the transcendence of Pi.

Pi is a transcendental number. (Transcendental means= Not capable of being determined by any combination of a finite number of equations with rational integral coefficients.)

Pi is a 'transcendental' number. This means that it is not the solution to any finite polynomial (eg: lots of numbers added in a series) with whole number coefficients. This is why it is impossible to square the circle.

It is easy to prove that if you have a circle that fits exactly inside a square, then pi = 4 times (Area of circle) / (Area of square)