Ferguson                    1946                   620  
Ferguson                    Jan. 1947              710  
Ferguson and Wrench         Sep. 1947              808  
Smith and Wrench            1949                 1,120  
Reitwiesner et al. (ENIAC)  1949                 2,037  
Nicholson and Jeenel        1954                 3,092  
Felton                      1957                 7,480  
Genuys                      Jan. 1958           10,000  
Felton                      May  1958           10,021  
Guilloud                    1959                16,167  
Shanks and Wrench           1961               100,265  
Guilloud and Filliatre      1966               250,000  
Guilloud and Dichampt       1967               500,000  
Guilloud and Bouyer         1973             1,001,250  
Miyoshi and Kanada          1981             2,000,036  
Guilloud                    1982             2,000,050  
Tamura                      1982             2,097,144  
Tamura and Kanada           1982             4,194,288  
Tamura and Kanada           1982             8,388,576  
Kanada, Yoshino and Tamura  1982            16,777,206  
Ushiro and Kanada           Oct. 1983       10,013,395  
Gosper                      1985            17,526,200  
Bailey                      Jan. 1986       29,360,111  
Kanada and Tamura           Sep. 1986       33,554,414  
Kanada and Tamura           Oct. 1986       67,108,839  
Kanada, Tamura, Kubo et al  Jan. 1987      134,217,700  
Kanada and Tamura           Jan. 1988      201,326,551  
Chudnovskys                 May  1989      480,000,000  
Chudnovskys                 Jun. 1989      525,229,270  
Kanada and Tamura           Jul. 1989      536,870,898  
Kanada and Tamura           Nov. 1989    1,073,741,799  
Chudnovskys                 Aug. 1989    1,011,196,691  
Chudnovskys                 Aug. 1991    2,260,000,000  
Chudnovskys                 May  1994    4,044,000,000  
Takahashi and Kanada        Jun. 1995    3,221,225,466  
Takahashi and Kanada        Aug. 1995    4,294,967,286
Takahashi and Kanada        Oct. 1995    6,442,450,938

Bailey, Borwein, Plouffe    Nov. 1995    40,000,000,000 (hexa 921C73C6838FB2)
Bellard                     Jul. 1996   200,000,000,000 (hexa 1A10A49B3E2B82A4404F9193AD4EB6) 
Bellard                     Oct. 1996   400,000,000,000 (hexa 9C381872D27596F81D0E48B95A6C46)

Pi Facts

I bet you don't know this much about pi!

• Pi is the number of times a circle's diameter will fit around its circumference
• Pi occurs in hundreds of equations in many sciences including those describing the DNA double helix, a rainbow, ripples spreading from where a raindrop fell into water, superstrings, Einstein's Gravitational Field Equation, normal distribution, distribution of primes, geometry problems, waves, navigation....
• It is easy to prove that if you have a circle that fits exactly inside a square, then

p = 4 x (Area of circle) / (Area of square)

• Pi does not have to be written in decimal (base 10) notation (3.14159265....). Here it is in binary (base 2) notation:

11.0010010000111111011010101000100010000101101000110000100011010011

You can do lots more stuff with Pi when it is in binary format - like drawing weird pictures of it, or even listening to it. As Pi has an infinite number of places, it is quite possible that any message you liked could be heard somewhere in Pi. It has even been suggested it contains the VOICE OF GOD. In Carl Sagan's book 'Contact' the places of Pi (in base 11) are found to contain a message from the beings that built the universe

• Satan does not appear in Pi too quickly: The first time '666' appears is at position 2440
• Half the circumference of a circle with radius 1 is exactly Pi. The area inside that circle is also exactly Pi!
• It is impossible to 'square the circle'. i.e.: You can't draw a square with the same area as a circle using standard / Euclidean straightedge and compass construction in a finite number of steps. The Greeks were obsessed with trying to do this
• In ancient Greece the symbol p was used to denote the number 80
• The digits of Pi appear 'random' and yet they describe something essential to the universe - No pattern emerges and yet it is predestined. The sequence of digits has so far passed all known tests for randomness. 'Approximate Entropy' can be used to establish just how random a number is - it turns out that Pi is more random than the square root of 2, which itself is more random than the square root of 3.
• There is no zero in the first 31 digits of Pi
• The fraction (22 / 7) is a well-used approximation of Pi. It is accurate to 0.04025%
• Another fraction used as an approximation to Pi is (355 / 113) which is accurate to 0.00000849%
• A more accurate but less easily remembered fraction is (104348 / 33215) as an approximation to Pi. This is accurate to 0.00000001056%
• If you have a uniform grid of parallel lines, unit distance apart and you drop a needle length k < 1 onto the grid, then the probability the needle falls across a line is 2k / p
• Taking the first 6,000,000,000 decimal places of Pi, this is the distribution:

0 occurs 599963005 times

1 occurs 600033260 times

2 occurs 599999169 times

3 occurs 600000243 times

4 occurs 599957439 times

5 occurs 600017176 times

6 occurs 600016588 times

7 occurs 600009044 times

8 occurs 599987038 times

9 occurs 600017038 times

This shows NO unusual deviation from expected 'random' behavior

• The first 39 digits of Pi suffice for any application imaginable. i.e.: Calculations involving circles the size of the Universe will have an accuracy to the size of a PROTON if Pi is taken to 39 decimal places
• Pi is irrational. An irrational number is a number that cannot be expressed in the form (a / b) where a and b are integers
• Pi is a 'transcendental' number. This means that it is not the solution to any finite polynomial (e.g.: lots of numbers added in a series) with whole number coefficients. This is why it is impossible to square the circle
• It is not known if Pi is 'normal'. That is if all numbers appear equally often forever. For all we know after a Billion Billion decimal places we get a random series of 1's and 0's (00101011101011000010111010....) No one has proved that Pi is not normal so people generally assume that it is
• There are MANY irrational numbers very close to integers of the form ep ^n0.5where n is an integer. For example e p ^1630.5 is seriously close to 262,537,412,640,768,744
• The probability of 2 large random numbers having no common factor (coprime) is 6/(p ²)
• Indiana State Legislature House Bill No. 246, 1897, set the value of Pi to a to 3. Duh! Actually the wording of the bill is really unclear and can interpreted in various ways. Other interpretations include 3.2, 4 and 9.2376. The bill goes on in a confused way to contradict other areas of elementary geometry, and to contradict itself. It got passed all the way up to Senate level, then a Mathematics Professor noticed it by chance, informed the Senate, and it was postponed indefinitely at its second reading.
• Pi was calculated to 2,260,321,363 decimal places in 1991 by the Chudnovsky brothers in New York
• The Babylonians found the first known value for Pi in around 2000BC -They used (25/8)
• The Egyptians used Pi = 3 but improved this to (22 / 7). They also used (256/81). If you imagine a circle in the Great Pyramid at Giza in Egypt like this:

Then the circle's circumference is twice the base length of the pyramid, and the circle's area is equal to the pyramid's vertical sectional area through the peak. The ratio of the perimeter of the base of the Great Pyramid to its height is twice Pi. The same ratio for the Pyramid of the Sun in Mexico is four times Pi. Both are built to an accuracy of a few inches.

• The D&M Pyramid (a geological feature on Mars) is situated at 40.868 degrees North, which is exactly equal to arcTan (^e/p ). Many of its internal angles also share trigonometric configurations of e and Pi.
• 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
• The Bible uses a value of Pi of 3. Here is a verse from I Kings 7,23:

And he made a molten sea, ten cubits from one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about

• The first person to use the Greek letter p 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
• The Pi memory champion is Hiroyoki Gotu (21 years old) who memorized an amazing 42,000 digits. Woooooooooooooooooh!
• There are many formulae for Pi. They are completely arithmetical despite the fact that Pi arises from Geometry:

Wallis: p = (8 / 3) . (4.4.6.6.8.8.10.10.12...) / (3.5.5.7.7.9.9.11.11...)

This is very slow to converge - After 7000 terms it is accurate to 7 decimal places (averaging terms 7000 and 7001)

Gregory (also attributed to Leibniz): p = 4 - (4/3) + (4/5) - (4/7) + (4/9) - (4/11) + ...

This is about the same - After 7000 terms it is also accurate to 7 decimal places

p = Sqrt ( 12 - (12/2²) + (12/3²) - (12/4²) + ...

This is slightly faster - After just 1100 terms it is accurate to at least 9 decimal places

There are many other such formulas

·         Ramanujan developed a formula for Pi which adds 8 decimal places each term!

This is the formula generally used by supercomputers to calculate Pi

·         David Bailey, Peter Borwein and Simon Plouffe recently computed the ten billionth digit in the hexadecimal expansion of Pi. They used an astonishing formula:

which enables one to calculate the n-th digit of Pi without being forced to calculate all the preceding n -1 digits. No one had previously even conjectured that such a digit-extraction algorithm for Pi was possible

• Choose 2 random numbers x and y between 1 and -1. Do this N times. If for M of those N times x² + y² < 1 then as N approaches infinity, p = 4(M/N)
• The area of a circle is p r²
• The volume of a sphere is ⁴/_3p r³ and its surface area is 4p 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)
• Most people would say that a circle has no corners - but it is more accurate to say that it has an infinite number of corners
• Pi is approximately ⁶/₅ Æ ² where Æ is the GOLDEN MEAN - another interesting number that appears all over nature and in art
• Pi is of course the ratio of a circle's circumference to its diameter. If we 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
• If you take 10 million random digits, statistically on average you would expect 200 cases where you get 5 digits in a row the same. If you take 10 million digits of Pi - guess what... you get exactly 200
• Euler showed that e^ip + 1 = 0 Where i is the square-root of -1
• Landau showed that p /2 is the value of x between 1 and 2 for which cos x vanishes
• p = n sin(180/n). Try it with different values of n. Your calculator will need to be in degrees (not radians) mode. This equation gets more accurate with larger values of n
• The following are all NEARLY Pi:

10^1/2

Cube root of 31

⁶⁶⁶/₂₁₂

¹⁰/p

(97 + ⁹/₂₂)^1/4

⁹/₅ + (⁹/₅)^1/2

(19 (7)^1/2) / 16

(2)^1/2 + (3)^1/2

1.1 x 1.2 x 1.4 x 1.7

(²⁹⁶/₁₆₇) ²

• Kochansky found that Pi is NEARLY a root of the equation 9x⁴ - 240x² + 1492
• A year is about p x10⁷ seconds
• 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
• It is not known if the following are 'irrational':

p + e

p /e

lnp

• If you approximate the circle with a radius of 1 as a 100 sided polygon, then its area is only accurate to 1 decimal place or 0.0658%
• At position 762 there are six nines in a row. This is known as the Feynman Point
• Pi in base Pi is 10
• p = 4(¹/₂)!²
• All permutations of 3 arbitrary digits appear somewhere in Pi
• In 1931 a Cleveland businessman published a book announcing that Pi is exactly 256/81
• Starting with the conventional 5-by-5 magic square, and then substituting the nth digit of pi for each number n in the square, we obtain a new array of numbers. The sum of the numbers in every column is duplicated by a sum of numbers in every row.
• Write the letters of the English alphabet, in capitals, clockwise around a circle, and cross out the letters that have right-left symmetry, A, H, I, M, etc. The letters that remain group themselves in sets of 3, 1, 4, 1, 6"
• At one time it was thought there was an illness attached to trying to 'square a circle' called Morbus Cyclometricus.
• After saying (correctly) that Pi / 2 is the value of x between 1 and 2 for which Cos x vanishes, Edmund Landau was dismissed from his position in 1934 for teaching in an 'un-German' style.
• In the following series of natural numbers, constructed by taking successively larger strings of digits from the beginning of the decimal expansion of the number Pi: 3, 31, 314, 31415, 314159, 3141592, etc. The first thousand numbers of the series include only 4 primes.
• The earliest known reference to Pi is on a Middle Kingdom papyrus scroll, written around 1650 BC by Ahmes the scribe.
• Decimal places 7, 22, 113 and 355 of Pi are all the number 2. (22/7 and 355/113 are good approximations of Pi)
• The sequence 314159 re-appears in the decimal expansion of Pi at place 176451. This sequence appears 7 times in the first 10 million places (not including right at the start)
• A.C. Aitken from Edimbourg University is able, for the first 2000 decimals of Pi, to tell you what number appears in position x
• (p + 20)ⁱ is almost equal to -1 (where i is the square-root of minus one)
• If you approximate the circle as a square then the value you get for Pi is about 10% out. It just goes to show that you shouldn't approximate the circle as a square. Well you wouldn't make square wheels would you?
• Here's a Pi limerick:

Three point one four one five nine two

Its been around forever - it's not new

It appears everywhere

In here and in there

Its irrational I know but its true!

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