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'''Pi''', or '''Archimedes' constant''', is an important mathematical constant defined as the ratio of the circumference of a [[circle]] to its diameter, and represented by the 16th letter of the Greek alphabet, '''<big><math>\pi</math></big>'''.  It was first used with its current meaning in 1706 by a Welsh mathematician William Jones.<ref name="hist">{{cite web

'''Pi''', or '''Archimedes' constant''', is an important [[mathematics|mathematical]] constant defined as the [[ratio]] of the [[circumference]] of a [[circle]] to its [[diameter]], and represented by the 16th letter of the Greek alphabet, '''<big><math>\pi</math></big>'''.  It was first used with its current meaning in 1706 by a Welsh mathematician William Jones.<ref name="hist">{{cite web

| url = http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html  

| url = http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Pi_through_the_ages.html  

| title = A history or Pi

| title = A history or Pi

| accessdate = 2012-02-11}}</ref>, who selected  '''<big><math>\pi</math></big>''' because it is the first letter of the Greek word for [[perimeter]] (''περίμετρος''), i.e. the circumference of a circle is its perimeter.  A Swiss mathematician, Leonhard Euler, brought the concept into general use in 1737.

| accessdate = 2012-02-11}}</ref>, who selected  '''<big><math>\pi</math></big>''' because it is the first letter of the Greek word for [[perimeter]] (''περίμετρος''), i.e. the circumference of a circle is its perimeter.  A Swiss mathematician, [[Leonhard Euler]], brought the concept into general use in 1737.

The value of '''<big><math>\pi</math></big>''' is approximately 3.1416, or 22/7. The exact value cannot be expressed as a fraction or as a decimal number, regardless of how many digits are used. Johann Heinrich Lambert proved this in 1761 by showing that '''<big><math>\pi</math></big>''' is an [[irrational number]], which means that it can't be expressed as a fraction.<ref>{{cite web

The value of '''<big><math>\pi</math></big>''' is approximately 3.1416, or 22/7. The exact value cannot be expressed as a [[fraction]] or as a [[decimal]] number, regardless of how many [[digit]]s are used. Johann Heinrich Lambert proved this in 1761 by showing that '''<big><math>\pi</math></big>''' is an [[irrational number]], which means that it can't be expressed as a fraction.<ref>{{cite web

| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lambert.html

| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lambert.html

| title = Biography of Johann Heinrich Lambert

| title = Biography of Johann Heinrich Lambert

| accessdate = 2012-02-12}}</ref> In 1882, Carl Louis Ferdinand von Lindeman proved that '''<big><math>\pi</math></big>''' is also a [[transcendental number]], which means that it can't be expressed as the solution to any simple equation.<ref>{{cite web

| accessdate = 2012-02-12}}</ref> In 1882, Carl Louis Ferdinand von Lindeman proved that '''<big><math>\pi</math></big>''' is also a [[transcendental number]], which means that it can't be expressed as the solution to any simple [[equation]].<ref>{{cite web

| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lindemann.html

| url = http://turnbull.mcs.st-and.ac.uk/~history/Biographies/Lindemann.html

| title = Biography of Carl Louis Ferdinand von Lindemann

| title = Biography of Carl Louis Ferdinand von Lindemann

Mathematicians have worked for centuries to calculate '''<big><math>\pi</math></big>''' to more and more decimal places. To some extent, the progress of mathematics, or at least of computation, can be gauged by the progress in the number of digits to which '''<big><math>\pi</math></big>''' has been calculated.  

Mathematicians have worked for centuries to calculate '''<big><math>\pi</math></big>''' to more and more decimal places. To some extent, the progress of mathematics, or at least of computation, can be gauged by the progress in the number of digits to which '''<big><math>\pi</math></big>''' has been calculated.  

Some ancients expressed '''<big><math>\pi</math></big>''' by using fractional approximations. The Rhind, or Ahmes Papyrus (''c.'' 1650 B.C.)<ref>http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html</ref><ref>https://www.math.tamu.edu/~don.allen/history/egypt/node3.html</ref> has shown that the ancient Egyptians had determined the value for '''<big><math>\pi</math></big>''' to be 3.1605. The Babylonian value from the same era was 3 1/8 = 3.125.<ref>Boyer, ''A History of Mathematics'', 2nd edition</ref>, coming to within 1 percent accuracy for both<ref>https://www.maa.org/press/periodicals/convergence/mathematical-treasure-old-babylonian-area-calculation</ref>.

Some ancients expressed '''<big><math>\pi</math></big>''' by using fractional approximations. The Rhind, or Ahmes Papyrus (''c.'' 1650 B.C.)<ref>http://www.math.buffalo.edu/mad/Ancient-Africa/mad_ancient_egyptpapyrus.html</ref><ref>https://www.math.tamu.edu/~don.allen/history/egypt/node3.html</ref> has shown that the [[ancient Egypt]]ians had determined the value for '''<big><math>\pi</math></big>''' to be 3.1605. The [[Babylonia]]n value from the same era was 3 1/8 = 3.125.<ref>Boyer, ''A History of Mathematics'', 2nd edition</ref>, coming to within 1 percent accuracy for both<ref>https://www.maa.org/press/periodicals/convergence/mathematical-treasure-old-babylonian-area-calculation</ref>.

[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of '''<big><math>\pi</math></big>'''.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref>, using regular polygons with a total of 96 sides, within and circumscribing a circle, and in about 225 B.C. he came up with a formula between the folowing numbers:  

[[Archimedes]] of Syracuse (287-212 BC) carried out "the first theoretical calculation" of '''<big><math>\pi</math></big>'''.<ref>[http://veling.nl/anne/templars/Pi_through_the_ages.html Pi through the ages]</ref>, using regular polygons with a total of 96 sides, within and circumscribing a circle, and in about 225 B.C. he came up with a formula between the folowing numbers:  

[[Gottfried Wilhelm von Leibniz]] (1646-1716) published a series for '''<big><math>\pi</math></big>''' in 1673, coming up with the formulas:

[[Gottfried Leibniz|Gottfried Wilhelm von Leibniz]] (1646-1716) published a series for '''<big><math>\pi</math></big>''' in 1673, coming up with the formulas:

<center><math>\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0.78539 \, 81633 \, 9744 \ldots</math><ref>https://oeis.org/A003881</ref></center>

<center><math>\dfrac \pi 4 = 1 - \dfrac 1 3 + \dfrac 1 5 - \dfrac 1 7 + \dfrac 1 9 - \cdots \approx 0.78539 \, 81633 \, 9744 \ldots</math><ref>https://oeis.org/A003881</ref></center>

</math></center>

</math></center>

In 1706, John Machin, secretary of England's Royal Society, developed a quickly converging formula for '''<big><math>\pi</math></big>''' and used it to calculated the first 100 digits. In 1844, the idiot savant Johann Dase of Hamburg used Machin's formula to calculate 200 digits in less than two months.<ref>Beckmann</ref> In contrast, William Shanks spent twenty years calculating '''<big><math>\pi</math></big>''' to 707 places, a task he completed in 1873. In 1945, it was discovered that only the first 527 of Shanks's digits were correct.

In 1706, John Machin, secretary of [[England]]'s Royal Society, developed a quickly converging formula for '''<big><math>\pi</math></big>''' and used it to calculated the first 100 digits. In 1844, the idiot savant Johann Dase of [[Hamburg]] used Machin's formula to calculate 200 digits in less than two months.<ref>Beckmann</ref> In contrast, William Shanks spent twenty years calculating '''<big><math>\pi</math></big>''' to 707 places, a task he completed in 1873. In 1945, it was discovered that only the first 527 of Shanks's digits were correct.

ENIAC, the first electronic computer, took seventy hours to calculate 2,037 digits in 1949. In 2008, the first million digits of '''<big><math>\pi</math></big>''' were published on Project Gutenberg.<ref>Hemphill, Scott, ''[https://www.gutenberg.org/ebooks/50 Pi to 1,000,000 places]''.</ref> In 2014, the anonymous programmer Houkouonchi calculated the first 13.3 trillion digits of '''<big><math>\pi</math></big>''' in 208 days.<ref>Yee, Alexander, "[http://www.numberworld.org/y-cruncher/ y-cruncher - A Multi-Threaded Pi-Program]"</ref> This result has not been published.

ENIAC, the first electronic [[computer]], took seventy hours to calculate 2,037 digits in 1949. In 2008, the first million digits of '''<big><math>\pi</math></big>''' were published on [[Project Gutenberg]].<ref>Hemphill, Scott, ''[https://www.gutenberg.org/ebooks/50 Pi to 1,000,000 places]''.</ref> In 2014, the anonymous programmer Houkouonchi calculated the first 13.3 trillion digits of '''<big><math>\pi</math></big>''' in 208 days.<ref>Yee, Alexander, "[http://www.numberworld.org/y-cruncher/ y-cruncher - A Multi-Threaded Pi-Program]"</ref> This result has not been published.

=='''<big><math>\pi</math></big>''' in mathematics==

=='''<big><math>\pi</math></big>''' in mathematics==

It's impossible to overestimate the importance of '''<big><math>\pi</math></big>''' (and ''[[e]]'') for mathematics. These values are tied by [[Euler's identity]]:

It's impossible to overestimate the importance of '''<big><math>\pi</math></big>''' (and ''[[e]]'') for mathematics. These values are tied by [[Euler's Formula|Euler's identity]]:

:<math>e^{\pi \imath} +1 = 0</math>.

:<math>e^{\pi \imath} +1 = 0</math>.