The First Thousand Decimal Digits of Pi

3.141592653 5897932384 6264338327 9502884197 1693993751 0582097494 4592307816 4062862089 9862803482 5342117067 9821480865 1328230664 7093844609 5505822317 2535940812 8481117450 2841027019 3852110555 9644622948 9549303819 6442881097 5665933446 1284756482 3378678316 5271201909 1456485669 2346034861 0454326648 2133936072 6024914127 3724587006 6063155881 7488152092 0962829254 0917153643 6789259036 0011330530 5488204665 2138414695 1941511609 4330572703 6575959195 3092186117 3819326117 9310511854 8074462379 9627495673 5188575272 4891227938 1830119491 2983367336 2440656643 0860213949 4639522473 7190702179 8609437027 7053921717 6293176752 3846748184 6766940513 2000568127 1452635608 2778577134 2757789609 1736371787 2146844090 1224953430 1465495853 7105079227 9689258923 5420199561 1212902196 0864034418 1598136297 7477130996 0518707211 3499999983 7297804995 1059731732 8160963185 9502445945 5346908302 6425223082 5334468503 5261931188 1710100031 3783875288 6587533208 3814206171 7766914730 3598253490 4287554687 3115956286 3882353787 5937519577 8185778053 2171226806 6130019278 7661119590 9216420198

The mathematical constant, pi (π), is an irrational number; that is, its exact value cannot be expressed as the ratio of two integers. That didn't stop my using 22/7 as a value of pi in the 7th grade. This ratio, 22/7 = 3.14285714286, is quite close to the value of pi (3.14159265358...), differing by just 0.04%, so using this ratio gives only a very small error. Although 10 trillion digits of pi have been calculated, very few digits are needed in practice. In fact, the circumference of the universe can be calculated from its diameter to one inch using just nine decimal places of pi. This interesting pi fact, and many others, can be found in my book,

53938/17169 = 3.14159 24049 15836... (2.49×10The square root of 2 (√2, 1.414213562373095...) is another irrational number with a plethora of rational approximations. The simple fraction, 99/70, gives the √2 to 72 ppm, while 665,857/470,832, which has a square of 2.0000000000045..., is extremely accurate (1.6×10^{-7})

87308/27791 = 3.14159 26019 21485... (5.17×10^{-8})

103638/32989 = 3.14159 26520 96153... (1.49×10^{-9})

208341/66317 = 3.14159 26534 67436... (1.22×10^{-10})

312689/99532 = 3.14159 26536 18936... (2.91×10^{-11})

3126535/995207 = 3.14159 26535 88650... (1.14×10^{-12})

*There's something about this portrait of Leonhard Euler that screams, "savant."Euler named these numbers the Pell numbers, since he mistakenly assigned priority to English mathematician, John Pell (1611-1685).(A 1753 pastel on paper portrait of Euler by Jakob Emanuel Handmann (1718-1781), via Wikimedia Commons. The portrait is in the Kunstmuseum Basel.)*

The Pell numbers grow exponentially, as can be seen from the first few terms, 0, 1, 2, 5, 12, 29, 70, 169, 408, 985, 2378, 5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210, 6625109, 15994428, 38613965, 93222358, 225058681, 543339720, 1311738121, 3166815962, 7645370045, 18457556052, 44560482149, 107578520350, 259717522849. A small number of Pell numbers are prime (2, 5, 29, 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701..., OEIS A086383), and these would be a tool to test your GCD algorithm. Since Pell numbers relate to the √2, computer scientist, Donald Knuth, noted that the Pell series can be used to construct approximate regular octagons with vertices (±(P

*Knuth Pell octagon based on P _{n} = 70, P_{n-1} = 29.*

The vertex points are (99, 0), (70, 70), (0, 99), (-70, 70), (-99, 0), (-70, -70), (0, -99), (70,-70).

(Graphed using Gnumeric.)

- Friedrich Lenz, "The Ratio of Proton and Electron Masses," Phys. Rev., vol. 82, no. 4 (May, 1951), pp. 554–554.
- Dev Gualtieri, "The Amazing Number Pi," Paperback, December 28, 2013, 110 pages, ISBN: 978-0985332587 (via Amazon).
- Lawrence Turner, "Rational Approximations to π," Southwestern Adventist University Web Site.
- Donald E. Knuth,"Leaper graphs," The Mathematical Gazette, vol. 78, no. 483 (November, 1994), pp. 274-297, doi:10.2307/3620202 (via arXiv).