“Enrichment beyond the classroom”
GREECE and ITALY
join a Mediterranean math-collaboration!
etwinning project 2015-2016
Pi Day is celebrated on March 14th (3/14) around the world. Pi (Greek letter “π”) is the symbol used in mathematics to represent a constant — the ratio of the circumference of a circle to its diameter — which is approximately 3.14159.
Pi has been calculated to over one trillion digits beyond its decimal point. As an irrational and transcendental number, it will continue infinitely without repetition or pattern. While only a handful of digits are needed for typical calculations, Pi’s infinite nature makes it a fun challenge to memorize, and to computationally calculate more and more digits.
This page lists a number of infinite expressions of . Proofs are not provided here.
John Wallis (1655) took what can now be expressed as
and without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula for to be
William Brouncker (ca. 1660's) rewrote Wallis' formula as a continued fraction, which Wallis and later Euler (1775) proved to be equivalent. It is unknown how Brouncker himself came up with the continued fraction,
James Gregory (1671) & Gottfried Leibniz (1674) used the series expansion of the arctangent function,
and the fact that arctan(1) = /4 to obtain the series
Unfortunately, this series converges to slowly to be useful, as it takes over 300 terms to obtain a 2 decimal place precision. To obtain 100 decimal places of , one would need to use at least 10^50 terms of this expansion!
History books credit Sir Isaac Newton (ca. 1730's) with using the series expansion of the arcsine function,
and the fact that arctan(1/2) = /6 to obtain the series
This arcsine series converges much faster than using the arctangent. (Actually, Newton used a slightly different expansion in his original text.) This expansion only needed 22 terms to obtain 16 decimal places for .
Leonard Euler (1748) proved the following equivalent relations for the square of ,
Ko Hayashi (1989) found another infinite expression for in terms of the Fibonacci numbers,
This page lists a number of infinite expressions of PI.
Proofs are not provided here.
John Wallis (1655) took what can now be expressed as
and without using the binomial theorem or integration (not invented yet) painstakingly came up with a formula for to be
William Brouncker (ca. 1660's) rewrote Wallis' formula as a continued fraction, which Wallis and later Euler (1775) proved to be equivalent. It is unknown how Brouncker himself came up with the continued fraction,
James Gregory (1671) & Gottfried Leibniz (1674) used the series expansion of the arctangent function,
and the fact that arctan(1) = /4 to obtain the series
Unfortunately, this series converges to slowly to be useful, as it takes over 300 terms to obtain a 2 decimal place precision. To obtain 100 decimal places of , one would need to use at least 10^50 terms of this expansion!
History books credit Sir Isaac Newton (ca. 1730's) with using the series expansion of the arcsine function,
and the fact that arctan(1/2) = /6 to obtain the series
This arcsine series converges much faster than using the arctangent. (Actually, Newton used a slightly different expansion in his original text.) This expansion only needed 22 terms to obtain 16 decimal places for .
Leonard Euler (1748) proved the following equivalent relations for the square of ,
Ko Hayashi (1989) found another infinite expression for in terms of the Fibonacci numbers,
by Lykeio Schimatariou, Greece
...our cinema aproach in youtube...
It is known that number pi, the real number that can be defined as the ratio of the circumference's length of a circle to its diameter in Euclidean geometry , and which is very often used in mathematics, physics and engineering.
It is also known ... as Archimedes' constant (not to be confused with the number of Archimedes). Archimedes determined the first scientifically proven method by which the number is calculated.
Typically approximately n ≈ 3,14 is used. The p first 50 decimal places is:
3.14159 26 535 89 793 23 846 26 433 83 279 50 288 41 971 69 399 37510
What about music and pi?
Watch the video.
M.K.Schimatari, Greece
"What a day!
We loved every moment of online- collaborating! "
Your Greek partners
What about Ramanujan and pi?
Srinivasa Ramanujan FRS was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions.
An equation for me has no meaning,
unless it represents a thought of God.
- Srinivasa Ramanujan
On the 125th anniversary of his birth, India declared the birthday of Ramanujan, December 22, as 'National Mathematics Day.' The declaration was made by Dr. Manmohan Singh in Chennai on December 26, 2011.
Once of the favourite pastimes of Sri Ramanujan was the construction of magic squares, Playing with numbers. This one is built round his date of birth, December 22, 1887. Added any way, vertically, horizontally or diagonally, the numbers add up to 139.
tHE LINK...
qr FOR THE MAGAZINE
