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A Brief History of Pi

Pi is a name given to the ratio of the circumference of a circle to the diameter. That means, for any circle, you can divide the circumference (the distance around the circle) by the diameter and always get exactly the same number. It doesn't matter how big or small the circle is, Pi remains the same. Pi is often written using the symbol and is pronounced "pie", just like the dessert.

A Brief History of Pi
Ancient civilizations knew that there was a fixed ratio of circumference to diameter that was approximately equal to three. The Greeks refined the process and Archimedes is credited with the first theoretical calculation of Pi.

In 1761 Lambert proved that Pi was irrational, that is, that it can't be written as a ratio of integer numbers.

In 1882 Lindeman proved that Pi was transcendental, that is, that Pi is not the root of any algebraic equation with rational coefficients. This discovery proved that you can't "square a circle", which was a problem that occupied many mathematicians up to that time. (More information on squaring the circle.)

How many digits are there? Does it ever end?
Because Pi is known to be an irrational number it means that the digits never end or repeat in any known way. But calculating the digits of Pi has proven to be an fascination for mathematicians throughout history. Some spent their lives calculating the digits of Pi, but until computers, less than 1,000 digits had been calculated. In 1949, a computer calculated 2,000 digits and the race was on. Millions of digits have been calculated, with the record held (as of September 1999) by a supercomputer at the University of Tokyo that calculated 206,158,430,000 digits.
(first 1,000 digits)

More about the History of Pi can be found at the Mac Tutor Math History archives.

Approximation of Pi
Archimedes calculated that Pi was between 3 10/71 and 3 1/7 (also written 223/71 < < 22/7 ). 22/7 is still a good approximation. 355/113 is a better one.

Pi Web Sites
Pi continues to be a fascination of many people around the world. If you are interested in learning more, there are many web sites devoted to the number Pi. There are sites that offer thousands, millions, or billions of digits, pi clubs, pi music, people who calculate digits, people who memorize digits, Pi experiments and more. Check this Yahoo page for a complete listing.

A Cool Pi Experiment
One of the most interesting ways to learn more about Pi is to do pi experiments yourself. Here is a famous one called Buffon's Needle.

In Buffon's Needle experiment you can drop a needle on a lined sheet of paper. If you keep track of how many times the needle lands on a line, it turns out to be directly related to the value of Pi.

Buffon's Needle Simulation Applet (Michael J. Hurben)
Buffon's Needle (George Reese, Office for Mathematics, Science and Technology Education University of Illinois Champaign-Urbana)

Digits of Pi

First 100 digits

3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 ...

First 1000 digits
3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989

Formulas for Pi

Vieta's Formula

2/PI = 2/2 * ( 2 + 2 )/2 * (2 + ( ( 2 + 2) ) )/2 * ...c

Leibnitz's Formula

PI/4 = 1/1 - 1/3 + 1/5 - 1/7 + ...

Wallis Product

PI/2 = 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...

2/PI = (1 - 1/2²)(1 - 1/4²)(1 - 1/6²)...

Lord Brouncker's Formula

4/PI = 1 + 1

----------------

2 + 3²

------------

2 + 5²

---------

2 + 7² ...

(PI²)/8 = 1/1² + 1/3² + 1/5² + ...

(PI²)/24 = 1/2² + 1/4² + 1/6² + ...

Euler's Formula

(PI²)/6 = (n = 1..) 1/n² = 1/1² + 1/2² + 1/3² + ...

(or more generally...)

(n = 1..) 1/n^(2k) = (-1)^(k-1) PI^(2k) 2^(2k) B_(2k) / ( 2(2k)!)

B_(k) = the k ^th Bernoulli number. eg. B₀=1 B₁=-1/2 B₂=1/6 B₄=-1/30 B₆=1/42 B₈=-1/30 B₁₀=5/66. Further Bernoulli numbers are defined as (n 0)B₀ + (n 1)B₁ + (n 2)B₂ + ... + (n (n-1))B_(N-1) = 0 assuming all odd Bernoulli #'s > 1 are = 0. (n k) = binomial coefficient = n!/(k!(n-k)!)

See Power Summations #2 for simplified expressions (without the Bernoulli notation) of these sums for given values of k.