Pi Day – 3.14 Tanya Khovanova

Definition  The mathematical constant Π is an irrational real number, approximately equal to , which is the ratio of a circle's circumference to its diameter in Euclidean geometry.  Π =

Pi jokes  What do you get when you take the sun and divide its circumference by its diameter?  Pi in the sky.

Pi Mnemonics (word length)‏  How I wish I could calculate pi.  How I wish I could enumerate Pi easily, since all these horrible mnemonics prevent recalling any of pi's sequence more simply.  How I want a drink, alcoholic of course, after the heavy chapters involving quantum mechanics. One is, yes, adequate even enough to induce some fun and pleasure for an instant, miserably brief.  Do you see a problem with mnemonics?

Pi Mnemonics  The problem is with digit zero  Luckily first zero occurs at 33 digit  A Pi story: /zimaths/pimnem.htm  t.htm

Limerick  There once was a fellow from Greece, Who forgot pi's last decimal piece. So he used electronics To collect pi mnemonics... Now he's hooked, and there is no release.

Calculating Pi  The length of the circumference is between the lengths of the perimeters of an outside square and inside square. Let us try to calculate:

Let’s calculate  Outside perimeter: 4.  Inside perimeter: 2√2 =  The average is

Archimedes Calculation  Archimedes of Syracuse (third century BC)  Perimeters of 96-sided polygons inscribing a circle and inscribed by it  π is between 223⁄71 and 22⁄7  The average is

Devoted his life to 35 digits of Pi  By 1610, the German mathematician Ludolph van Ceulen computed the first 35 decimal places of π.  He was so proud of this accomplishment that he had them inscribed on his tombstone.

Current Record  December 2002, Yasumasa Kanada of Tokyo University correctly computed π to 1.24 trillion digits  How can they check correctness without someone else recomputing it?

Checking Correctness  Use two different power series for calculating the digits of π.  For one, any error would produce a value slightly high.  For the other, any error would produce a value slightly low.  As long as the two series produced the same digits, there was a very high confidence that they were correct.

Some Famous Formulae  Which one converges the fastest?

Do we need that many digits?  While the value of pi has been computed to billions of digits, practical science and engineering will rarely require more than 100 digits.  As an example, computing the circumference of a circle the size of the Milky Way with a value of pi truncated at 40 digits would produce an error margin of less than the diameter of a proton.  But we can use them for testing computers

Simplifying Pi  'Tis a favorite project of mine  A new value of pi to assign.  I would fix it at 3  For it's simpler, you see,  Than 3 point

Pi and the Bible  A little known verse of the Bible reads  And he made a molten sea, ten cubits from the 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. (I Kings 7, 23)

Indiana Pi bill  Dr. Edwin J. Goodwin, M.D., was a mathematical hobbyists trying to square the circle. Dr. Goodwin thought he had succeeded, and decided that the State of Indiana should be the first beneficiary of this "new mathematical truth."  In 1897, Dr. Goodwin wrote a bill incorporating his new ideas, and persuaded his State Representative to introduce it.  Dr. Goodwin had copyrighted his solution to squaring the circle, and his idea was to allow Indiana to use these new facts in its schools free of charge. People in the rest of the country and the world would have to pay him a royalty.  What do you think happened next?

Bill Conclusion  Towards the end the bill says "the ratio of the diameter and circumference is as five-fourths to four." That is, Pi is 3.2.  A nice, round, wrong number.

Lucky Chance  On February 5, the head of the Purdue University Mathematics Department, Professor Waldo, was in the Statehouse lobbying for the University's budget appropriation.  He was astonished to find the General Assembly debating mathematical legislation. Naturally, he listened in. Naturally, he was horrified. He heard a Representative speak for the bill:  The case is perfectly simple. If we pass this bill which establishes a new and correct value of pi, the author offers our state without cost the use of his discovery and its free publication in our school textbooks, while everyone else must pay him a royalty.  After the debate, a Representative offered to introduce him to Dr. Goodwin. Professor Waldo replied that he was already acquainted with as many crazy people as he cared to know.

Approximation by Fractions  Usually when we try to approximate a number as a fraction m/n with a given n, our precision is?  1/2n.  Continued fractions allow us to be much more precise.  m/n approximation by continued fraction has precision 1/n^2.

Approximation by Continued Fractions  3  22/7 = … – accuracy %;  333/106  355/113 – accuracy %;  /33102  /33215 – accuracy %.

Area  Area of a circle with radius r =π r 2  Another way to calculate π: Draw an imaginary circle of radius r on the grid centered at the origin. Calculate the number of grid points inside the circle – a Π is approximated as a/r 2

Area  Said the Mathematician, "Pi r squared.“  Said the Baker, "No! Pie are round, cakes are square!"

Another way of calculating Pi  Drop a needle of length 1 repeatedly on a surface containing parallel lines drawn 1 units apart.  If the needle is dropped n times and x of those times it comes to rest crossing a line, then one may approximate π using: 2n/x  Buffon’s Needle -

Randomness of Pi Digits  So far the digits of Pi passed all statistical randomness tests.  What does it tell us about the statistics?

Randomness  Property of randomness – can’t compress  Digits of Pi highly non-random – we can compress to one Greek letter

Pi digits statistics  But it is not even proved that all the digits occur infinitely many times

Celebrating Pi Day  Pi Minute – 3/14 1:59  Pi Moment – 3/14 1:59:27  In Europe Pi Approximation Day – 22 July.  Albert Einstein’s birthday  Any stories?  Happy Pi Day!