Pi Day – 3.14

Tanya Khovanova

Definition

The mathematical constant Π is an irrational real number, approximately equal to 3.14159, which is the ratio of a circle's circumference to its diameter in Euclidean geometry.

Π = 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209...

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:

http://www.uz.ac.zw/science/maths/zimaths/pimnem.htm

http://users.aol.com/s6sj7gt/cadtext.htm

Limerick

There once was a fellow from Greece,Who forgot pi's last decimal piece.So he used electronicsTo 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 = 2.82842712 The average is 3.41421356

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 3.1419.

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 1 4 1 5 9

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 = 3.142857… – accuracy

0.04025%; 333/106 355/113 – accuracy 0.00000849%; 103993/33102 104348/33215 – accuracy

0.00000001056%.

Area

Area of a circle with radius r =π r2

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/r2

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 - http://www.angelfire.com/wa/hurben/buff.html

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!