To those who like math, some interesting history

----------------- #1023 - The Number Pi

- Our history tells us that the number “Pi” was first approximated in 1,900 years before the birth of Christ. “Pi” was known to be the ratio of the circumference of any circle to its diameter. The calculation was done to the base 60 not to the base 10. To the base 10 their calculation was a little greater than 3. Pi has a long history of discovery. It was not until 1761 that Pi was recognized as an Irrational Number. There are so many equations used to describe nature and natural laws that have this unique number included.

- The Babylonians had Pi as the ratio of 25/8, or 3.125.

- The Egyptian scribe, Ahmes, got it to 256 / 81 which is within 1% of the accurate number. The Ahmes ratio is 3.16049.
- In the Bible, 1 Kings 7:23 says “Pi” is around the 30-cubit circumference and a 10-cubit diameter. So, the Bible’s ratio for Pi was 3.0

- In the year 263 the Chinese mathematician, Liu Hui, put Pi at 3.14014.

- In the year 463 the Indian mathematician, Aryabhata, got it to 339/108, or 3.1416

- Archimedes in 250 years before Christ used regular polygons, one inner and one outer, with the circle in between them. He was the first to develop a mathematical proof for Pi. Pi was number in between 223/71 to 22/7, which is 3.108 to 3.14286.

- In 1400 the Persian astronomer, Kasyani, calculated Pi correctly to 16 digits.

- Correctly calculated Pi = 3.14159 26535 89.………… Pi is the Greek word for perimeter. If you use Pi with 11 decimals that is accurate enough to calculate the circumference of the Earth to within 1 millimeter. If you go to 39 decimal places you can calculate the Observable Universe to the accuracy of the size of the hydrogen atom.

- In 1761 Johann Lambert calculated that Pi was an irrational number. An Irrational number means there is no ratio of two integers that exactly equals Pi. As an irrational number its decimal representation never ends, its numbers never repeat.

------------ Pi = the ratio of circumference / diameter = “c / d” = but no other ratio of
“ m / n” of two integers will exactly equal this ratio of c / d. Pi is also the ratio of the area of the circle divided by the radius squared.

- The circumference of a circle with radius 1 equals 2*Pi. 180 degrees equals 1 Pi. We call this measure of angles, radians. 360 degrees of arc = 2*Pi radians. The radian was first used in 1870.

- Mathematicians began to use geometric series to calculate Pi.

-------------- Pi = 4/1 - 4/3 + 4/5 - 4/7 + 4/9 - 4/11 …. etc , etc, etc

This series goes on to infinity but it converges very slowly on the accurate number for Pi, eventually it does if you calculate long enough.

- In the year 1424 another series was calculated that converges on Pi much more rapidly:

------------- Pi = square root of 12 * ( 1 - 1/3^2 + 1/5*3^2 - 1/7*3^2 + 1/9*3^2.…….. etc, etc, etc……

- In 1593 another series was calculated:

----------- Pi / 2 = 2/1*2/3*4/3*4/5*6/5*6/7*8/7*8/9* ……….. etc, etc, etc, …..

- In 1707:

---------------- Pi / 4 = 4* arctan 1/5 - arctan 1 / 239

---------------- where: arctan x = x - x^3/3 + x^5/5 - x^7 / 7 + ………… etc, etc, etc,

- In 1900’s William Shanks calculated Pi to 707 digits. It took him 15 years but he made a mistake and only 527 digits were correct.

- Another series was developed:

---------------- Pi^2 /6 = 1/1^2 + ½^2 + 1/3^2 + ¼^2 + ………… etc, etc, etc…..

This series was used to prove that Pi was a transcendental number.

- In 1949 they started using the computer to calculate Pi and got to 2,037 digits. It took 70 hours of computation.

- In 1973 computers got to 1,000,000 digits.

- In 1989 computers got to 1,000,000,000 digits.

- Some people have memorized Pi. A 24 year old graduate student in China recited Pi for 24 hours and 4 minutes out to 67,890 digits without an error!!! One of the memory techniques is to compose words whose letters correspond to the number of each digit. “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.” This memory trick takes Pi out to 15 places.

- The most famous equation for Pi among mathematicians is Euler‘s equation ( See note 1):

------------- e^i * Pi + 1 = 0

------------- where e is the base of the natural logarithms, e = 2.71828.……

“ little ^i” is the imaginary number = the square root of -1. The reason mathematicians like this equation is it shows off so much math, so simply. The base of natural logarithms, “e”, is an “ irrational number” raised to the power of Pi , another irrational number, multiplied by the square root of -1, an imaginary number, plus an integer 1 equals zero. It all cancels out to nothing.

- The geometric series for e = 1/0! + 1/! + ½! + 1/3! + ………. etc. etc. ,etc.

“!” means factorial , 3! = 3 factorial = 3*2*1 = 6

- There are many other famous equations that have Pi:

- The cosmological constant = (8*Pi*G / 3*c^2 ) * density of the Universe.
G = the gravitational constant. c = constant speed of light. The cosmological constant is the rate of expansion of the Universe. It is the repulsive force that counteracts the attractive force of gravity.

- Or, Critical Density = 3*Ho^2 / 8*Pi*G. Critical Density is the balance between the Universe expanding or collapsing. “Ho” is Hubble’s Constant for expansion which is 47,000 miles per hour per million lightyears of distance. The reciprocal of this number is the Age of the Universe, 13.7 billion years old.

- Heisenberg Uncertainty Principle:

------------------------------ delta x * delta p > h / 4*Pi.

Where “x” is the position of an atomic particle, “p” is the momentum, which is the mass of the particle times its velocity must always be greater than Planck’s Constant / 4*Pi. That is, knowing more about the actual position of an electron means you must know less about its actual velocity, and vise versa.

- Einstein’s Field equations for General Relativity.

- Coulomb’s law for electric force

- Magnetic permeability of free space.

- Kepler’s law for orbits, period squared = radius cubed.

- Probability for a Normal Distribution in statistics.

- The Indiana Pi Bill. Yes, the state legislatures in Indiana in 1897 tried to pass a law making Pi equal to 3.2, thus simplifying a lot of routine calculations. The bill passed the Committee on Education. It passed the House unanimously. Some college professors coached some senators overnight and the next day the Senate postponed the bill indefinitely. Kinda like the California Budget in 2009.

- Politicians do not understand numbers unless they are poll numbers. There was this politician who was a king. He so loved his servant that he granted the servant one wish. The servant used his math and asked the King for simply one grain of rice and double it on each square of a chess board, two grains placed on the first square, 4 grains on the second square, 8 grains on the third square and so on. So, “Granted” said the King. It cost him 18,446,744,073,709,551,615 grains of rice. This was equivalent to 671,000,000 tons of rice and so bankrupted the kingdom. Reminds you of the Stimulus Package 0f 2009 doesn’t it?? Politicians do not understand math. Pi to politicians is something you divide up among your lobbyist. or is that pork?

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(1) Leonard Euler (1707 - 1783) Received his masters degree in math at age 16 from University of Basel. He became professor of math in St. Petersburg, Russia. He lost his right eye looking too long at the Sun attempting to work out a system of time determination. Euler was a prolific writer, writing on every subject in math. He published 800 papers, probably. He introduced “e” for the base of the natural logarithms and “little i” for the square root of -1 in imaginary numbers. He calculated the orbit of Uranus in 1783. His equation was first: e^ix = cos x + i*sin x Then, when x = Pi you get Euler’s other equation.

(2) See Review #63 “ Nuclear Reactions and Pi from Breadsticks” if you want to learn more about Pi.

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707-536-3272, [email protected] Monday, December 27, 2010

#3 Very interesting change of focus