Concerning Pi

Yesterday March 14 was dubbed Pi Day. It also happens that I started this website and blog one year ago ,and it’s Albert Einstein ‘s day of birth.

Here are some facts ,relations and formulas related to the number Pi.

Approximate value of Pi to 1500 digits (done with Mathematica);

The Euler identity or formula ,which mathematicians like very much and find beautiful ,containing five math constants:

Here is a quick derivation of a known formula involving Pi:

where ln is the natural logarithm with base e = 2.7182818284590452354.

Two integrals involving Pi (done and verified with Mathematica):

And here is a brief study I made of a series of formulas involving Pi and the inverse trigonometric function arctangent (done with the Texas Instruments TI 92 Plus calculator and Mathematica)  :

The general form for these relations  is:

Or:

A plot of the function  gives:

Using the Mathematica function FindRoot , we find two values for f(x) (with x >-1 and x< -1):

Thus we may deduce the following  values :

These pi formulas resemble or seem to be related to the Machin-Like Formulas or to the more general formulas :

which have been studied in the past.

I have also considered the study of other formulas involving Pi and other inverse trigonometric functions . Maybe I will publish or write more about these subjects in the future.

About two book series

Throughout the years of reading and studying I have read many books of the French
encyclopedic series  “Que sais-je?” . Each book of the series consists of about 128 pages
and deals with a particular subject or field of study and is written by a specialist . The number
of books I’ve read in this series amounts to about one hundred , books related to various
subjects : science , mathematics , physics , astronomy , history , philosophy , religion ,
education , psychology , etc , or sometimes a combination of these subjects.

The first two books I read (and worked out) in the series were ‘Histoire de l’astronomie
classique ‘ (History of classical astronomy) and  ‘La Relativité’ (Relativity) by the late
astronomer Paul Couderc. I was 15 years of age at that time.

The history of astronomy book contains interesting information , but I remember  what
impressed me most was the author’s praise of the achievements of Isaac Newton and of his
great work “Philosophiæ Naturalis Principia Mathematica” or “Mathematical Principles of
Natural Philosophy”.
Here are  sample images of some of the books I have read in the “Que sais-je ?” series:

I have also read a few books of  the more recent  “Very Short Introductions” series.

The “Que sais-je ?” series has more than 3500 titles in 2014 , whereas The “Very Short
Introductions” has between 300 and 400 titles .
Another difference I have noticed between the two collections is that when it comes to
scientific subjects  the  “Very Short Introductions” series gives a  general introduction and
a good overview of the subject  ,  while the “Que sais-je?” series delves deeper into the
subject and gives more advanced details with mathematical proofs and equations when
necessary.