March 19, 2018 Monday
Bedtime Story
Probability Theory and 𝑒
So the logarithm of any real number x to
base b, which mathematically is represented as logbx is the unique
real number y such that by = x.
As you can see, almost any real number can
be the base of logarithm, but out of so many only three have been the favorite
in mathematics: b = 10 (common logarithm), b = e (natural logarithm) and b = 2
(in binary logarithm).
Probably one of the first mathematicians
who saw the clear relation between the constant e and logarithms was the
Scottish mathematician and astronomer James Gregory who is perhaps better known
for designing the first reflecting telescope (most of these great brains/minds
are largely forgotten, especially those who came before the dawn of the
European civilization i.e. before 1600).
James Gregory was born in 1638 in the
village of Drumoak, Aberdeenshire, Scotland and lived a very short life dying
at the age of 36 from stroke.
It has to be stressed once again that today
logarithm is seen first of all as a function and secondly it is seen as an
inverse function of exponentiation.
That was not how it used to be thought earlier
by its pioneers; they simply understood logarithm as a smart numerical
technique that aided complicated calculations; They saw logarithms purely as a
computing devise just as we might view electronic calculator.
That is why the mathematical constant e did
not find its association with logarithms for so many decades and perhaps even a
century as the constant e kept coming up as a kind of function either in
compounding function or in exponential growth.
So mentally it was hard to perceive that a
calculating device could have any link with a mathematical constant that
appeared in some functions.
In fact, besides finding the constant e via
solving the compounding problem, Jacob Bernoulli had come across this constant
by yet another path of probability theory.
You may ask of me what probability theory
is.
Well, I may not know much about it but
Jacob Bernoulli had given a long and hard thought on it.
In his book ‘The Art of Conjecturing’ that
was published in 1713, eight years after his death, we get a glimpse of Bernoulli’s
thought on probability theory:
“…probability as a measurable degree of
certainty; necessity and chance; moral vs. mathematical expectation; a priori
and a posteriori probability; expectation of winning when players are divided
according to dexterity; regard of all available arguments, their valuation, and
their calculable evaluation; law of large numbers…”
This phrase is a jumble of lot of profound
ideas and yet it may not explain Bernoulli’s contribution to the probability
theory.
Bernoulli, even though a mathematician,
thought about chances or probability in a more practical manner.
He asked the following question to himself:
What would be probability for a gambler to lose if he tries his hands on a slot
machine that is programmed to pay out once in n times and the gambler plays it
n times?
We shall consider this interesting problem n
the nights to come.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.com
Good night mon ami and my fellow cousin ape.
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Another great educator and a teacher that I am aware of is
Professor Subhashish Chattopadhyay in Bangalore, India.
While I narrate stories, Professor Subhashish an electronic
engineer and a former professor at BARC, does and teaches real mathematics and
physics.
He started the participation of Indian students at the
International Physics Olympiad.
Do visit him here:
All his books can be downloaded for free through this link:
For edutainment and English education of your children, I
recommend this large collection of Halloween Songs for Kids:
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