March 18, 2018 Sunday
Bedtime Story
The Elusive Constant 𝒆
Last night we were dealing with exponential
functions where we saw that the constant of proportionality of this
relationship is the natural logarithm of base b.
The constant e is the unique base for which
the constant of this proportionality is 1.
From the study of this exponential
function, Huygens saw the emergence of logarithm to base 10 of e and he
calculated its value to 17 decimal places.
Yet even Huygens did not get the number e
as an independent constant on which entire logarithmic tables can be based
upon.
In 1668 a German mathematician by the name
of Nicholas Mercator came out with a book on logarithms that he called Logarithmo-technia in which he describes
a series that is obviously now called the Mercator series.
The Mercator series goes like this:
ln(1+x) = x – x²/2
It was in this book by Mercator that the
term natural logarithm for the first time came on print though of course, since
the language of science of those times was either Latin or French, the term that
is found in the original untranslated book is ‘log naturalis’.
Yet that number e once again fails to
appear on its own and stays hidden though it is being referred indirectly again
and again and yet so far, none has given it its due exclusivity that it
deserves.
Surprisingly enough, when the discovery of
the mathematical constant e came, it came through the study of compound
interest as we saw earlier, despite its strong relationship with logarithms and
constantly getting referred indirectly in logarithmic works of many mathematicians.
Even bigger oddity or rather incongruity
about the discovery of the number e is that even when Jacob Bernoulli did
discover it independently and on its own right through his compounding problem,
he totally missed out on its deep connection with the logarithms which in those
days were equivalent to today’s super computers.
Today we see logarithm as the inverse
operation to exponentiation, just as division is seen as inverse of operation.
So if 5 raised to the power of 3 is equal
to 125, then the logarithm to base 5 of 125 is 3.
So 53 = 125
Then log5125 = 3
The beauty is that since the exponentiation
function allows any positive real number to be raised to any real power, then
by the definition of logarithm it also means that the logarithm can be
calculated for any two positive real numbers b and x where b is not equal to 1.
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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