March 22, 2018 Thursday
Bedtime Story
Origins of Takakazu/Bernoulli Numbers
As we saw last night, Jacob Bernoulli was
neither the first one nor the only one to have discovered the numbers that have
now come to have been named after him.
Oddly enough, just like in the case of
Jacob Bernoulli, so in the case of Seki Takakazu, the numbers were published
posthumously.
Seki Takakazu lived between 1642 and 1708
and thus he was a contemporary of Newton, Leibniz and Huygens.
Those days communications across continents
was probably more difficult than sending messages across planets in the solar
system would be today.
So to even contemplate that the Japanese
mathematician would have plagiarized the works of Europeans would be a
reflection of one’s ignorance of those times.
Not only did Takakazu discover Bernoulli
numbers independently but he worked on the subject of infinitesimal calculus as
well that was totally new then.
It is so strange to me that there was such
a great controversy between Newton and Leibniz regarding the invention of
calculus but Takakazu finds almost no mention anywhere; it truly seems that
there is no justice in history.
Due to his many contributions to Japanese
mathematics, Takakazu is often spoken to as “Japan’s Newton”.
Now let us what are these strange numbers
that both Takakazu and Bernoulli discovered?
The question that perhaps is more pertinent
to ask is not what these numbers are but rather from whence do these Bernoulli
numbers have their origins from?
Well, great mathematicians from time
immemorial such as Pythagoras, Archimedes, Aryabhata, al-Karaji and al-Haytham
had always wondered to the solutions of problems like these:
1 + 2 + 3 + 4 + 5 + …
12 + 22 + 32
+ 42 + 52 + …
13 + 23 + 33
+ 43 + 53 + …
In short, these problems relate to computation
of sums of integer powers of which for a long time mathematicians could never
arrive at a complete general formula.
These men who I mentioned had come up with
descriptive “solutions” to some extent, but there was no known general formula
even as late as 1600.
It was somewhere around 1600 that
mathematicians in Europe such as Thomas Harriot of England, Johann Faulhaber
(Faulhaber’s formula) of Germany, Pierre de Fermat and Blaise Pascal of France started
to make some contributions to the solution to these problems.
It will be too big and unwanted a
digression if I were to give you a brief description of the contribution made
by each of them but as a demonstration, let me show you Faulhaber’s formula
which is sometimes also known as Bernoulli’s formula.
Faulhaber’s formula is an interesting one and
it is a great demonstration of the kind of thinking that goes inside the minds
of mathematical brains.
We shall consider the Faulhaber’s formula
in 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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