August 04, 2018 Saturday
Bedtime Story
Dr. Paolo Ruffini Investigates Polynomials
Last night I had introduced into our
ongoing story on polynomials yet another Italian who like Gerolamo Cardano
happened to be both a mathematician and a doctor – Paolo Ruffini.
It was just that Ruffini was born nearly
190 years after the death of Cardano.
Ruffini was taking an interest in an
obsolete, largely-ignored 1771 paper of Lagrange that had something to do with
the theory of solubility of algebraic equations.
This was one of the many seminal papers
that came out during Lagrange’s twenty years in Berlin and it was obviously
overshadowed by his major work of reformulation of classical mechanics that we
know all know as Lagrangian Mechanics.
Mathematicians like Lagrange and Ruffini were
probably – for much unknown reasons – wondering at the same question: What was
it that made some polynomials solvable and others not?
Could there be something there to explore
on that subject and come out with a method of knowing the answer beforehand before
tackling any polynomial head on.
That paper of Lagrange had in it subtle
hints that the solutions to equations of the fifth and higher degrees might me
impossible.
Lagrange’s paper had introduced the
technique of resolvent for the first time in abstract algebra that was widely
used later by Évariste Galois.
It is almost impossible for me to simplify
the idea of resolvent in abstract algebra and Galois Theory and so for now we
need to simply accept that it was something seminal that Lagrange had
introduced only to be largely ignored.
In science and in mathematics, it often
happens that no great work goes ignored for ever as there is always someone
somewhere in some time period that will come out sniffing for it and pick it up
where it was left.
We know that something similar happened
with Gregor Mendel’s work (who incidentally happened to have been born the same
year in Austrian Empire as the year that Ruffini passed away not very far in
Italy) who cross bred pea plants and noted how its seven traits manifested in
successive generations.
Like mathematicians, Mendel too was looking
for patterns when for most human apes there existed none.
It is said that the first person to
conjecture about insolubility of quintics and higher polynomials on print was
Gauss who in his 1798 book that he wrote at the age of 21 titled ‘Arithmetical
Investigations’ made the following point:
“There is little doubt this problem does
not so much defy modern methods of analysis as that it proposes the
impossible.”
Gauss had made an extensive study of
polynomials and in the theory of polynomials there exists Gauss’s lemma which consists
of two related statements.
We are not going to go into those lemmas at
present but suffice it is to know that Gauss had an intuitive feeling about the
insolubility of polynomials higher than quartics.
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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