August 03, 2018 Friday
Bedtime Story
Solution to Quartic is too Unwieldy
Last night we were discussing the solution
proposed by Ferrari to the depressed quartic equation of the form:
y4 + py2 + qy + r = 0
In the last step variable m was added to
the left side and it was squared.
This variable m can have any arbitrary
value, we can choose such a value so as to get an equation in which the value
of the right hand side would be 0.
To achieve this would require the use of
technique of discriminant which allows arriving at values of the roots without
computing them.
This is a very convoluted mathematics which
is not easy either for me either to work it out or to explain it to you.
In fact, writing down the complete solution
to the watered-down general quartic equation x4 + ax3 + bx2
+ cx + d = 0 would probably take up this entire page.
So we can leave it at that and you have to
just take it as given that the quartic polynomial equation has a general
solution expressible in radicals which perhaps may not be very useful for most
practical purposes.
Among the many eminent mathematicians who were
engaged with this problem of polynomials there was also one Italian by the name
of Paolo Ruffini who was born in Valentano, Italy in 1865 and who incredibly by
the age of 23 had earned a degree in mathematics, philosophy and medicine/surgery!
Mind boggling achievement no matter how
primitive each subject would have been those days.
If you think he was merely educated with
all these degrees, you are most certainly wrong because this man turned out to
be practicing physician (along the lines of Cardano) along with holding a
formal position of professor of mathematics at the University of Modena in
Italy (it still is a full-fledged functioning university today).
He even published a paper on typhus, an
infectious disease that is caused by the bacteria Rickettsia prowazekii but
spread by the bites of body lice Pediculus humanus corporis.
Yet, in spite being a practicing doctor his
true fame (if at all mathematicians can ever claim to achieve it) came from his
work in mathematics.
He took up the work done by Lagrange on
permutations.
You might vaguely be familiar with the idea
of permutations which is an act of arranging or rearranging of elements of a
set in certain order.
The word tuple is often associated with the
process of permutations.
In 1770 Lagrange had published a paper
titled ‘Thoughts on the algebraic solutions of equations’.
In this paper Lagrange tried to understand
why the polynomials of third and fourth degrees offer general formulas to their
solutions.
The paper was a sort of unification or
review of all the known ways to solve equations that were known so far.
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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