April 05, 2017 Wednesday
Bedtime Story
Casus Irreducibilis or The Irreducible Case
From this concept of constructible numbers, you can see so clearly
how closely related are the three branches of mathematics.
You can extend the concept of constructible point to constructible
number.
A number is constructible if and only if you can write it down
using four basic arithmetic operations and the extraction of square roots but
of no higher-order roots.
This transformation of geometric questions of compass and
straightedge into algebra lead to the solution of some of the oldest questions
that stood as a challenge for centuries to the human intellect.
Seen from this point, the root of 2 is a constructible number as
all you need to do is make two lines of one unit at right angle to each other
and then join them.
The hypotenuse of this unit right angle triangle will be equal to
root of 2.
By definition, root of 3 is also constructible.
You can try to construct it if you have time and nothing better to
do.
Pierre Wantzel used abstract algebra of theory of fields and filed
extensions to solve these age old geometric problems.
What Pierre Wantzel actually proved or showed now goes under the
title of casus irreducibilis which is a Latin term for “the irreducible case”.
It has something to do with cubic polynomial equations that look something
like this:
ax3 + bx2 + cx + d = 0
When such a cubic polynomial equation is irreducible over the
rational number and has three real roots, then in order to express the roots
with radicals, one has to introduce complex-valued expressions.
Imaginary numbers need to be invoked if the roots are needed to be
expressed with radicals.
The strange thing is that for 50 years after its publication in
1843, this theorem lay dormant and unnoticed.
The third problem of squaring the circle is a challenge of
constructing a square that is of same area as a given circle by using a compass
and an unmarked straightedge only in finite steps.
The problem does not seem to be a terribly difficult one.
Imagine a circle of unit 1.
Then its area would be ๐
A square of the same area then will have its side as √๐
So apparently it is a doable thing.
The only tricky part is the root of pi.
We shall try to tackle this menace
tomorrow.
Stay tuned to the voice of an average story storytelling
chimpanzee or login at http://panarrans.blogspot.in/
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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