January 04, 2017 Wednesday
Bedtime Story
The Pigeonhole Principle
The pigeonhole principle, just like the axiom of choice, at the
first glance appears an extremely ordinary, good-for-nothing, obvious
statement.
This is how it goes.
If n objects are distributed over m places, and if n > m, then
some place receives at least 2 objects.
So what?
Even an average ape like me knows this childishly simple truism.
It is just like saying that in a set of 3 shoes, there will at
least be two left shoes or two right shoes.
Well, let us look into it.
It was the great German mathematician Peter Dirichlet (who
incidentally happened to be born in 1805 when Prussian territory was under the
yoke of the First French Empire) who first treated this idea formally.
Formal treatment in mathematics refers to a concept wherein a
subject is made abstract using symbols, well-formed formulas (wwf), axioms and
inference rules.
Moreover, there has to complete strictness or mathematical rigor
in the application of these rules, symbols and formulas.
Dirichlet published a paper in 1834 titled Schubfachprinzip.
The word Schubfach is German that translates into drawer which the
dictionary states as an open-topped box that can be slid in and out of the
cabinet that contains it.
So the paper that Dirichlet published literally translates as the
“drawer principle”.
Yet with time, the word gradually changed to pigeonhole that denotes
an array of compartments for sorting out mails, and posts. (As an aside, Dirichlet’s
father was a postmaster and perhaps that is truly what Dirichlet had in mind
and not a typical drawer).
The pigeonhole principle classically and traditionally deals with
finite sets and does not apply to a problem like that of the Grand Hotel with
infinite rooms.
By the way, the pigeonhole principle provides the solution to the
birthday problem.
The birthday problem asks of us the probability in a set of n
randomly chosen people of two people having the same day of birth such January
the first.
Think about it mon ami.
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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