Tuesday, December 27, 2011

69: Interesting Numbers

    Due to an unfortunate Internet service outage during the time I
normally research & write the script for this podcast, I was stuck at
home this weekend trying to think of an interesting topic off the top
of my head.  Somehow the corny proof that all numbers are interesting,
repeated ad nauseum each year by nerdy college professors trying to
recruit potential math majors, came to mind.  In case you haven't
heard it, it goes like this: 
    Suppose not all numbers are interesting.  Then there must be a
lowest non-interesting number.  Call this number n1.  But n1 has the
property of being the lowest non-interesting number-- and this surely
makes it quite interesting.  Therefore our premise is contradicted,
and there is no lowest non-interesting number.  Hence, all numbers are
    Thinking about this proof brought to mind an anecdote about the
famous Indian mathematician Srinivasa Ramanujan, told by G.H.Hardy.
When Ramanujan was sick in the hospital, Hardy visited him one day,
and realized he didn't have much to talk about.  So he mentioned that
he was disappointed to have driven to the hospital in a cab numbered
1729, which was a rather dull number.  Ramanujan replied, "No, it is a
very interesting number:  it's the smallest number expressible as a
sum of two cubes in two different ways."
    Ramanujan was actually a rather colorful character in general.
Born to a high-caste but poor family in India in 1887, he received a
standard education, but at the age of 15 a friend got him a loan of an
outdated college math textbook, Carr's "Synopsis of Pure
Mathematics".  This text only covered developments through about 1860,
but Ramanujan continued from there and ended up proving contemporary
European theorems in many domains, and going beyond them in some
areas, though he didn't realize it.  Lacking interest or skills in
English, he failed the scholarshiup exams, and eventually got a job as
a clerk.  In 1913, he worked up the courage to write to the famous
G.H.Hardy, who was then at Cambridge, with a few of his theorems to
see if they were of any value.  At first, Hardy was dismissive of this
odd letter from a distant Indian clerk, but once he read the details,
he was astonished.
    Some of the theorems turned out to replicate classical or
well-known results, though Ramanujan obviously did not have much
access to current literature.  But some were deep and quite
surprising to Hardy.  To give you a flavor of the kind of math in the
letter,  one theorem was that if u equals the infinite series value of
x over one plus x^5 over one plus x^10 and so on, and v = x^1/3 plus
one over x plus one over x squared and so on, he could specify the
value of v to the 5th in terms of powers of u.  Hardy remarked that
this defeated him completely, and he had never seen anything like it
before.  Oddly enough, in addition to his many valuable results, in
some areas Ramanujan was just totally wrong, such as in the theory of
prime numbers. 
    Hardy managed to get Ramanujan to Cambridge, where he attempted to
catch him up with the academic community.  In just a few years he was
elected to the Royal Society and received a prestigious Trinity
Fellowship.  Unfortunately, the climate of Britain did not really
agree with him, and he fell ill and died by 1920.
    And this has been your math mutation for today.

Review of 'The World of Mathematics', my main source on Ramanujan.

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