Thursday 4 August 2016

THE WRINKLE OF SERENDIPITY

Serendipity: "the faculty or phenomenon of finding valuable or agreeable things not sought for".

The theorem of Pythagoras is taught in school.  I find it remarkable for two reasons.  It is a beautiful mathematical concept. Most kids have no trouble grasping it.

Let me state it for the record.  "In a right angled triangle, the sum of the squares of the lengths of the two sides of the right angle equals the square of the length of the hypotenuse."

Kids get put off by such mathematical statements.  So let us simplify this.

 


As per the theorem, in this triangle :

 Or, 9 + 16 = 25.
This is true for all right angled triangles.  If you draw a right angle with any arbitrary lengths instead of 3 and 4, the third side will "adjust" itself as per the rule given above.

Because this is such a unique feature of a right triangle, such groups of three numbers are called Pythagorean triplets.  "Pythagorean" because they follow from the theorem of Pythagoras.  Before the nationalists among us get all worked up and say "Indians discovered this rule and Pythagoras misappropriated it", it needs to be mentioned that the theorem is named after Pythagoras not because he discovered it, but because he was the first one to state its "proof".  In fact, the concept of "mathematical proof" was originated by Pythagoras.  So let us see what we mean by this...

One can draw any number of right triangles, measure the three sides and verify the rule of squares  described above.  But how many such triangles will one do it for?  The number of such right triangles is infinite.  Every time one comes across a right triangle with new dimensions, one can verify the rule.  But how about the next one?  "Mathematical proof" is a technique which in a generic way establishes that the theorem is true for any right triangle.  That was the greatness of Pythagoras and hence the theorem is rightly named after him.

Let us come back to the Pythagorean triplets.
9 + 16 = 25, hence 3,4,5 are a triplet (9 being square of 3, 16 being square of 4, 25 being square of 5)
25 + 144 = 169, hence 5, 12, 13 are a triplet (5 square 25, 12 square 144, 13 square 169)
36 + 64 = 100, hence 6, 8, 10 are a triplet (6 square 36, 8 square 64, 10 square 100)
These are some more triplets ... these appear in most school text books.
9,12,15
10,24,26
18,24,30

What I did not hear my teachers ask us or discuss with us is - "How many such triplets exist?". Since numbers are boundless or infinite in number, one would assume there are infinite such triplets. However the practice of Mathematics requires such claims to be proven.  Going one step beyond (or maybe stepping one step behind), one wonders - Can I produce such triplets at will?  The answer is Yes.

So let's go back to the proof bit for a while.  Those with some mathematical inclination must have or if not, definitely should read a book titled FERMAT'S LAST THEOREM written by Simon Singh.  A newspaper called EVENING STANDARD in its review of the book said, "It's a magnificent story, one told with infectious enthusiasm."  The book is a bit like the book A BRIEF HISTORY OF TIME by Stephen Hawkings.  Such books last one a lifetime.  One cannot possibly grasp every detail of every topic discussed.  But these books haunt us, they send even the most mathematically and scientifically disinclined among us on an enjoyable journey.

In the book, Simon Singh tells us that the Greek mathematician Euclid stated a proof for the belief that there exist an infinite number of Pythagorean triplets.  It is a simple proof which anybody who has completed ten years of school can understand by applying their minds slightly.  So really,  understanding the proof is straightforward.  What fills one with humility is the fact that Mathematics feels the absolute need to prove such things. What gives one a sense of mathematical history is the simplicity and certainty of the proof itself.

The real stunner for me however was different.  Very recently it came to my notice that an eleven year old boy "stumbled" upon this proof.  It has been six years since the kid discovered the proof and thereby taught himself a way to "manufacture" Pythagorean triplets.  The kid did not feel the need to "announce" this.  When I happened to ask the kid "Hey did you see Euclid's proof for infinite Pythagorean triplets?" he casually told me "Yeah I stumbled upon it in seventh standard when I was playing around with numbers in my mind."

It is tempting to conclude that such a kid is a genius.  However if one reads a bit about the history of Mathematics and some of the discoveries made, one is bound to conclude that such serendipity must dwell within tens if not hundreds of school kids all over the world. Being innocent and entirely devoid of an appetite for fame and recognition, they must stumble upon such facts and get on with their lives - friends, football, cricket, television.

And yet only a handful of kids go on to become "mathematicians".  So what happens along the way?  One could say serendipity dies a natural death as age advances.  I don't buy that.  Not entirely anyways.  Though it is true that Mathematics is a field for youth - it is said that hardly any mind boggling discoveries have been made by elder mathematicians, I can't believe that such serendipity dies a natural death in school.  So what looks upon such serendipity as a wrinkle and bears down upon with the scalding weight of a hot iron to smother it out?  I leave it to readers to draw their own conclusions.

For those interested, I present here the kid's method to manufacture Pythagorean triplets...

Take any odd number - say 37
Find its square - 37 squared = 1369
Find those two consecutive numbers which add up to 1369 (Divide 1369 by 2 to get 684.5 - then 684 and 685 are the numbers you are looking for - 684 + 685 = 1369)
Voila! ... 37, 684, 685 are your Pythagorean triplet!

Let us verify:-
37 square = 1369
684 square = 467856
685 square = 469225

467856 + 1369 is indeed = 469225!  Which means:

thereby proving that 37, 684 and 685 are indeed a Pythagorean triplet.

Now let's look at how this amazing "method" came about (more or less identical to Euclid's proof).  It is based on the following facts and observations
- Any natural number can be squared (1 square is 1, 2 square is 4, 3 square is 9 etc.)
- If you subtract each of these squares from the previous square, you get a sequence of odd numbers viz. 3,5,7,9,11,13,15... (Subtract 2 square from 1square, 3 square from 2 square etc.)
- Look at the sequence of odd numbers viz. 3,5,7,9,11,13,15,17,19,21,23,25... Some of them are perfect squares e.g. 9, 25, 49, 81....
- The million dollar observation: Each of the odd numbers in the above sequence are the sum of the original two consecutive natural numbers whose squares were subtracted

These statements put together and some close scrutiny of the numbers lead to the trick to produce Pythagorean triplets.

When I first read it and heard about it, it appeared simple.  And it is simple.  But I spent half a day trying to manufacture Pythagorean triplets myself - I used to get it right and then at times I used to get hopelessly confused.  But then - I possess neither serendipity nor high intelligence.












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