Irrational Action
A)
PYTHAGORAS THEOREM: There is no need
to elaborate much on the famous theorem of Pythagoras as the theorem and its
proof are quite well known even at schoolboy level. In fact, the theorem and its
applications were known to many diverse civilizations – Egyptian, Indian,
Chinese and even Babylonians- much before Pythagoras’ time. (But these poor
fellows were applying the theorem in every day life without realizing that it
had not yet been mathematically proved.)
In this chronology of Mathematics, we will avoid the proof so that readers will not get bored; but it will be a great misdemeanor to avoid even stating the theorem.
"For a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two smaller sides."
And this theorem works both ways. Not only that all right angled triangles need to obey this rule (whether they like it or not) but also that, if a triangle exhibits this property then we can say that it is a right angled triangle without measuring it or even seeing it.
It is said that Pythagoras was so thrilled by his
discovery that he promptly sacrificed oxen to the local deity. (Luckily this
did not set a trend and make any dangerous precedent for mathematicians who
followed. First of course, is the question of affordability. Secondly imagine
what would have happened if Euclid had decided to follow these worthy
footsteps- with his 13 volumes of Elements to follow, Euclid might have made a
serious dent on the bovine population of Alexandria.)
B) PYTHAGORAN BELIEFS: To Pythagoras and his followers “Everything was a number”.** Whether it is the music emanating from a stringed instrument or the orbit of planets everything in the universe seemed to indicate the harmony of numbers.They were not merely fascinated by them;they literally worshipped them. To them the numbers consisted of two types: a) the counting numbers or positive integers as we call them today and b) fractions which have integers in both numerators as well as denominators or rational numbers as we call them today.
C) PYTHAGOREAN TRIPLETS: In the right- angled triangle and the Pythagoras theorem we have seen just now, there are countless number of triplets- i.e., a set of three integers forming three sides of a right angled triangle. For example, numbers 3,4 and 5 is one such set.
(In fact Euclid, 200 years later, gave a simple and elegant proof that there is indeed infinite number of Pythagorean triplets. Readers who are interested in seeing this proof can read “ Euclid’s Proof Regarding Infinite Pythagorean Triplets” and come back to this essay or go and read it later at leisure.)
D) IRRATIONAL NUMBER: This story would have been a pleasant one, had not one of Pythagoras’ disciples, Hipposus, in his infinite curiosity while exploring various combinations of the three sides of right angled triangles,stumbled on the simplest of them all. He found that for a right angled triangle of unit length for two of its shorter sides the hypotenuse turned out to be a number he could not handle at all-√2.
√2 is obviously not an integer. Not it can be expressed as a ratio of two integers- a rational number, as we call them today. Then what is it? Does such a number exist? For a person 2500 back, this is a daunting question indeed. The existence of this number cannot be denied as it flows from his master’s famous theorem.
It is said that since this unpardonable discovery could not be laid to rest with the newfound system of logic of the Brotherhood, Pythagoras handled it the only he could. To his eternal shame, Pythagoras ordered that Hipposus be drowned. Thus to my knowledge, Mathematics got its first martyr, long before any other physical sciences could claim such a distinction.
Thus in
Mathematics was born another concept –irrational number- not as a result of any
happy turn of events but as an unwanted offspring, which its own father tried to
disown.
(Again, it is our Euclid, who has given a
wonderful proof that √2 is an irrational number. Interested reader can go to "Proof of irrationality for √2".)
**The contributions made by Pythagoras to Mathematics cannot be denied and he is indeed the first great Mathematician.However,he was an incorrigible mystic.Maybe,some of it on account of his visit to the East.Whatever be the cause,some of his beliefs on numbers bordered on lunacy.We have not gone into this mumbo jumbo as it would have deviated us from our main theme here.
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