The Gift
Called
Numbers
By Prof Arulalan Rajan
I
nvoking Kronecker, I write this small article on the gift
from God - the Numbers - both, the Giver and Gift being Omnipresent! While we know common properties of many
numbers, we look at some of them, integers and non integers, which are unique in unique sense! The facts mentioned
below have been taken from various online resources.
5: x5 has the same units place digit as x.
8, 9: The only consecutive integers that are perfect powers - Proved by Preda Mihailescu in 2002!
13: Smallest emirp number, a prime number whose reverse is also prime number. More interesting: 132 = 169 and 312 =
961.
16: Only number of the form xy= yx with x≠y
18: Only positive number that is twice the sum of its
digits!
26: Only integer whose neighbours are perfect square
and perfect cube. It is the only solution of the Diophantine equation x3 - y2 = 2. Diophantine equation is a polynomial equation in
more unknowns such that only integer solutions are studied.
40: The only number, which when written in word (FORTY) has alphabets in lexical order.
144: Largest square in Fibonacci Sequence
Golden Ratio φ= (1+√5)/2: symmetry and φ are synonymous.
– The only number of the form x2=x+1 and x-1=1/x.
– Any higher polynomial of φ turns out to be a linear polynomial of the form aφ+b.
– The values of a and b are again elements of classical Fibonacci sequence itself!
– Has a continued fraction expansion
Woah! The list above just gives a flavour of the subtle beauty
that every number has; a subtlety that has not been paid much attention to! More so by engineers! With modern engineering, study of
numbers has taken back seat, with exceptions being cryptography
and coding theory. However, even in these areas, more emphasis
had always been on the applications and never on the associated
beauty of numbers. Engineers should take a while to stand back and
look at numbers, just for the sake of sheer beauty. For, as the great
philosopher Bertrand Russel says “Mathematics, rightly viewed,
possesses not only truth, but supreme beauty - a beauty cold and
austere, like that of sculpture, without appeal to any part of our
weaker nature, without the gorgeous trappings of paintings or music, yet sublimely pure and capable of a stern perfection such as
only the greatest art can show”
.
Dr. Arulalan Rajan is an Assistant professor at
the Electronics and Communication Depart
at NITK. He did his PhD from IISc on Integers, Sequences and Applications. Dr. Rajan is
known to be one of the most friendly and approachable professors and is famous for his
numerous treats and for hosting lunches at his
home. He is immensely passionate about numbers and believes in learning by application.
70
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