AMTI Two Mathematical Offerings Articles on Mathematics by Dr. V G Tikekar and P L Bhatnagar ( 100 ) Birth Anniversary S Ramanujan ( 125 )

57 Pages·2018·13.73 MB·English

by  Dr. V G Tikekar P L Bhatnagar

TWO
MATHEMATi ca,
oFFERINGS

125th Birth
TWO MATHEMATICAL
OFFERINGS

100th
Birth anniversary

THE ASSOCIATION OF
MATHEMATICS TEACHERS OF INDIA
8-19, Vijay Avenue,85/37, Venkatarangam Pillai Street,
Triplicane, Chennai - 600005.

Telephone: [phone]
E-mail: [email protected] Web site: amtionline.com
PYeNnr

CONTENTS

Between us
Report
Indispensable Mathematics

. Ten Commandments in Mathematics

33
BETWEEN US

Dear Reader,

We are happy to place a new book in your hands. This is a special
publication in honor of the 125th birth anniversary of Srinivasa
Ramanujan and the birth centenary of Prof. P.L. Bhatnagar, Founder
Professor of Applied Math at |.1.Sc., Bangalore and Former President
of the AMT.

While we can say, with pardonable pride, that all that we do in The
AMTI are to perpetuate the memory of the mathematical genius
Srinivasan Ramanujan, in spotting out and encouraging talents. We
also joined the country with special programmes during the year
2012. A report of the same is separately included in this book.

‘Two of the significant items of the celebration are published in this
book. One is the talk given by Prof. V.G. Tikekar, Retd. Professor &
Chairman, Mathematics Dept. Sc. Bangalore-560012 on
“Indispensable Mathematics”. The other is the very popular talk of
Late Prof. P.L. Bhatnagar, (as indicated above) on the “Ten
‘Commandments in Mathematics". (We are happy to reproduce this
earlier publication of the same in our journal The Mathematics
Teacher, Vol.1(6), 1965).

We hope and trust that our well-wishers will receive this publi
also with enthusiasm.

With kind regards / best wishes,
Yours sincerely,

(M. MAHADEVAN)
Report on the celebration of National Mathematics Year-2012

1, AMTI celebrated National Mathematics Year on 25" and 26”
October 2012 with a special programme involving 38 students and
8 teachers from 7 schools. The new building in Perungalathur was
the venue. Though we suspected difficulties of persons reaching
the place, the attendance was encouraging indeed.

2. The programme started after breakfast to all with prayer followed
by welcome and introductory remarks by the general secretary. Sri
R.Athmaraman whose brain wave was this plan, formally
inaugurated the same.

3. Prof. R.Sivaraman of DGVaishnav College, Chennai gave a talk titled
Ramanujan’s Influence- Life and Work -aided by power point
projection.

4, After tea Prof. V.G.Tikekar gave an interesting lecture on
“indispensable Mathematics” drawing the attention of the
audience to significant aspects of mathematics. (The full article is
published as part of this special book)

5, After lunch there was written quiz written by all participants —
students and teachers.

6. After tea all went back to their residences.

7. The next day programme started after break-fast with oratorical
competition for students with 16 participating adjudged by Prof.
V.G.Tikekar, Sri S.RSanthanam and Ku. R.Vijayalakshmi. Three of
them were adjudged best speakers on the topic ‘What type of
Math education would | like to have’. Each was given Rs. 1000/- as
reward.

8 This was followed by a talk by Ku. R. Vijayalakshmi, titled
"Centenary Tribute to Prof. P.L.Bhatnagar”. She talked for more
than one hour. She eulogized Prof. Bhatnagar by citing some of his
views, thoughts and works particularly his favourite "Ten
‘Commandments in Mathematics". (Full article on that title is
published as part of this special book). Then lunch was announced.

9 In the post lunch session, in the absence of Prof. Poolan Prasad
who was expected to join us, Prof. M. Tamban Nair of IIT Madras
gave an interaction session with the children on “infinity”
interspersed with power point presentations. It was well received.

410. Then oral quiz was conducted for the selected 4 teams of 4 each
based on their written quiz performance. Sri Sadagopan Rajesh was
the quiz master and the winning team was given a prize of Rs.
1000/- each and others Rs. 500/- each.

11. Prof. G.Rangan, former executive chairman, Prof.
K.SRamachandran, former secretary and Prof. J.Pandurngan,
Present executive chairman joined the valedictory function and
distributed prizes. The children who won prizes in May workshop
related test conducted on 19" August 2012 also received them that
day.

12. With vote of thanks by the general secretary and national anthem
the function came to a successful end.

Formal mathematics is like spelling and
grammar -- a matter of correct application of
the local rules. Meaningful mathematics is like
Journalism -- it tells us an interesting story.
Unlike some journalism, the story has to be
true. The best mathematics is like literature -- it
brings story to life before your eyes and involves
you in it, intellectually and emotionally.
-an Stewart
INDISPENSABLE MATHEMATICS"

»
Dr. V.G. Tikekar,
ets Professor & Chairman, Mathematics Dept. Se Bangoloe 560012

1, PREAMBLE

We have all assembled here, under the banner of the AMTI, to
mark the 125" Birth Anniversary of the great Ramanujan,
India’s Genius Mathematician. The current year 2012 is also the
birth centenary year of one of the past presidents of the AMTI,
namely Prof. P.L. Bhatnagar, mathematician of a great ability
and achievement. Professor P.L. Bhatnagar founded, in 1956,
the Dept. of Applied Mathematics (now renamed as Dept of
Mathematics) at the Indian Institute of Science, Bangalore. This
Professor of Applied Mathematics never applied for any job in
his life; he was invited with honour to all the positions that he
adorned. All his students, colleagues, and well- wishers are
grateful to him for all he did for the growth and progress, of all
the organizations he served with distinction and dedication,
and for the knowledge and help he imparted to them, and also
for the inspiration that he kindled in them. In fact, if they are
not grateful to him, then they will be great fools; such was his
magnanimity and the greatness of his mind. | salute the great
Ramanujan and my guru Prof. P.L. Bhatnagar.

* The lecture delivered at the AMTI in Chennai, on
Oct.25",2012 as a part of the celebration of the National
‘Mathematics Year 2012.
2. INTRODUCTION

Mathematics has grown immensely in many directions over
many centuries, and it has attracted the attention of a large
number of minds from all walks of human activity. As such, the
field of mathematics has different shades, and it is classified in
various ways: Pure mathematics, Applied mathematics,
Applicable mathematics, Industrial mathematics, Algorithmic
mathematics, Discrete mathematics, Continuous mathematics,
Ancient mathematics, Modern mathematics, Analog
mathematics, Dialectical mathematics, Analytic mathematics,
and so on. Out of this big Storage of mathematical knowledge
of all hues, which part can be considered as dispensable is a
moot point. Because of the extensive period of its existence
mankind, could some of its parts have become old-
fashioned, or non ~ rigorous, or not of that much use in the
present- day advanced techno — savvy world and hence could
be dispensed with? | shall discuss here, a problem (or call it an
example) and from that discussion, shall point out as to what
type of mathematics can just not be dispensable. The problem
itself may not be considered by many as mathematical, but the
underlying thinking that will be used in the discussion while
getting at its solution, will throw sufficiently good light on the
type of mathematics that will be necessary and useful in any
branch of mathematics, and hence can be considered and taken
ispensable.

3. PROBLEM

There are in all n persons. Denote them by 1,2,3, ne
Each of these n persons has one distinct piece of information,
that is, no two pieces of information are the same. We shall
denote these distinct pieces of information by the same
numbers 1,2, sn. This way we shall take it that the
person 1 has the piece of information 1, the person 2 has the
piece of information 2,.. .., person n has the piece of
information n. Now these persons pass on these pieces of
information to each other by making phone calls among
themselves. During any such phone call, each person on each
side of the conversation, passes on to the other side, all the
pieces of information (such as those which he/she might have
gathered because of the phone calls that have taken place
earlier to the current phone call under consideration) that
he/she has at the time of making that phone call. What is the
minimum number of phone calls required so that at the end of
all of them, all the n persons will have all the pieces of
information with them?

4. NOTATION

We shall use the symbol (m, p) to denote the phone call
between the persons m and p. (m, p = Lye..esM)

‘Suppose that (1,2) is the first phone call. Then at the end of it,
person | will have two pieces of information, namely 1 and 2.
-We shall denote this fact by writing 1<{1,2}. Similar
representation will be employed in respect of the outcome of
any phone call. So, if (1,2), (2,6) are the first two phone calls
made in that order, then at the end of these two phone calls,
we shall have 1-{ 1,2}, 2<-{2,1,6,} and 6<{6, 1,2}.
NOTE 1

3, then the
achieve the

ifn = 3, ie, if the total number of persons
following 3 phone calls made in that order wi
objective:

(1,2), (2.3), 3,1),

This means that, at the end of these 3 phone calls, made in that
order, all the 3 persons will possess all the three distinct pieces
of information that they had with them ( with ith person having
ith piece of information, i = 1,2,3,) to start with. The reader
should satisfy himself / herself with the correctness of this
claim. We further claim that when n = 3, no two phone calls,
whatever their order, will achieve the desired objective, and
invite the reader to construct, if possible, a counter - example
to disprove our claim. We guarantee that he/she will not be
able to do that.

5. A SOLUTION TO THE GENERAL PROBLEM ( not necessarily
the optimal)

Generally we attack any problem step by step, each step
leading to achieving a sub-goal so that , at the end of all the
steps, we get a solution to the problem, on hand, which we
started with. We shall do the same thing in respect of the
problem of n persons with n pieces of information (as stated in
section 3 above)

Let our first sub-goal be: person 1 gets all the n pieces of
formation. It is easy to see that this sub-goal can be achieved
by the following phone calls made in that order:

(1,2), (1,3), (1,4), s(n)

1)

10