maths newsletter issue 20

INDIAN MATHEMATICS VOLUME – 20
[Continued from Issue 19] 10/12/2020

In the 1830s Charles Whish, an English civil servant in the He went on to use the same mathematics to obtain infinite
Madras establishment of the East India Company, brought to series expressions for the sine formula, which could then be
light a collection of manuscripts from a mathematical school used to calculate the sine of any angle to any degree of
that flourished in the central part of Kerala, between accuracy, as well as for other trigonometric functions like
Kozhikode and Kochi, starting from late fourteenth century cosine, tangent and arctangent. He laid the foundations for the
and continuing at least into the beginning of the seventeenth later development of calculus and analysis, and either he or
century. The school is seen to have originated with Madhava, his disciples developed an early form of integration for simple
to whom his successors have attributed many results functions. Among his other contributions, Madhava
presented in their texts. Since the middle of the 20th century discovered the solutions of some transcendental equations by
Indian scholars have worked on these manuscripts and the a process of iteration, and found approximations for some
contents of the manuscripts have been studied. Apart from transcendental numbers by continued fractions.
Madhava, Nilakantha Somayaji was another leading
personality from the School. There were other major advances in Kerala at around this
time. Citra Bhanu was a sixteenth century mathematicians
Although almost all of Madhava’s original work is lost, he is from Kerala who gave integer solutions to twenty-one types of
referred to in the work of later Kerala mathematicians as the systems of two algebraic equations. These types are all the
source for several infinite series expansions (including the possible pairs of equations of the following seven forms:
sine, cosine, tangent and arctangent functions and the value x +y =a, x−y = b, xy = c, x2+y2 = d, x2−y2 = e, x3 + y3 = f, and
of π), representing the first steps from the traditional finite x3 – y3 = g.
processes of algebra to considerations of the infinite, with its For each case, Citra Bhanu gave an explanation and
implications for the future development of calculus and justification of his rule as well as an example. Some of his
mathematical analysis. explanations are algebraic, while others are geometric. That
there would be essentially no progress between the
Madhava was more than ⋯happy to play around with contributions of Bhaskara II and the innovations of Madhava,
infinity, particularly infinite series. He showed how, although who was far more innovative than any other Indian
one can be approximated by adding a half plus a quarter plus mathematician producing a totally new perspective on
an eighth plus a sixteenth, etc, (as even the mathematics, seems unlikely.
ancient Egyptians and Greeks had known), the exact total of The work of the Kerala mathematicians anticipated the
one can only be achieved by adding up infinitely many calculus as it developed in Europe later, and in particular it
fractions. involves manipulations with indefinitely small quantities (in
the determination of circumference of the circle etc.)
Some of the remarkable discoveries a formula for the ecliptic; reminiscent of the infinitesimals in calculus; it has also been
the Newton-Gauss interpolation formula; the formula for the argued by some authors that the work is indeed calculus
sum of an infinite series; Lhuillier’s formula for the already. The overall context raises a question of possible
circumradius of a cyclic quadrilateral. Of particular interest is transmission of ideas from Kerala to Europe, through some
the approximation to the value of π which was the first to be intermediaries.
made using a series. Madhava's result which gave a series Kim Plofker, the author of Mathematics in India writes,
for π, translated into the language of modern mathematics, “Hundreds of thousands of manuscripts in India and
reads elsewhere attest to this tradition, and a few of its highlights –
decimal place value numerals, the use of negative numbers,
π R = 4R - 4R + 4R − ⋯ ⋯ ⋯ solutions to indeterminate equations, power series in the

35 pg. 1

This formula, as well as several others referred to above, were
rediscovered by European mathematicians several centuries
later. Madhava also gave other formulae for π, one of which
leads to the approximation 3.14159265359.

Madhava linked the idea of an infinite series with geometry
and trigonometry. He realized that, by successively adding
and subtracting different odd number fractions to infinity, he
could home in on an exact formula for π. Through his
application of this series, Madhava obtained a value
for π correct to an astonishing 13 decimal places.

Kerala school – have become standard episodes in the story and 4 + 3 = 7 = "A" making 76 the only other 2-digit
told by general histories of mathematics.” automorphic number whose square is 5776.

[Will be continued] The nth square number ending in 25 can be derived directly
from N(n)2 = 100 n (n - 1) + 25.
INTERESTING MATHEMATICS
Mathematics in sports
1. A straight line has dimension 1, a plane - 2. Fractals
have mostly fractional dimension. Athletics is an exclusive collection of sporting events that
involve competitive running, jumping, throwing, and
2. The numerical digits we use today such as 1, 2 and 3 walking. The most common types of athletics competitions are
are based on the Hindu-Arabic numeral system track and field, road running, cross country running, and race
developed over 1000 years ago. walking. Athletics events were depicted in the Ancient
Egyptian tombs in Saqqara, with illustrations of running at the
3. You can almost perfectly convert miles and kilometers Heb-Sed festival and high jumping appearing in tombs from
using the Fibonacci sequence. as early as of 2250 BC.

4. 73 is the 21st prime number. Its mirror, 37, is the 12th If a runner ran one lap of the track in the second lane, how far
and its mirror, 21, is the product of multiplying 7 and 3 did she / he run? Solution: You might think that the answer is
and in binary 73 is a palindrome, 1001001, which 400 meters, since it is a 400-meter track. To calculate the
backward is 1001001. distance around the turns, we use the formula for the
circumference (C) of a circle with radius (r): = 2 since the
5. International Paper Sizes (e.g. A4) use a 1: √2 ratio. If two semicircles on the inside rail form a circle with a
you cut them in half crosswise, the same ratio will be circumference of 200 m, we can find the radius of the inside
maintained. It's great for scaling up or down. rail semicircles. = 2 ⇒ 200 = 2 ≈ 31.83 Now when you
run in the second lane, the semicircle turns have a radius that
AUTOMORPHIC NUMBERS is one meter longer or 32.83 m Thus, distance around both
turns in the second lane is; = 2 = 2 32.83 ≈ 206.2 . Thus,
Automorphic numbers are numbers of "n" digits whose a lap in the second lane has a total distance of 406.2 m, 200
squares end in the number itself. Such numbers must end in 1, meters for the straight-aways plus 206.2 m for the turns.
5, or 6 as these are the only numbers whose products produce
1, 5, or 6 in the units place. For instance, the square of 1 is 1; T. A. Sarasvati Amma
the square of 5 is 25; the square of 6 is 36. It is well known that
all 2-digit numbers ending in 5 result in a number ending in 25 Born in 1918, in Tekkath Amayankoth
making 25 a 2-digit automorphic number with a square of 625. Kalam, Kerela, Saraswathi Amma is the
No other 2-digit numbers ending in 5 will produce an second daughter of Kuttimalu Amma and
automorphic number. Marath Achutha Menon. A renowned
scholar in the field of geometry of ancient
Is there a 2-digit automorphic number ending in 1? We and medieval India, she has contributed
know that the product of 10A + 1 and 10A + 1 is 100A2 + 20A towards creating one of the most
+ 1. "A" must be a number such that 20A produces a number canonical works in the field of
whose tens digit is equal to "A". For "A" = 2, 2 x 20 = 40 and 4 Mathematics of India from the Vedic
is not 2. For "A" = 3, 3 x 20 = 60 and 6 is not 3. Continuing in
this fashion, we find no 2-digit automorphic number ending in
1.

Is there a 2-digit automorphic number ending in 6?
Again, we know that the product of 10A + 6 and 10A + 6 is
100A2 + 120A + 36. "A" must be a number such that 120A
produces a number whose tens digit added to 3 equals "A".
For "A" = 2, 2 x 120 = 240 and 4 + 3 = 7 which is not 2. For "A"
= 3, 3 x 120 = 360 and 6 + 3 = 9 which is not 3. Continuing in
this manner through A = 9, for "A" = 7, we obtain 7 x 120 = 840

pg. 2

period, all the way down to the 17th century AD. comprehensive and shows her remarkable competence in
dealing with Sanskrit and Malayalam mathematical texts. In
On completion of her MA in English from the Bihar University, 1962, Sarasvati’s article on “Mahāvīra’s Treatment of
Saraswathi worked in the Madras University for a period of Series” as well as her long paper related to Jaina
around three years (1957-1960) as a scholar of the Government mathematics were published.
of India (Department of Education) in the Sanskrit Sarasvati became a member of the Indian Society for History
Department. It was here that she worked under the of Mathematics (Delhi) and started reading its journal,
distinguished Sanskrit scholar Dr V Raghavan, who guided the Gaṇita Bhāratī and discovered that “many people were
her in her thesis and most prominent work: Geometry in doing their Ph.D. research on Indian mathematics”.
Ancient and Medieval India. In this period, she taught at Sarasvati must have been very happy to see the second
multiple institutes like Shree Kerala Varma College, Thrissur, (revised) edition of her book (Delhi, 1999) and must have felt
the Maharaja College, Ernakulam and finally the Ranchi much satisfaction that errors/misprints in the first (1979)
Women’s College where she was appointed a lecturer in edition were all corrected. Unfortunately, she had been
Sanskrit. suffering from bone cancer since long and after 27 August
1999, she was bedridden due to a broken bone in her left leg.
She submitted her 300-page doctoral thesis on Indian She breathed her last on 15 August 2000. She came into this
contributions to mathematics in 1963 during her tenure at world with a purpose which she fulfilled. Her life and work
the Ranchi University, which, after an examination and viva will stand out as a memorable episode in the history of
voce was approved for the award of PhD degree by Dr. R.S. mathematics in India.
Mishra and Dr. A. Narasinga Rao, both eminent
mathematicians. During this period, she also guided the APPLICATIONS OF MATHEMATICS
doctoral thesis of R.C. Gupta on Trigonometry in Ancient and Internet and Phones
Medieval India.
Both internet and phone lines form a gigantic network which
Additionally, she wrote several papers for academic journals allows users to exchange data – whether websites or calls. All
including, Journal of Oriental Research, Journal of Ganganath Jha users are connected by countless links which have a
Research Institute, Journal of Ranchi Institute and Indian Journal of certain capacity. When you make a phone call or request a
History of Science, and also presented a paper on The cyclic website network operators have to find a way to connect
quadrilateral in Indian Mathematics at the Proceedings of the All- sender and receiver without exceeding the capacity of any
India Oriental Conference. individual link.
Without the mathematics of queuing theory, it would be
From 1973, Dr Saraswathi served as the Principal of Shree impossible to guarantee a reliable service. Mathematical
Shree Lakshmi Narain Trust Mahila Mahavidyalaya, models using Poisson processes all but guarantee that you will
Dhanbad, Bihar, but resigned in 1980 as she believed that the hear a dial tone when making a phone call. Routing internet
burden of her administrative duties was keeping her from connections is much more difficult – requests arrive at an
focusing on her research work. In one of her letters to unpredictable rate and have a more variable duration. This led
R.C. Gupta, she says, “I do not do any useful work now-a-days, to the development of packet-switching: all data is split into
immersed as I am in the squabbles and problems of an affiliated small “packets” which are transmitted independently. Some
college accustomed to tactics to which I am not accustomed.” believe that the mathematics of Fractals can help create a
much more reliable internet service in the future.
Her work also throws light on the interaction between Greek Queueing theory has its origins in the research of the Danish
and Hindu mathematics in the development of trigonometry. mathematician A. K. Erlang (1878–1929). While working for
Together with three other works: Hindu Geometry, Hindu the Copenhagen Telephone Company, Erlang was interested
Astronomy, and chapter III of the book Mathematics of Ancient
and Medieval India, all produced around the same time, Dr pg. 3
Saraswathi’s thesis is an indispensable and rich literature on
Hindu Geometry for students and teachers alike. Dealing with
Sanskrit and Prakrit scientific and quasi-scientific literature,
the book breaks the myth around Indian mathematics: that it
is purely algebraic and computational. It deals in detail with
the Sulbasutras in the Vedic literature, with the mathematical
parts of Jaina Canonical works and of the Hindu Siddhantas
and with the contributions to geometry made by the
astronomer mathematicians Aryabhata I & II, Sripati,
Bhaskara I & II, Sangamagrama Madhava, Parameswara,
Nilakantha, his disciples and a host of others.

The cyclic quadrilateral is a figure of pride in Indian geometry,
and has an eventful history. Brahmagupta’s formulas for its
area and diagonals are considered to be among the most
beautiful results of 7th century mathematics. Sarasvati’s paper
“Cyclic Quadrilateral in Indian Mathematics” is quite

in determining how many circuits and switchboard operators December 22, the birth anniversary of India’s famed
were needed to provide an acceptable telephone service. This mathematician Srinivasa Ramanujan, is celebrated as National
investigation resulted in his seminal paper “The Theory of Mathematics Day.
Probabilities and Telephone Conversations,” which was
published in 1909. Erlang proved that the arrivals for such If we go back to the history of National Mathematics Day
queues can be modeled as a Poisson process, which celebrations, then it was first celebrated in 2012 on
immediately made the problem mathematically tractable. 22nd December. It was declared by our Prime
In 1953, the English mathematician and statistician D. G. Minister Manmohan Singh on 26 February 2012 to celebrate
Kendall suggested a convenient notation for the mathematical Ramanujan’s birth anniversary as National Mathematics Day
description of queues. According to Kendall’s notation, a nationally. It was also declared that the year 2012 is celebrated
queue with arrival time distribution A, service time as the National Mathematics Year.
distribution B, and c servers is called an A/B/c queue. Thus, a
single-server queue in which the arrival and service times From his childhood, he has a huge passion for mathematics.
both follow an exponential distribution is called So much so, that at the age of 12, he had mastered
an M/M/1 queue (the M denotes the “memory-less” property trigonometry and developed many theorems on his own with
of the exponential distribution). no assistance. He was awarded a scholarship to study at
Government Arts College, Kumbakonam, but he eventually
The Danish and US researchers instead are solving the lost it due to his abysmal performance in other subjects. He
problem with a special kind of network coding that utilizes ran away from home and enrolled himself at Pachaiyappa’s
clever mathematics to store and send the signal in a different College in Madras.
way. The advantage is that errors along the way do not
require that a packet be sent again. Instead, the upstream With the support of Mathematician Ramaswamy Iyer, he got a
and downstream data are used to reconstruct what is job as a clerk at the Madras Port Trust. His breakthrough
missing using a mathematical equation. finally came in 1913, when Ramanujan wrote to G H Hardy.
The British mathematician, on realizing Ramanujan’s genius
[ Mathigon] wrote back to him, invited him to London. Hardy then got
Ramanujan into Trinity College, Cambridge and what began
Mathematics Olympiad Questions was a captivating saga of success. In 1917, Ramanujan was
elected to be a member of the London Mathematical Society.
1. From a square with sides of length 5, triangular pieces In 1918, he also became a Fellow of the Royal Society,
becoming the youngest person to achieve the feat.
from the four corners are removed to form a regular The London weather and the poor eating habits slowly
octagon. Find the area removed to the nearest integer? affecting the health of Ramanujan, and he breathed his last in
Kumbakonam at an age of 32. The mathematics wizard made
2. An ant leaves the anthill for its morning exercise. It a significant contribution to mock theta function that
walks 4 feet east and then makes a 160° turn to the right generalises the form of the Jacobi theta functions, while
and walks 4 more feet. It then makes another 160° turn preserving their general properties.
to the right and walks 4 more feet. If the ant continues National Mathematics Day is celebrated in various schools,
this pattern until it reaches the anthill again, what is the colleges, universities, and educational institutions in India.
distance in feet it would have walked? Even the International Society UNESCO (United Nations
Educational, Scientific and Cultural Organization) and India
3. Five persons wearing badges with numbers 1, 2, 3, 4, 5 had agreed to work together to spread mathematics learning
are seated on 5 chairs around a circular table. In how and understanding. Along with this, various steps were taken
many ways can they be seated so that no two persons to educate the students in mathematics and spread knowledge
whose badges have consecutive numbers are seated to the students and learners all over the world.
next to each other?
Carolina Araujo awarded 2020 Ramanujan
4. Let ABC be a triangle and let Ω be its circumcircle. The Prize for Young Mathematicians
internal bisectors of angles A, B and C intersect Ω at A1,
B1 and C1 respectively and the internal bisectors of The year 2020 Ramanujan Prize for Young
angle A1, B1 and C1 of the triangle A1B1C1 intersect Ω
at A2, B2 and C2 respectively. If the smallest angle of Mathematicians was awarded to Dr Carolina
triangle ABC is 40°, what is the magnitude of the
smallest angle of triangle A2B2C2 in degrees? Araujo, Mathematician from the Institute for Pure and

5. Find the smallest positive integer n ≥ 10 such that n + 6 Applied Mathematics (IMPA), Rio de Janeiro, Brazil. Her
is a prime and 9n + 7 is a perfect square.
work area focuses on birational geometry, which aims to
[Answers in the last page]
classify and describe the structure of algebraic varieties.
National Mathematics Day.

The prize is awarded annually to a researcher from a
developing country funded by the Department of Science
and Technology of the Government of India in association
with ICTP (International Centre for Theoretical Physics), and
the International Mathematical Union (IMU) was given for
her outstanding work in algebraic geometry.

pg. 4

circle: o. This applies to left/right, up/down touching only -
there are no diagonal markers.
All possible o and o are given, so if there is no circle between
two touching squares then they are not consecutive and one
is not double the value of the other. Standard Sudoku rules
also apply: place 1 to 9 once each into every row, column and
bold-lined box.

Araujo specializes in algebraic geometry, including birational
geometry and foliations. She has been a Simons Associate with
ICTP since 2015, and is the vice president of the Committee for
Women in Mathematics at the International Mathematical
Union. is the first non-Indian women mathematician to
receive this prize.

Paper Folding Instructions –
A Tadpole

Answers: 1) 4 2) 36 3) 10 4) 55 5) 53

TRICKY MATCHSTICK EQUATION
ADD 3 matchsticks to make the equation true.
Negative numbers are not allowed and no number

is greater than 9.

Brain Teaser

Kropki Sudoku Managing Editor: LEELA K M
Aswanth [12A Outgoing Student] – Logo
Kropki Sudoku is one of the Sudoku type which will not
contain any hint but only set of dots in the grid which will
give enough information to solve a Kropki Sudoku puzzle
uniquely and logically. Kropki Sudoku puzzles add black (o)
and white (o) circles to a regular Sudoku puzzle.

All pairs of touching squares which contain consecutive
values (such as 2 & 3 or 6 & 7) are marked with a white
circle: o. Also, all pairs of touching squares which contain
values where one number is exactly twice the value of the
other (such as 2 & 4, or 4 & 8) are marked with a black

pg. 5