Ramanujan was an Indian mathematician who made extraordinary contributions to mathematical analysis, number theory, infinite series, and continued fractions. Some of his major accomplishments included proving that any integer can be expressed as the sum of at most four prime numbers, discovering properties of the partition function, and formulating the Ramanujan prime and the Ramanujan theta function. He developed his own mathematical research in isolation in India and his unorthodox formulas and insights inspired further research. His work introduced new areas of study and unexpected relationships between different branches of mathematics and analysis.

1. great MATHEMATICIANS

3.  He was born on 22nd of December 1887 in a small village of Tanjore
district, Madras. He sent a set of 120 theorems to Professor Hardy of
Cambridge. As a result he invited Ramanujan to England.
Ramanujan showed that any big number can be written as sum of
not more than four prime numbers. He showed that how to divide the
number into two or more squares or cubes. when Mr. Litlewood came
to see Ramanujan in taxi number 1729, Ramanujan said that 1729 is
the smallest number which can be written in the form of sum of cubes
of two numbers in two ways, i.e. 1729 = 93 + 103 = 13 + 123 since
then the number 1729 is called Ramanujan’s number.

4.  In mathematics, there is a distinction between having an insight and
having a proof. Ramanujan's talent suggested a plethora of formulae
that could then be investigated in depth later. It is said that
Ramanujan's discoveries are unusually rich and that there is often more
to them than initially meets the eye. As a by-product, new directions of
research were opened up. Examples of the most interesting of these
formulae include the intriguing infinite series for π, one of which is
given below

5.  Although there are numerous statements that could bear the
name Ramanujan conjecture, there is one statement that
was very influential on later work. Ramanujan
conjecture is an assertion on the size of the tau function,
which has as generating function the discriminant modular
form Δ(q), a typical cusp form in the theory of modular
forms. It was finally proven in 1973, as a consequence
ofPierre Deligne's proof of the Weil conjectures. The
reduction step involved is complicated. Deligne won
a Fields Medal in 1978 for his work on Weil conjectures.

6.  The number 1729 is known as the Hardy–Ramanujan number after
a famous anecdote of the British mathematician G. H. Hardy
regarding a visit to the hospital to see Ramanujan. In Hardy's
words.
 “I remember once going to see him when he was ill at Putney. I had
ridden in taxi cab number 1729 and remarked that the number
seemed to me rather a dull one, and that I hoped it was not an
unfavorable omen. "No," he replied, "it is a very interesting number;
it is the smallest number expressible as the sum of two cubes in two
different ways."”
 The two different ways are these:
1729 = 13 + 123 = 93 + 103Generalizations of this idea have created
the notion of "taxicab numbers". Coincidentally, 1729 is also
a Carmichael number.

7. • Aryabhatta was born in 476A.D in Kusumpur,
India. Aryabhata is the first well known Indian
mathematician. Born in Kerala, he completed his
studies at the university of Nalanda. He was the
first person to say that Earth is spherical and it
revolves around the sun. He gave the formula (a
+ b)2 = a2 + b2 + 2ab

8. HIS CONTRIBUTIONS
• He made the fundamental advance in finding the
lengths of chords of circles He gave the value of as
3.1416, claiming, for the first time, that it was an
approximation. He also gave methods for extracting
square roots, summing arithmetic series, solving
indeterminate equations of the type ax -by = c, and
also gave what later came to be known as the table
of Sines. He also wrote a text book for astronomical
calculations, Aryabhatasiddhanta. Even today, this
data is used in preparing Hindu calendars
(Panchangs). In recognition to his contributions to
astronomy and mathematics, India's first satellite

9.  Bhāskara (also known as Bhāskara
II and Bhāskarāchārya ("Bhāskara the
teacher"), (1114–1185), was
an Indian mathematician and astronomer.
He was born near Vijjadavida (Bijāpur in
modern Karnataka). Bhāskara is said to
have been the head of an
astronomical observatory at Ujjain, the
leading mathematical center of ancient
India. He lived in the Sahyadri region.

10.  He was the first to give that any number
divided by 0 gives infinity (00). He has
written a lot about zero, surds, permutation
and combination. He wrote, “The hundredth
part of the circumference of a circle seems to
be straight. Our earth is a big sphere and
that’s why it appears to be flat.” He gave
the formulae like sin(A ± B) = sinA.cosB ±
cosA.sinB

11. . His famous book Siddhanta Siromani is divided
into four sections -Leelavati , Bijaganita ,
Goladhayaya, and Grahaganita. Leelavati
contains many interesting problems and was a
very popular text book. Bhaskara introduced
chakrawal, or the cyclic method, to solve
algebraic equations.. Bhaskara can also be
called the founder of differential calculus. He
gave an example of what is now called
"differential coefficient" and the basic idea of
what is now called "Rolle's theorem".
Unfortunately, later Indian mathematicians did
not take any notice of this. Five centuries later,
Newton and Leibniz developed this subject.

12.  He was born in 598 AD
in Bhinmal city in the state
of Rajasthan. He is renowned
for introduction of negative
numbers and operations on zero
into arithmetic. He was an Indian
mathematician and astronomer
who wrote many important
works on mathematics and
astronomy.
Brahmagupta (598 A.D. -665 A.D.)

13.  His main work was Brahmasphutasiddhanta, which
was a corrected version of old astronomical treatise
Brahmasiddhanta. This work was later translated into
Arabic as Sind Hind. He formulated the rule of three
and proposed rules for the solution of quadratic and
simultaneous equations. He gave the formula for the
area of a cyclic quadrilateral as where s is the semi
perimeter. He gave the solution of the indeterminate
equation Nx²+1 = y². He is also the founder of the
branch of higher mathematics known as "Numerical
Analysis".

14.  Brahmagupta's theorem states that if a cyclic
quadrilateral is orthodiagonal (that is,
has perpendicular diagonals), then the perpendicular
to a side from the point of intersection of the
diagonals always bisects the opposite side.
 More specifically, let A, B, C and D be four points
on a circle such that the lines AC and BD are
perpendicular. Denote the intersection
of AC and BD by M. Drop the perpendicular from
M to the line BC, calling the intersection E.
Let F be the intersection of the line EM and the
edge AD. Then, the theorem states that F is the
midpoint AD.

15.  Mahavira was a 9th-century Indian
mathematician from Gulbarga who
asserted that the square root of a
negative number did not exist. He
gave the sum of a series whose terms
are squares of an arithmetical
progression and empirical rules for
area and perimeter of an ellipse. He
was patronised by the great
Rashtrakuta king Amoghavarsha.
Mahavira was the author of Ganit
Saar Sangraha.

16.  He separated Astrology from Mathematics.
He expounded on the same subjects on which
Aryabhata and Brahmagupta contended, but
he expressed them more clearly. He is highly
respected among Indian Mathematicians,
because of his establishment of terminology
for concepts such as equilateral, and isosceles
triangle; rhombus; circle and semicircle.
Mahavira’s eminence spread in all South
India and his books proved inspirational to
other Mathematicians in Southern India.

17. 
Varāhamihira (505–587 CE), also called
Varaha or Mihira, was
an Indian astronomer, mathematician,
and astrologer who lived in Ujjain.[1] He is
considered to be one of the nine jewels
(Navaratnas) of the court of legendary
ruler Vikramaditya

18.  Varahamihira (505-587) produced the Pancha
Siddhanta (The Five Astronomical Canons). He made
important contributions to trigonometry, including sine and
cosine tables to 4 decimal places of accuracy and the
following formulas relating sine and cosine functions:

19. • Shakuntala Devi is a calculating prodigy who was born on November 4, 1939
in Bangalore, India. Her father worked in a "Brahmin circus" as a trapeze
and tightrope performer, and later as a lion tamer and a human cannonball.
Her calculating gifts first demonstrated themselves while she was doing card
tricks with her father when she was three. They report she "beat" them by
memorization of cards rather than by sleight of hand. By age six she
demonstrated her calculation and memorization abilities at the University of
Mysore. At the age of eight she had success at Annamalai University by doing
the same. Unlike many other calculating prodigies, for example Truman Henry
Safford, her abilities did not wane in adulthood. In 1977 she extracted the
23rd root of a 201-digit number mentally. On June 18, 1980 she demonstrated
the multiplication of two 13-digit numbers 7,686,369,774,870 x
2,465,099,745,779 picked at random by the Computer Department of Imperial
College, London. She answered the question in 28 seconds. However, this time
is more likely the time for dictating the answer (a 26-digit number) than the
time for the mental calculation (the time of 28 seconds was quoted on her own
website). Her correct answer was 18,947,668,177,995,426,462,773,730.
Shakuntala Devi

20. This event is mentioned on page 26 of the 1995
Guinness Book of Records ISBN 0-553-56942-2. In
1977, she published the first study of homosexuality in
India. According to Subhash Chandra's review of Ana
Garcia-Arroyo's book The Construction of Queer Culture
in India: Pioneers and Landmarks, For Garcia-Arroyo
the beginning of the debate on homosexuality in the
twentieth century is made with Shakuntala Devi's book
The World of Homosexuals published in 1977.
[...] Shakuntala Devi's (the famous mathematician) book
appeared. This book went almost unnoticed, and did not
contribute to queer discourse or movement. [...] The
reason for this book not making its mark was
becauseShakuntala Devi was famous for her
mathematical wizardry and nothing of substantial
import in the field of homosexuality was expected from
her. Another factor for the indifference meted out to the
book could perhaps be a calculated silence because the
cultural situation in India was inhospitable for an open
and elaborate discussion on this issue. In 2006 she has
released a new book called In the Wonderland of
Numbers with Orient Paperbacks which talks about a
girl Neha and her fascination for numbers.

21. Narayana Pandit Narayana was the son of Nrsimha (sometimes written Narasimha). We
know that he wrote his most famous work Ganita Kaumudi on
arithmetic in 1356 but little else is known of him. His mathematical
writings show that he was strongly influenced by Bhaskara II and he
wrote a commentary on the Lilavati of Bhaskara
IIcalled Karmapradipika. Some historians dispute that Narayana is the
author of this commentary which they attribute to Madhava.
 In the Ganita Kaumudi Narayana considers the mathematical operation
on numbers. Like many other Indian writers of arithmetics before him he
considered an algorithm for multiplying numbers and he then looked at
the special case of squaring numbers. One of the unusual features of
Narayana's work Karmapradipika is that he gave seven methods of
squaring numbers which are not found in the work of other Indian
mathematicians.
 He discussed another standard topic for Indian mathematicians namely
that of finding triangles whose sides had integral values. In particular
he gave a rule of finding integral triangles whose sides differ by one unit
of length and which contain a pair of right-angled triangles having
integral sides with a common integral height. In terms of geometry
Narayana gave a rule for a segment of a circle. Narayana

22.  Narayana also gave a rule to calculate
approximate values of a square root. He did this
by using an indeterminate equation of the
second order, Nx2 + 1 = y2, where N is the
number whose square root is to be calculated.
If x and y are a pair of roots of this equation
with x < y then √N is approximately equal
to y/x. To illustrate this method Narayana
takes N = 10. He then finds the solutions x =
6, y = 19 which give the approximation 19/6 =
3.1666666666666666667’
which is correct to 2 decimal places. Narayana
then gives the solutions x = 228, y = 721 which
give the approximation 721/228 =
3.1622807017543859649, correct to four places.
Finally Narayana gives the pair of solutions x =
8658, y = 227379 which give the approximation
227379/8658 = 3.1622776622776622777,
correct to eight decimal places. Note for
comparison that √10 is, correct to 20 places,
3.1622776601683793320

24. Euclid also known as Euclid of Alexandria, was a Greek
mathematician, often referred to as the "Father of
Geometry“. His Elements is one of the most influential
works in the history of mathematics. Euclid deduced the
principles of what is now called Euclidean geometry from a
small set of axioms. Euclid also wrote works on
perspective, conic sections, spherical geometry, number
theory and rigor. Euclid may have been a student of
Aristotle. He founded the school of mathematics at the
great university of Alexandria. He was the first to prove that
there are infinitely many prime numbers; he stated and
proved the unique factorization theorem; and he
devised Euclid's algorithm for computing gcd. He
introduced the Mersenne primes and observed
that (M2+M)/2 is always perfect (in the sense of
Pythagoras) if M is Mersenne. Among several books
attributed to Euclid are The Division of the Scale, The
Optics, The Cartoptrics. Several of his masterpieces have
been lost, including works on conic sections and other
advanced geometric topics. Apparently Desargues'
Homology Theorem was proved in one of these lost works;
this is the fundamental theorem which initiated the study of
projective geometry

25. Euler may be the most influential mathematician who ever
lived he ranks #77 on Michael Hart's famous list of the
Most Influential Persons in History. His notations and
methods in many areas are in use to this day. Just as
Archimedes extended Euclid's geometry to marvelous
heights, so Euler took marvelous advantage of the
analysis of Newton and Leibniz. He gave the world
modern trigonometry. He invented graph theory. Euler
was also a major figure in number theory, proving that
the sum of the reciprocals of primes less than x is approx.
(ln ln x). Euler was also first to prove several interesting
theorems of geometry, including facts about the 9-point
Feuerbach circle; relationships among a triangle's
altitudes, medians, and circumscribing and inscribing
circles; and an expression for a tetrahedron's area in terms
of its sides. Euler was first to explore topology, proving
theorems about the Euler characteristic. he settled an
arithmetic dispute involving 50 decimal places of a long
convergent series. Four of the most important constant
symbols in mathematics (π, e, i = √-1, and γ =
0.57721566...) were all introduced or popularized by Euler.

26. THANK YOU
PREPARED BY:
VAISHNAV.S
XI STD
KENDRIYA VIDYALAYA
MAHE