Srinivasa Ramanujan was a famous Indian mathematician who made significant 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 not more than 4 prime numbers, discovering the Ramanujan prime, and formulating the Ramanujan conjecture. Other notable Indian mathematicians discussed in the document include Aryabhata, Bhaskara II, Brahmagupta, and Mahavira, who all made important advances in areas such as calculus, trigonometry, algebra, and astronomy.

2. MATHS PROJECT

6. Srinivasa Ramanujan

7. Srinivasa Ramanujan
 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.

8.  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

9.  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 of Pierre 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.

10. • 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.

11.  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

12. 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 was named Aryabhata.

13.  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.

14.  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

15.  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.

16.  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.)

17.  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".

18. • 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.

19.  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.

20.  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.