This ppt about the Indian mathemation

1. ….And their contributions

2. name:M Charan raju
Cse A
Ht.no:22911A0528

3. ▣ Indian mathematics emerged in the Indian subcontinent
from 1200 BC until the end of the 18th century. In the
classical period of Indian mathematics (400 AD to 1200
AD), important contributions were made by scholars
like Aryabhata, Brahmagupta, and Bhaskara II.
The decimal number system in use today was first
recorded in Indian mathematics. Indian mathematicians
made early contributions to the study of the concept
of zero as a number, negative numbers, arithmetic,
and algebra. In addition, trigonometry was further
advanced in India, and, in particular, the modern
definitions of sine and cosine were developed there.
These mathematical concepts were transmitted to
the Middle East, China, and Europe and led to further
developments that now form the foundations of many
areas of mathematics.

4.  Baudhayana
 Katyayana
 Panini, ca. 5th c. BC, Algebraic grammarian
 Yajnavalkya, credited with authorship of
the Shatapatha Brahmana, which contains
calculations related to altar construction.

5. Post-Vedic Sanskrit to Pala period mathematicians (5th c. BC
to 11th c. AD)
 Aryabhata - Astronomer who gave accurate calculations
for astronomical constants, 476AD-520AD
 Aryabhata II
 Bhaskara I
 Brahmagupta - Helped bring the concept of zero into
arithmetic (598 AD-670 AD)
 Bhāskara II
 Mahavira
 Pavuluri Mallana - the first Telugu Mathematician
 Varahamihira
 Shridhara (between 650-850) - Gave a good rule for
finding the volume of a sphere.

6.  Narayana Pandit
 Madhava of Sangamagrama some elements of
Calculus hi
 Parameshvara (1360–1455), discovered drk-ganita,
a mode of astronomy based on
observations, Madhava's Kerala school
 Nilakantha Somayaji,1444-1545 - Mathematician
and Astronomer, Madhava's Kerala school
 Mahendra Suri (14th century)
 Shankara Variyar (c. 1530)
 Raghunatha Siromani, (1475–1550), Logician,
Navadvipa school

8. Aryabhata (475 A.D. -550 A.D.) is the first well known
Indian mathematician. Born in Kerala, he completed his
studies at the university of Nalanda. In the
section Ganita (calculations) of his astronomical treatise
Aryabhatiya (499 A.D.), he made the fundamental advance
in finding the lengths of chords of circles, by using the
half chord rather than the full chord method used by
Greeks. He gave the value of as 3.1416, claiming, for the
first time, that it was an approximation. (He gave it in the
form that the approximate circumference of a circle of
diameter 20000 is 62832.) 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 Aryabhatta.
About-

9.  Aryabhatta is the first writer on astronomy to whom
the Hindus do not allow the honour of a divine
inspiration. Writers on mathematical science
distinctly state that he was the earliest uninspired
and a merely human writer on astronomy. This is a
notice which sufficiently proves his being an
historical character.
 He also ascribed to the epicycles, by which the
motion of aplanet is represented, aform varying
from the circle and nearly elliptic.
 His text specifies the earth's diameter, 1050yojanas;
and the orbit orcircumference of the earth's wind
[spiritus vector] 3393yojanas; which, as the scholiast
rightly argues, is no discrepancy.
His contributions….

11. The great 7th Century Indian mathematician and
astronomer Brahmagupta wrote some important
works on both mathematics and astronomy. He was
from the state of Rajasthan of northwest India (he is
often referred to as Bhillamalacarya, the teacher from
Bhillamala), and later became the head of the
astronomical observatory at Ujjain in central India.
Most of his works are composed in elliptic verse, a
common practice in Indian mathematics at the time,
and consequently have something of a poetic ring to
them. It seems likely that Brahmagupta's works,
especiallyhismostfamoustext,the“Brahmasphut-
asiddhanta”, were brought by the 8th Century
Abbasid caliph Al-Mansur to his newly founded
centre of learning at Baghdad on the banks of the
Tigris, providing an important link between Indian
mathematics and astronomy and the nascent upsurge
in science and mathematics in the Islamic world.
About-

12.  In his work on arithmetic, Brahmagupta explained how to find
the cube and cube-root of an integer and gave rules facilitating
the computation of squares and square roots.
 He also gave rules for dealing with five types of combinations
of fractions. He gave the sum of the squares of the first n natural
numbers as n(n + 1)(2n + 1)⁄ 6and the sum of the cubes of the
first nnatural numbers as (n(n + 1)⁄2)².
 Furthermore, he pointed out, quadratic equations (of the
type x2 + 2 = 11, for example) could in theory have two possible
solutions, one of which could be negative, because 32= 9 and -
32= 9.
 In addition to his work on solutions to general linear equations
and quadratic equations, Brahmagupta went yet further by
considering systems of simultaneous equations (set of equations
containing multiple variables), and solving quadratic equations
with two unknowns, something which was not even considered
in the West until a thousand years later, when Fermat was
considering similar problems in 1657.

14. ▣ Bhaskara (1114A.D. -1185A.D.) orBhaskaracharaya is the most
well known ancient Indian mathematician. He was born in 1114
A.D. atBijjada Bida (Bijapur, Karnataka) in the Sahyadari Hills.
He was the first to declare that any number divided by zero is
infinity and that the sum of any number and infinity is also
infinity. He is famous for his book Siddhanta Siromani(1150
A.D.). It is divided into four sections -Leelavati (a book on
arithmetic),Bijaganita (algebra), Goladhayaya (chapter on
sphere -celestial globe), andGrahaganita (mathematics of the
planets). Leelavati contains many interesting problems and was
a very popular textbook. Bhaskara introducedchakrawal, orthe
cyclic method, to solve algebraic equations. Six centurieslater,
European mathematicians like Galois, Euler and Lagrange
rediscovered this method and called it "inverse cyclic". 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. As an astronomer, Bhaskara is renowned
for his concept of Tatkalikagati(instantaneous motion).
About-

15.  Terms for numbers
In English, the multiples of 1000aretermed as thousand, million, billion,
trillion, quadrillion etc. These terms were named recently in English, but
Bhaskaracharya gave the terms for numbers in multiples of ten which are as
follows: eka(1),dasha(10),shata(100), sahastra(1000), ayuta(10,000),
laksha(100,000),prayuta (1,000,000=million), koti(107),Kutarbuda(108),
abja(109=billion), kharva (1010),nikharva (1011),mahapadma (1012=trillion),
shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and
parardha(1017).

Kuttak
Kuttak according to modern mathematics is 'indeterminate equation of first
order'. In the western world, the method of solving such equations was called as
'pulverizer'. Bhaskara suggested a generalized solution to get multiple answers
for these equations.

Simple mathematical methods
Bhaskara II suggested simple methods to calculate the squares, square roots,
cube, and cube roots of big numbers. The Pythagoras theorem was proved by
him in only two lines. Bhaskara's 'Khandameru'is the famous Pascal Triangle.

17. Srinivasa Ramanujan (1887-1920) hailed as an all-time great
mathematician, like Euler, Gauss or Jacobi, for his natural genius, has left
behind 4000 original theorems, despite his lack of formal education and a
short life-span. In his formative years, after having failed in his F.A. (First
examination in Arts) class at College, he ran from pillar to post in search of
a benefactor. It is during this period, 1903-1914,he kept a recordof the final
results of his original research work in the form of entries in two large-sized
Note Books. These were the ones which he showed to Dewan Bahadur
Ramachandra Rao (Collector of Nellore), V. Ramaswamy Iyer (Founder of
Indian Mathematical Society), R. Narayana Iyer (Treasurer of IMS and
Manager, Madras Port Trust), and to several others to convince them of his
abilities as a Mathematician. The orchestrated efforts of his
admirers, culminated in the encouragement he received from Prof. G.H.
Hardy of Trinity College, Cambridge, whose warm response to the historic
letter of Ramanujan which contained about 100 theorems, resulted in
inducing the Madras University, to its lasting credit, to rise to the occasion
thrice - in offering him the first research scholarship of the University in
May 1913 ; then in offering him a scholarship of 250 pounds a year for five
years with 100 pounds for passage by ship and for initial outfit to go to
England in 1914 ; and finally, by granting Ramanujan 250 pounds a year as
an allowance for 5 years commencing from April 1919 soon after his
triumphant return from Cambridge ``with a scientific standing and
reputation such as no Indian has enjoyed before''.
About-

18.  Ramanujan's arrival atCambridge was the beginning of a very
successful five-year collaboration with Hardy. In some ways the two
made an odd pair: Hardy was a great exponent of rigor in
analysis, while Ramanujan's results were (asHardy put it) "arrived at
by a process of mingled argument, intuition, and induction, of which
he was entirely unable to give any coherent account". Hardy did his
best to fill in the gaps in Ramanujan's education without discouraging
him. He was amazed by Ramanujan's uncanny formal intuition in
manipulating infinite series, continued fractions, and the like: "I have
never met his equal, and can compare him only
with Euleror Jacobi."One remarkable result of the Hardy-Ramanujan
collaboration was a formula for the number p(n) of partitions of a
number n. A partition of a positive integer n is just an expression
for n as a sum of positive integers, regardless of order. Thus p(4) = 5
because 4 can be written as 1+1+1+1, 1+1+2, 2+2, 1+3, or 4.
 Besides his published work, Ramanujan left behind several
notebooks, which have been the object of much study. The English
mathematician G. N. Watson wrote a long series of papers about them.
More recently the American mathematician Bruce C. Berndt has written
a multi-volume study of the notebooks. In 1997The Ramanujan
Journal was launched to publish work "in areas of mathematics
influenced by Ramanujan".

20. ▣ Calyampudi Radhakrishna Rao was born to C.D. Naidu and A.
Laxmikantamma on 10September 1920in Huvvina Hadagalli in
present day Karnataka. He was the eighth in a family of 10children.
After his father’sretirement, the family settled down in
Vishakapatnam in Andhra Pradesh. From his earliest years, Rao
had an interest in mathematics. After completing high school he
joined the Mrs. A.V.N. College at Vishakapatnam for the
Intermediate course. He received his M.A. in Mathematics with first
rank in 1940.Rao decided to pursue a research career in
mathematics, but was denied a scholarship on the grounds of late
submission of the application.
▣ He then went to Kolkata for an interview for a job. He did not get
the job, but by chance he visited the Indian Statistical Institute,
then located in a couple of rooms in the Physics Department of the
Presidency College, Kolkata. He applied for aone-year training
course at the Institute and was admitted to the Training Section of
the Institute from 1January 1941. In July 1941he joined the M.A
Statistics program of the Calcutta University. By the time he passed
the M.A. exam in 1943, winning the gold medal of the University,
he had already published some research papers!
About-

21. ▣ The living legend and doyen of Indian Statistics, 91 year old Prof. Calyampudi
Radhakrishna (C. R.) Rao was awarded the Guy Medal in Gold of the Royal
Statistical Society, UK on the 29thof June, 2011"For his fundamental
contributions to statistical theory and methodology, including unbiased
estimation, variance reduction by sufficiency, efficiency of estimation,
information geometry, as well as the application of matrix theory in linear
statistical inference", the announcement stated.
▣ The Gold Medal is awarded by the Royal Statistical Society (triennially, except
the war period) and named after William Guy. There areSilver and Bronze
Medals too, C. R. Rao already obtained the Silver Medal in 1965. Since 1892he is
the 34threcipient of the Gold Medal. Previously, R. A. Fisher (1946), E. S.
Pearson (1955),J. Neyman (1966),M. S. Bartlett (1969),H. Cramér (1972),and D.
Cox (1973) received this prize, just to mention a few. Among the recipients only
H. Cramér and J. Neyman were outside Great Britain. C. R. Rao is the first non-
European and non-American to receive the award. I believe that he has long
deserved this prize. His formulae and theory include "Cramer -Rao inequality",
"Fischer -Rao theorem" and "Rao - Blackwellisation". . In 1980,18thJune she
solved the multiplication of 13 digit number 7,686,369,774,870 and
2,465,099,745,779picked up by the computer science department of imperial
college, London. Shakuntala solved the question in a flash and took 28seconds
to solve the entire problem, and her answer was
18,947,668,177,995,426,462,773,730.This amazing incident helped her get a place
in the Guinness book of world record

23. ▣ Shakuntala Devi was born on 4th of November, 1939 in
Bengaluru in a well-known Brahmin priest family. She
did card tricks with her father when she was only three.
Shakuntala Devi received her early lessons in
mathematics from her grandfather. By the age of 5,
Shakuntala Devi became an expert in complex mental
arithmetic and was recognised as a child prodigy. She
demonstrated her talents to a large assembly of students
and professors at the University of Mysore a year later.
And when she was 8 years old, she demonstrated her
talents at Annamalai University. Shakuntala Devi has
authored a few books. She shares some of the methods of
mental calculations in her world famous book, Figuring:
The Joy of Numbers. Puzzles to puzzle You, More Puzzles
to puzzle you, The Book of Numbers, Mathability:
Awaken the Math Genius in Your Child, Astrology for
you, Perfect Murder, In the Wonderland of Numbers are
some of the popular books written by her. Her book, In
the Wonderland of Numbers, talks about a girl Neha, and
her fascination for numbers.
About-

24.  Shakuntala Devi was a genius and once in 1977 she
mentally solved the 23rd root of a 201 digit number
without any help from mechanical aid.
 She shares some of the methods of mental calculations in
her world famous book, Figuring: The Joy of Numbers.
Puzzles to puzzle You, More Puzzles to puzzle you, The
Book of Numbers, Mathability: Awaken the Math Genius
in Your Child, Astrology for you, Perfect Murder, In the
Wonderland of Numbers are some of the popular books
written by her. Her book, In the Wonderland of Numbers,
talks about a girl Neha, and her fascination for numbers.
 She has been travelling around the globe performing for
the student community, Prime Ministers, Presidents,
Politicians and Educationalists.

25.  The most fundamental contribution of ancient
India in mathematics is the invention of decimal
system of enumeration, including the invention
of zero. The decimal system uses nine digits (1 to
9)and the symbol zero (for nothing) to denote all
natural numbers by assigning a place value to the
digits. The Arabs carried this system to Africa
and Europe.
 Indians have significantly contributed in the
field of mathematics and ,if God wills, they will
do the same in the near future.