it is the mathematics projest for class 10th term 1 mostly based on CBSE format

1. MATHEMATICS PROJECT
BY:-ATISHAY JAIN
CLASS:- X-C
ROLL NO.:-10336

2. ACKNOWLEDGEMENT:-
• I would like to express my special thanks of gratitude to my
teacher Mr.Vikram Yadav as well as our principal Mrs.R.R Dhaiya
who gave me the golden opportunity to do this wonderful project
on the topic Mathematician’s Biography, which also helped me in
doing a lot of Research and I came to know about so many new
things I am really thankful to them.
• Secondly I would also like to thank my parents and friends who
helped me a lot in finalizing this project within the limited time
frame.

3. TOPICS:-
•1-Biography:-Aryabhata
•2-Biography:-Ramanujan

4. BIOGRAPHY:-
ARYABHATA

5. ARYABHATA
NAME:-
While there is a tendency to
misspell his name as "Aryabhatta"
by analogy with other names
having the "bhatta" suffix, his name
is properly spelled Aryabhata every
astronomical text spells his name
thus, including Brahmagupta's
references to him "in more than a
hundred places by name".
Furthermore, in most instances
"Aryabhatta" would not fit the
metre either.

6. TIME AND PLACE OF BIRTH:-
ARYABHATA MENTIONS IN THE ARYABHATIYA THAT IT WAS COMPOSED
3,600 YEARS INTO THE KALI YUGA, WHEN HE WAS 23 YEARS OLD. THIS
CORRESPONDS TO 499 CE, AND IMPLIES THAT HE WAS BORN IN 476.
ARYABHATA PROVIDES NO INFORMATION ABOUT HIS PLACE OF BIRTH. THE
ONLY INFORMATION COMES FROM BHĀSKARA I, WHO DESCRIBES
ARYABHATA AS ĀŚMAKĪYA, "ONE BELONGING TO THE AŚMAKA COUNTRY."
DURING THE BUDDHA'S TIME, A BRANCH OF THE AŚMAKA PEOPLE SETTLED
IN THE REGION BETWEEN THE NARMADA AND GODAVARI RIVERS IN
CENTRAL INDIA; ARYABHATA IS BELIEVED TO HAVE BEEN BORN THERE.

7. EDUCATION:-
IT IS FAIRLY CERTAIN THAT, AT SOME POINT, HE WENT TO
KUSUMAPURA FOR ADVANCED STUDIES AND LIVED THERE FOR SOME
TIME. BOTH HINDU AND BUDDHIST TRADITION, AS WELL AS BHĀSKARA
I (CE 629), IDENTIFY KUSUMAPURA AS PĀṬALIPUTRA, MODERN PATNA.
A VERSE MENTIONS THAT ARYABHATA WAS THE HEAD OF AN
INSTITUTION (KULAPA) AT KUSUMAPURA, AND, BECAUSE THE
UNIVERSITY OF NALANDA WAS IN PATALIPUTRA AT THE TIME AND HAD
AN ASTRONOMICAL OBSERVATORY, IT IS SPECULATED THAT ARYABHATA
MIGHT HAVE BEEN THE HEAD OF THE NALANDA UNIVERSITY AS WELL.
ARYABHATA IS ALSO REPUTED TO HAVE SET UP AN OBSERVATORY AT
THE SUN TEMPLE IN TAREGANA, BIHAR.

8. MAJOR WORKS:-
HIS MAJOR WORK, ARYABHATIYA, A COMPENDIUM OF MATHEMATICS AND ASTRONOMY, WAS
EXTENSIVELY REFERRED TO IN THE INDIAN MATHEMATICAL LITERATURE AND HAS SURVIVED TO
MODERN TIMES. THE MATHEMATICAL PART OF THE ARYABHATIYA COVERS ARITHMETIC,
ALGEBRA, PLANE TRIGONOMETRY, AND SPHERICAL TRIGONOMETRY. IT ALSO CONTAINS
CONTINUED FRACTIONS, QUADRATIC EQUATIONS, SUMS-OF-POWER SERIES, AND A TABLE OF
SINES.
THE ARYA-SIDDHANTA, A LOST WORK ON ASTRONOMICAL COMPUTATIONS, IS KNOWN
THROUGH THE WRITINGS OF ARYABHATA'S CONTEMPORARY, VARAHAMIHIRA, AND LATER
MATHEMATICIANS AND COMMENTATORS, INCLUDING BRAHMAGUPTA AND BHASKARA I. THIS
WORK APPEARS TO BE BASED ON THE OLDER SURYA SIDDHANTA AND USES THE MIDNIGHT-DAY
RECKONING, AS OPPOSED TO SUNRISE IN ARYABHATIYA. IT ALSO CONTAINED A DESCRIPTION OF
SEVERAL ASTRONOMICAL INSTRUMENTS: THE GNOMON (SHANKU-YANTRA), A SHADOW
INSTRUMENT (CHHAYA-YANTRA), POSSIBLY ANGLE-MEASURING DEVICES, SEMICIRCULAR AND
CIRCULAR (DHANUR-YANTRA / CHAKRA-YANTRA), A CYLINDRICAL STICK YASTI-YANTRA, AN
UMBRELLA-SHAPED DEVICE CALLED THE CHHATRA-YANTRA, AND WATER CLOCKS OF AT LEAST
TWO TYPES, BOW-SHAPED AND CYLINDRICAL.

9. PLACE VALUE SYSTEM AND ZERO:-
THE PLACE-VALUE SYSTEM, FIRST SEEN IN THE 3RD-CENTURY BAKHSHALI
MANUSCRIPT, WAS CLEARLY IN PLACE IN HIS WORK. WHILE HE DID NOT USE A
SYMBOL FOR ZERO, THE FRENCH MATHEMATICIAN GEORGES IFRAH ARGUES
THAT KNOWLEDGE OF ZERO WAS IMPLICIT IN ARYABHATA'S PLACE-VALUE
SYSTEM AS A PLACE HOLDER FOR THE POWERS OF TEN WITH NULL
COEFFICIENTS.HOWEVER, ARYABHATA DID NOT USE THE BRAHMI NUMERALS.
CONTINUING THE SANSKRITIC TRADITION FROM VEDIC TIMES, HE USED LETTERS
OF THE ALPHABET TO DENOTE NUMBERS, EXPRESSING QUANTITIES, SUCH AS
THE TABLE OF SINES IN A MNEMONIC FORM.

10. APPROXIMATION OF PI:-
ARYABHATA WORKED ON THE APPROXIMATION FOR PI (PI), AND MAY HAVE COME TO THE
CONCLUSION THAT PI IS IRRATIONAL. IN THE SECOND PART OF THE ARYABHATIYAM
(GAṆITAPĀDA 10), HE WRITES:
CATURADHIKAM ŚATAMAṢṬAGUṆAM DVĀṢAṢṬISTATHĀ SAHASRĀṆĀM
AYUTADVAYAVIṢKAMBHASYĀSANNO VṚTTAPARIṆĀHAḤ.
"ADD FOUR TO 100, MULTIPLY BY EIGHT, AND THEN ADD 62,000. BY THIS RULE THE
CIRCUMFERENCE OF A CIRCLE WITH A DIAMETER OF 20,000 CAN BE APPROACHED."
THIS IMPLIES THAT THE RATIO OF THE CIRCUMFERENCE TO THE DIAMETER IS ((4 + 100) × 8
+ 62000)/20000 = 62832/20000 = 3.1416, WHICH IS ACCURATE TO FIVE SIGNIFICANT
FIGURES.
AFTER ARYABHATIYA WAS TRANSLATED INTO ARABIC (C. 820 CE) THIS APPROXIMATION
WAS MENTIONED IN AL-KHWARIZMI'S BOOK ON ALGEBRA.

11. TRIGONOMETRY:-
ARYABHATA DISCUSSED THE CONCEPT OF SINE IN HIS WORK BY THE NAME OF
ARDHA-JYA, WHICH LITERALLY MEANS "HALF-CHORD". FOR SIMPLICITY, PEOPLE
STARTED CALLING IT JYA. WHEN ARABIC WRITERS TRANSLATED HIS WORKS FROM
SANSKRIT INTO ARABIC, THEY REFERRED IT AS JIBA. HOWEVER, IN ARABIC
WRITINGS, VOWELS ARE OMITTED, AND IT WAS ABBREVIATED AS JB. LATER
WRITERS SUBSTITUTED IT WITH JAIB, MEANING "POCKET" OR "FOLD (IN A
GARMENT)". (IN ARABIC, JIBA IS A MEANINGLESS WORD.) LATER IN THE 12TH
CENTURY, WHEN GHERARDO OF CREMONA TRANSLATED THESE WRITINGS FROM
ARABIC INTO LATIN, HE REPLACED THE ARABIC JAIB WITH ITS LATIN
COUNTERPART, SINUS, WHICH MEANS "COVE" OR "BAY"; THENCE COMES THE
ENGLISH WORD SINE.

12. ALGEBRA:-
IN ARYABHATIYA, ARYABHATA PROVIDED ELEGANT RESULTS FOR THE
SUMMATION OF SERIES OF SQUARES AND CUBES:-
1^2 + 2^2 + CDOTS + N^2 = {N(N + 1)(2N + 1) OVER 6}
AND
1^3 + 2^3 + CDOTS + N^3 = (1 + 2 + CDOTS + N)^2
(SEE SQUARED TRIANGULAR NUMBER)

13. BIOGRAPHY:-
SRINIVASA RAMANUJAN

14. EARLY LIFE:-
RAMANUJAN WAS BORN ON 22 DECEMBER 1887 INTO A TAMIL
BRAHMIN FAMILY IN ERODE, MADRAS PRESIDENCY (NOW TAMIL
NADU), AT THE RESIDENCE OF HIS MATERNAL GRANDPARENTS.
HIS FATHER, K. SRINIVASA IYENGAR, WORKED AS A CLERK IN
A SARI SHOP AND HAILED FROM THANJAVUR DISTRICT. HIS
MOTHER, KOMALATAMMAL, WAS A HOUSEWIFE AND ALSO SANG AT
A LOCAL TEMPLE. THEY LIVED IN SARANGAPANI STREET IN A
TRADITIONAL
HOME IN THE TOWN OF KUMBAKONAM. THE FAMILY
HOME IS NOW A MUSEUM. WHEN RAMANUJAN WAS A YEAR AND
A HALF OLD, HIS MOTHER GAVE BIRTH TO A SON, SADAGOPAN, WHO
DIED LESS THAN THREE MONTHS LATER.

15. ON 1 OCTOBER 1892, RAMANUJAN WAS ENROLLED AT THE LOCAL
SCHOOL. AFTER HIS MATERNAL GRANDFATHER LOST HIS JOB
AS A COURT OFFICIAL IN KANCHIPURAM, RAMANUJAN AND
HIS MOTHER MOVED BACK TO KUMBAKONAM AND HE WAS
ENROLLED
IN THE KANGAYAN PRIMARY SCHOOL. WHEN HIS PATERNAL
GRANDFATHER DIED, HE WAS SENT BACK TO HIS MATERNAL
GRANDPARENTS, THEN LIVING IN MADRAS. HE DID NOT LIKE
SCHOOL IN MADRAS, AND TRIED TO AVOID ATTENDING. HIS FAMILY
ENLISTED A LOCAL CONSTABLE TO MAKE SURE THE BOY ATTENDED
SCHOOL. WITHIN SIX MONTHS, RAMANUJAN WAS BACK
IN KUMBAKONAM.

16. ATTENTIONTOWARDSMATHEMATICS:-
RAMANUJAN MET DEPUTY COLLECTOR V. RAMASWAMY AIYER,
WHO HAD RECENTLY FOUNDED THE INDIAN MATHEMATICAL
SOCIETY. WISHING FOR A JOB AT THE REVENUE DEPARTMENT
WHERE AYER WORKED, RAMANUJAN SHOWED HIM HIS MATHEMATICS
NOTEBOOKS. AS AYER LATER RECALLED:
I WAS STRUCK BY THE EXTRAORDINARY MATHEMATICAL
RESULTS CONTAINED IN IT [THE NOTEBOOKS].
I HAD NO MIND TO SMOTHER HIS GENIUS BY AN APPOINTMENT
IN THE LOWEST RUNGS OF THE REVENUE
DEPARTMENT.

17. AIYER SENT RAMANUJAN, WITH LETTERS OF INTRODUCTION, TO
HIS MATHEMATICIAN FRIENDS IN MADRAS. SOME OF THEM
LOOKED AT HIS WORK AND GAVE HIM LETTERS OF INTRODUCTION
TO R. RAMACHANDRA RAO, THE DISTRICT COLLECTOR FOR
NELLORE AND THE SECRETARY OF THE INDIAN MATHEMATICAL
SOCIETY. RAO WAS IMPRESSED BY RAMANUJAN’S RESEARCH
BUT DOUBTED THAT IT WAS HIS OWN WORK. RAMANUJAN
MENTIONED A CORRESPONDENCE HE HAD WITH PROFESSOR
SALDHANA, A NOTABLE BOMBAY MATHEMATICIAN, IN WHICH
SALDHANA EXPRESSED A LACK OF UNDERSTANDING OF HIS WORK
BUT CONCLUDED THAT HE WAS NOT A PHONY.

18. WORKS:-
RAMANUJAN WROTE HIS FIRST FORMAL PAPER FOR
THE JOURNAL ON THE PROPERTIES OF BERNOULLI NUMBERS.
ONE
PROPERTY HE DISCOVERED WAS THAT THE DENOMINATORS
(SEQUENCE
A027642 IN OEIS) OF THE FRACTIONS OF BERNOULLI
NUMBERS WERE ALWAYS DIVISIBLE BY SIX. HE ALSO DEVISED
A METHOD OF CALCULATING BN BASED ON PREVIOUS
BERNOULLI
NUMBERS. ONE OF THESE METHODS FOLLOWS:
IT WILL BE OBSERVED THAT IF N IS EVEN BUT NOT EQUAL TO
ZERO,

19. (I) BN IS A FRACTION AND THE NUMERATOR OF BN
N IN ITS LOWEST
TERMS IS A PRIME NUMBER,
(II) THE DENOMINATOR OF BN CONTAINS EACH OF THE FACTORS 2
AND 3 ONCE AND ONLY ONCE,
(III) 2N(2N - 1)BN
N IS AN INTEGER AND 2(2N - 1)BN CONSEQUENTLY
IS AN ODD INTEGER.
IN HIS 17-PAGE PAPER, “SOME PROPERTIES OF BERNOULLI’S
NUMBERS”, RAMANUJAN GAVE THREE PROOFS, TWO COROLLARIES
AND THREE CONJECTURES. RAMANUJAN’S WRITING INITIALLY
HAD MANY FLAWS.

20. LIFE IN ENGLAND:-
RAMANUJAN DEPARTED FROM MADRAS ABOARD THE S.S.
NEVASA ON 17 MARCH 1914. WHEN HE DISEMBARKED IN
LONDON ON 14 APRIL, NEVILLE WAS WAITING FOR HIM WITH
A CAR. FOUR DAYS LATER, NEVILLE TOOK HIM TO HIS HOUSE
ON CHESTERTON ROAD IN CAMBRIDGE. RAMANUJAN IMMEDIATELY
BEGAN HIS WORK WITH LITTLEWOODS AND HARDY. AFTER
SIX WEEKS, RAMANUJAN MOVED OUT OF NEVILLE’S HOUSE
AND TOOK UP RESIDENCE ON WHEW ELL'S COURT, A FIVE-MINUTE
WALK FROM HARDY’S ROOM. HARDY AND LITTLEWOODS BEGAN
TO LOOK AT RAMANUJAN’S NOTEBOOKS.

21. ILLNESS AND DEATH:-
THROUGHOUT HIS LIFE, RAMANUJAN WAS PLAGUED BY HEALTH
PROBLEMS. HIS HEALTH WORSENED IN ENGLAND. HE WAS DIAGNOSED
WITH TUBERCULOSIS AND A SEVERE VITAMIN DEFICIENCY,
AND WAS CONFINED TO A SANATORIUM. IN 1919 HE RETURNED TO
KUMBAKONAM, MADRAS PRESIDENCY, AND SOON THEREAFTER,
IN 1920, DIED AT THE AGE OF 32. HIS WIDOW, S. JANAKI AMMAL,
MOVED TO BOMBAY; IN 1950 SHE RETURNED TO CHENNAI
(FORMERLY MADRAS), WHERE SHE LIVED UNTIL HER DEATH IN
1994 AT AGE 95.
A 1994 ANALYSIS OF RAMANUJAN’S MEDICAL RECORDS AND
SYMPTOMS BY DR. D. A. B. YOUNG CONCLUDED THAT IT WAS
MUCH MORE LIKELY HE HAD HEPATIC AMOEBIASIS, AN ILLNESS
THEN WIDESPREAD IN MADRAS, RATHER THAN TB. HE HAD TWO
EPISODES OF DYSENTERY BEFORE HE LEFT INDIA. WHEN NOT
PROPERLY TREATED, DYSENTERY CAN LIE DORMANT FOR YEARS AND
LEAD TO HEPATIC AMOEBIASIS.ALTHOUGH DIFFICULT TO DIAGNOSE,
IT IS A READILY CURABLE DISEASE.

22. THE RAMANUJAN CONJECTURE:-
ALTHOUGH THERE ARE NUMEROUS STATEMENTS THAT COULD HAVE
BORNE THE NAME RAMANUJAN CONJECTURE, THERE IS ONE THAT
WAS VERY INFLUENTIAL ON LATER WORK. IN PARTICULAR, THE CONNECTION
OF THIS CONJECTURE WITH CONJECTURES OF ANDRÉ WEIL
IN ALGEBRAIC GEOMETRY OPENED UP NEW AREAS OF RESEARCH.
THAT 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 THAT WORK.