SREENIVASA RAMANUJAN

Early life

Ramanujan's home on Sarangapani Street, Kumbakonam

Ramanujan was born on 22 December 1887 in Erode, Madras Presidency (now

Pallipalayam, Erode, Tamil Nadu), at the residence of his maternal grandparents in a Brahmin

family. His father, K. Srinivasa Iyengar, worked as a clerk in a sari shop and hailed from the

district of Thanjavur. 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 named Sadagopan, who died less than three months later. In December 1889,

Ramanujan had smallpox and recovered, unlike thousands in the Thanjavur District who died

from the disease that year.[8]

He moved with his mother to her parents' house in Kanchipuram,

near Madras (now Chennai). In November 1891, and again in 1894, his mother gave birth to two

children, but both children died in infancy.

On 1 October 1892, Ramanujan was enrolled at the local school. In March 1894, he was

moved to a Tamil medium 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.[11]

When his paternal grandfather died, he was sent back to his

maternal grandparents, who were then living in Madras. He did not like school in Madras, and he

tried to avoid attending. His family enlisted a local constable to make sure he attended school.

Within six months, Ramanujan was back in Kumbakonam.

Since Ramanujan's father was at work most of the day, his mother took care of him as a

child. He had a close relationship with her. From her, he learned about tradition and puranas. He

learned to sing religious songs, to attend pujas at the temple, and to keep particular eating habits

– all of which are part of Brahmin culture. At the Kangayan Primary School, Ramanujan

performed well. Just before the age of 10, in November 1897, he passed his primary

examinations in English, Tamil, geography and arithmetic. With his scores, he stood first in the

district.[13]

That year, Ramanujan entered Town Higher Secondary School where he encountered

formal mathematics for the first time.

By age 11, he had exhausted the mathematical knowledge of two college students who

were lodgers at his home. He was later lent a book on advanced trigonometry written by S. L.

Loney. He completely mastered this book by the age of 13 and discovered sophisticated theorems

on his own. By 14, he was receiving merit certificates and academic awards which continued

throughout his school career and also assisted the school in the logistics of assigning its 1200

students (each with their own needs) to its 35-odd teachers. He completed mathematical exams in

half the allotted time, and showed a familiarity with geometry and infinite series. Ramanujan was

shown how to solve cubic equations in 1902 and he went on to find his own method to solve the

quartic. The following year, not knowing that the quintic could not be solved by radicals, he tried

to solve the quintic.

Adulthood in India

On 14 July 1909, Ramanujan was married to a nine-year-old bride, Srimathia Janki

(Janakiammal), (21 March 1899 – 13 April 1994). She came from Rajendram, a village close to

Marudur (Karur district) Railway Station. Ramanujan's father did not participate in the marriage

ceremony.

After the marriage, Ramanujan developed a hydrocele testis, an abnormal swelling of the tunica

vaginalis, an internal membrane in the testicle. The condition could be treated with a routine

surgical operation that would release the blocked fluid in the scrotal sac. His family did not have

the money for the operation, but in January 1910, a doctor volunteered to do the surgery for free.[

After his successful surgery, Ramanujan searched for a job. He stayed at friends' houses

while he went door to door around the city of Madras (now Chennai) looking for a clerical

position. To make some money, he tutored some students at Presidency College who were

preparing for their F.A. exam.

In late 1910, Ramanujan was sick again, possibly as a result of the surgery earlier in the

year. He feared for his health, and even told his friend, R. Radakrishna Iyer, to "hand these

[Ramanujan's mathematical notebooks] over to Professor Singaravelu Mudaliar [the mathematics

professor at Pachaiyappa's College] or to the British professor Edward B. Ross, of the Madras

Christian College."[33]

After Ramanujan recovered and retrieved his notebooks from Iyer, he took

a northbound train from Kumbakonam to Villupuram, a coastal city under French control.

Mathematical achievements

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 by G. H. Hardy 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

This result is based on the negative fundamental discriminant d = −4×58 = −232 with

class number h(d) = 2 (note that 5×7×13×58 = 26390 and that 9801=99×99; 396=4×99) and is

related to the fact that

Compared to Heegner numbers, which have class number 1 and yield similar formulae.

Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of

some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term

also gives the approximation for π, which is correct to six decimal places. See

also the more general Ramanujan–Sato series.

One of his remarkable capabilities was the rapid solution for problems. He was sharing a

room with P. C. Mahalanobis who had a problem, "Imagine that you are on a street with houses

marked 1 through n. There is a house in between (x) such that the sum of the house numbers to

the left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what

are n and x?" This is a bivariate problem with multiple solutions. Ramanujan thought about it

and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was

the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it.

"It is simple. The minute I heard the problem, I knew that the answer was a continued fraction.

Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan

replied.[85][86]

His intuition also led him to derive some previously unknown identities, such as

for all , where is the gamma function, and related to a special value of the Dedekind eta

function. Expanding into series of powers and equating coefficients of , , and gives some

deep identities for the hyperbolic secant.

In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a

non-convergent asymptotic series that permits exact computation of the number of partitions of

an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact

convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to

a powerful new method for finding asymptotic formulae, called the circle method.

He discovered mock theta functions in the last year of his life. For many years these

functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak

Maass forms.

The Ramanujan conjecture

Main article: Ramanujan–Petersson conjecture

Although there are numerous statements that could have borne the name Ramanujan

conjecture, there is one statement 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 his work on Weil conjectures.

Hardy–Ramanujan number 1729

Main article: 1729 (number)

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:[91]

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

1729 = 13 + 12

3 = 9

3 + 10

3.

Generalizations of this idea have created the notion of "taxicab numbers". Coincidentally,

1729 is also a Carmichael number.

ARYABHATA

Aryabhata (Sanskrit: आर्यभट; IAST: Āryabhata) or Aryabhata I (476–550 CE) was the

first of the major mathematician-astronomers from the classical age of Indian mathematics and

Indian astronomy. His works include the Āryabhatīya (499 CE, when he was 23 years old) and

the Arya-siddhanta.

Biography

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.

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.

Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of

which are lost. 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.

Mathematics

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.

Approximation of π

Aryabhata worked on the approximation for pi ( ), and may have come to the

conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikamśatamaṣṭaguṇamdvāṣaṣṭistathāsahasrāṇām

ayutadvayaviṣkambhasyāsannovṛttapariṇāhaḥ.

"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.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this

an approximation but that the value is incommensurable (or irrational). If this is correct, it is

quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by

Lambert.

After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was

mentioned in Al-Khwarizmi's book on algebra.

Trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."

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.

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find

integer solutions to Diophantine equations that have the form ax + by = c. (This problem was

also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese

remainder theorem.) This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when

divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general,

diophantine equations, such as this, can be notoriously difficult. They were discussed extensively

in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's

method of solving such problems, elaborated by Bhaskara in 621 CE, is called the kuttaka

method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a

recursive algorithm for writing the original factors in smaller numbers. This algorithm became

the standard method for solving first-order diophantine equations in Indian mathematics, and

initially the whole subject of algebra was called kuttaka -ganita or simply kuttaka Algebra

In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and

cubes:

and

(see squared triangular number)

Astronomy

Aryabhata's system of astronomy was called the audAyaka system, in which days are reckoned

from uday, dawn at lanka or "equator". Some of his later writings on astronomy, which

apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly

reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, he seems to

ascribe the apparent motions of the heavens to the Earth's rotation.

Brahmagupta

Brahmagupta was an Indian mathematician and astronomer.

He is the author of two early works on mathematics and astronomy: the

Brāhmasphudasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a

theoretical treatise, and the Khandakhādyaka ("edible bite", dated 665), a more practical text.

According to his commentators, Brahmagupta was a native of Bhinmal.

Brahmagupta was the first to give rules to compute with zero. The texts composed by

Brahmagupta were composed in elliptic verse in Sanskrit, as was common practice in Indian

mathematics. As no proofs are given, it is not known how Brahmagupta's results were derived.

Author

The text of the Brāhmasphudasiddhānta (24.7–8) states that Brahmagupta composed the

work at the age of 30 in Saka era 550 (i.e. CE 628), during the reign of King Vyāghramukha,

establishing CE 598 as Brahmagupta's year of birth. He is said to have been a native of

Bhillamala, a city in the state of Rajasthan,[3]

at the time the seat of power of the Gurjars. His

father's name is recorded as Jisnugupta.

He likely lived most of his life in Bhillamala during the reign (and possibly under the patronage)

of King Vyaghramukha.[5]

For this reason, Brahmagupta is also referred to as Bhillamalacharya,

that is, the teacher from Bhillamala. He was the head of the astronomical observatory at Ujjain,

and it was during his tenure there that he wrote his two surviving treatises, both on mathematics

and astronomy: the Brahmasphutasiddhanta in 628, and the Khandakhadyaka in 665.

Mathematics

Algebra

Brahmagupta gave the solution of the general linear equation in chapter eighteen of

Brahmasphutasiddhanta,

The difference between rupas, when inverted and divided by the difference of the

unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that

from which the square and the unknown are to be subtracted.

which is a solution for the equation equivalent to , where

rupas refers to the constants c and e. He further gave two equivalent solutions to the general

quadratic equation

18.44. Diminish by the middle [number] the square-root of the rupas multiplied by four

times the square and increased by the square of the middle [number]; divide the remainder by

twice the square. [The result is] the middle [number].

18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased

by the square of half the unknown, diminish that by half the unknown [and] divide

[theremainder] by its square. [The result is] the unknown.

which are, respectively, solutions for the equation equivalent to,

and

He went on to solve systems of simultaneous indeterminate equations stating that the

desired variable must first be isolated, and then the equation must be divided by the desired

variable's coefficient. In particular, he recommended using "the pulverizer" to solve equations

with multiple unknowns.

18.51. Subtract the colors different from the first color. [The remainder] divided by the

first [color's coefficient] is the measure of the first. [Terms] two by two [are] considered [when

reduced to] similar divisors, [and so on] repeatedly. If there are many [colors], the pulverizer [is

to be used].

Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Addition

was indicated by placing the numbers side by side, subtraction by placing a dot over the

subtrahend, and division by placing the divisor below the dividend, similar to our notation but

without the bar. Multiplication, evolution, and unknown quantities were represented by

abbreviations of appropriate terms. The extent of Greek influence on this syncopation, if any, is

not known and it is possible that both Greek and Indian syncopation may be derived from a

common Babylonian source.

Arithmetic

Four fundamental operations (addition, subtraction, multiplication and division) were

known to many cultures before Brahmagupta. This current system is based on the Hindu Arabic

number system and first appeared in Brahmasphutasiddhanta. Brahmagupta describes the

multiplication as thus “The multiplicand is repeated like a string for cattle, as often as there are

integrant portions in the multiplier and is repeatedly multiplied by them and the products are

added together. It is multiplication. Or the multiplicand is repeated as many times as there are

component parts in the multiplier”. [10]

Indian arithmetic was known in Medieval Europe as

"Modus Indoram" meaning method of the Indians. In Brahmasphutasiddhanta, Multiplication

was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta,

entitled Calculation, Brahmagupta details operations on fractions. The reader is expected to

know the basic arithmetic operations as far as taking the square root, although he explains how to

find the cube and cube-root of an integer and later gives rules facilitating the computation of

squares and square roots. He then gives rules for dealing with five types of combinations of

fractions, , , , , and .

Series

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.

12.20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased

by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these

with identical balls [can also be computed].

Here Brahmagupta found the result in terms of the sum of the first n integers, rather than

in terms of n as is the modern practice.

He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the

sum of the cubes of the first n natural numbers as (n(n+1)/2)².

Zero

Brahmagupta's Brahmasphuṭasiddhanta is the first book that mentions zero as a number,[14]

hence Brahmagupta is considered the first to formulate the concept of zero. He gave rules of

using zero with negative and positive numbers. Zero plus a positive number is the positive

number and negative number plus zero is a negative number etc. The Brahmasphutasiddhanta is

the earliest known text to treat zero as a number in its own right, rather than as simply a

placeholder digit in representing another number as was done by the Babylonians or as a symbol

for a lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his

Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first

describes addition and subtraction,

18.30. [The sum] of two positives is positives, of two negatives negative; of a positive

and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and

zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero.[...]

Here Brahmagupta states that and as for the question of where he did not

commit himself.[15]

His rules for arithmetic on negative numbers and zero are quite close to the

modern understanding, except that in modern mathematics division by zero is left undefined.