radian measure See
ANGLE
.
radius of convergence See
POWER SERIES
.
Ramanujan, Srinivasa Aiyangar (1887–1920) Indian
Number theory Born on December 22, 1887, in Erode,
Tamil Nadu state, India, Srinivasa Ramanujan is consid-
ered one of India’s greatest mathematical geniuses. With
an inexplicable talent for handling
SERIES
and
CONTIN
-
UED FRACTIONS
, Ramanujan made significant contribu-
tions to the field of
NUMBER THEORY
and offered a
whole host of new and fundamentally important formu-
lae that have since found applications in many different
branches of science. He is also remembered for his
famous collaboration with leading British mathemati-
cian G
ODFREY
H
AROLD
H
ARDY
(1877–1947).
Ramanujan excelled in the topic of mathematics
during his early school years. As a teenager he came
across a small mathematics text by G. C. Carr called
Synopsis of Elementary Results in Pure Mathematics.
Written in a terse style, and being not much more than
a list of 5,000 mathematical formulae, equations, and
results, with no proofs or explanations, Ramanujan
took it upon himself to work through each and every
result and provide his own explanation of it. From that
moment on Ramanujan cared only for mathematics. By
the age of 17 he had conducted his own investigations
on the properties of the
HARMONIC SERIES
and had cal-
culated E
ULER
’
S CONSTANT
to 15 decimal places.
In 1904 Ramanujan won a scholarship to attend
the Government College in Kumbakonam, but it was
revoked the following year because he devoted all his
time to mathematics and neglected his other courses.
He attended another college in Madras, but never grad-
uated, again for failing to attend to his nonmathemati-
cal courses.
As a self-taught scholar, Ramanujan began pub-
lishing results in the Journal of the Indian Mathemati-
cal Society. His brilliant 1911 paper, “Some Properties
of Bernoulli’s Numbers,” on the B
ERNOULLI NUMBERS
garnered him national attention. During this time
Ramanujan was supporting himself as a clerk in an
accounting office.
Ramanujan soon came to realize that he was work-
ing at a mathematical level that was beyond the exper-
tise of anyone he knew in India. Encouraged by friends,
he began writing to mathematicians in England for
guidance. The renowned number theorist Godfrey
Hardy of Trinity College, Cambridge, was the recipient
of one of Ramanujan’s letters and was suitably
impressed by the mathematical content it contained. In
1914 Hardy brought Ramanujan to England, and thus
began their extraordinary collaboration.
Hardy was impressed by the intuitive nature of
Ramanujan’s work. With no formal mathematical
education, Ramanujan had only a vague idea of what
constituted a mathematical proof, but nonetheless
had a profound internal sense of mathematical truth
and amazing insight into the workings of numbers.
One amusing story, for instance, claims that Hardy
435
R