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 considered one of India’s greatest mathematical geniuses. With
an inexplicable talent for handling SERIES and CONTINUED FRACTIONS, Ramanujan made significant contributions to the field of NUMBER THEORY and offered a
whole host of new and fundamentally important formulae that have since found applications in many different
branches of science. He is also remembered for his
famous collaboration with leading British mathematician GODFREY HAROLD HARDY (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 calculated EULER’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the Government College in Kumbakonam, but it was
revoked the following year because he devoted all his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 graduated, again for failing to attend to his nonmathematical courses.
As a self-taught scholar, Ramanujan began publishing results in the Journal of the Indian Mathematical Society. His brilliant 1911 paper, “Some Properties
of Bernoulli’s Numbers,” on the BERNOULLI 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 working at a mathematical level that was beyond the expertise 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
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 considered one of India’s greatest mathematical geniuses. With
an inexplicable talent for handling SERIES and CONTINUED FRACTIONS, Ramanujan made significant contributions to the field of NUMBER THEORY and offered a
whole host of new and fundamentally important formulae that have since found applications in many different
branches of science. He is also remembered for his
famous collaboration with leading British mathematician GODFREY HAROLD HARDY (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 calculated EULER’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 graduated, again for failing to attend to his nonmathematical courses.
As a self-taught scholar, Ramanujan began publishing results in the Journal of the Indian Mathematical Society. His brilliant 1911 paper, “Some Properties
of Bernoulli’s Numbers,” on the BERNOULLI 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 working at a mathematical level that was beyond the expertise 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