Math Project for 2013-2014
There is no doubt that the world today is
indebted to the contributions made by Indian
mathematicians. Some of the most
important contributions made by Indian
mathematicians were the introduction of
decimal system as well as the invention of
zero. Here are some the famous Indian
mathematicians, along with their
• Place value system and zero
The place-value system, was clearly in place in
Aryabhata’s work. While he did not use a symbol
for zero, the French mathematician Georges Ifrah
argues that knowledge of zero was suggested in
Aryabhata's place-value system as a place holder for
the powers of ten with null coefficients.
• Approximation of π
Aryabhata worked on the approximation for pi (π), and
may have come to the conclusion that it is irrational.
Aryabhata gave the area of a triangle as "for a triangle,
the result of a perpendicular with the half-side is
Brahmagupta's Brahmasphuṭasiddhanta is the first book
that mentions zero as a number, 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.
Brahmagupta's most famous geometrical work is
his formula for cyclic quadrilaterals. Given the lengths of
the sides of any cyclic quadrilateral, Brahmagupta gave an
approximate and an exact formula for the figure's area,
i.e., “the area is the square root from the product of the
halves of the sums of the sides diminished by each side of
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 that Ramanujan's
discoveries are unusually rich and that there is often
more to them than initially meets the eye. As a byproduct, 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
P.C. Mahalnobis asked Nelson Annandale the questions on
what factors influence the formation of European and
Indian marriages. He wanted to examine if the Indian side
came from any specific castes. He used the data collected
by Annandale and the caste specific measurements made
by Herbert Risley to come up with the conclusion that the
sample represented a mix of Europeans mainly with
people from Bengal and Punjab but not with those from
the Northwest Frontier Provinces or from Chhota Nagpur.
He also concluded that the intermixture more frequently
involved the higher castes than the lower ones. This
analysis was described by his first scientific paper in
1922. During the course of these studies he found a way
of comparing and grouping populations using a
multivariate distance measure. This measure, denoted
"D2" and now named Mahalanobis distance, is
independent of measurement scale.
Estimation theory is a branch of statistics that deals
with estimating the values of parameters based on
measured/empirical data that has a random
component. The parameters describe an underlying
physical setting in such a way that their value affects
the distribution of the measured data.
An estimator attempts to approximate the unknown
parameters using the measurements.
In estimation theory, two approaches are generally
The probabilistic approach assumes that the
measured data is random with probability
distribution dependent on the parameters of
The set-membership approach assumes that the
measured data vector belongs to a set which
depends on the parameter vector.
D.R. Kaprekar discovered a number of results in number
theory and described various properties of numbers
including Kaprekar Constant, Kaprekar Numbers, self
numbers, Harshad numbers and Demlo numbers.
Kaprekar discovered the Kaprekar constant or 6174 in
1949. He showed that 6174 is reached in the limit as
one repeatedly subtracts the highest and lowest
numbers that can be constructed from a set of four
digits that are not all identical. Thus, starting with 1234,
4321 − 1234 = 3087, then
8730 − 0378 = 8352, and
8532 − 2358 = 6174.
Repeating from this point onward leaves the same
number (7641 − 1467 = 6174). In general, when the
operation converges it does so in at most seven
He was influenced by the mathematicians Hermann
Weyl and Claude Chevalley. From 1950 to 1963 he was
at the Columbia University and worked on
representations of semisimple Lie groups. During this
period he established as his special area the study of
the discrete series representations of semisimple Lie
groups, which are analogues of the Peter–Weyl
theory in the non-compact case.
•Discrete Series Representation
In mathematics, a discrete series representation an
irreducible unitary representation of a locally
compact topological group G that is a
subrepresentation of the left regular representation of
G on L²(G). In the Plancherel measure, such
representations have positive measure. The name
comes from the fact that they are exactly the
representations that occur discretely in the
decomposition of the regular representation.
Satyendra Nath Bose
Bose showed that the contemporary theory was
inadequate, because it predicted results not in
accordance with experimental results. In the process of
describing this discrepancy, Bose for the first time took
the position that the Maxwell–Boltzmann
distribution would not be true for microscopic particles,
where fluctuations due to Heisenberg's uncertainty
principle will be significant. Thus he stressed the
probability of finding particles in the phase space, each
state having volume h3, and discarding the distinct
position and momentum of the particles. Bose's
interpretation is now called Bose–Einstein statistics.
Einstein adopted the idea and extended it to atoms. This
led to the prediction of the existence of phenomena
which became known as Bose–Einstein condensate, a
dense collection of bosons, which was demonstrated to
exist by experiment in 1995.
Some of Bhāskara II’s contributions are:
•A proof of the Pythagorean theorem by calculating the
same area in two different ways and then canceling out
terms to get a2 + b2 = c2.
•Solutions of quadratic, cubic and quartic indeterminate
equations are explained.
•Solutions of indeterminate quadratic equations (of the
type ax2 + b = y2).
•Integer solutions of linear and quadratic indeterminate
•Solved quadratic equations with more than one
unknown, and found negative and irrational solutions.
•In Siddhanta Shiromani, Bhaskara developed spherical
trigonometry along with a number of
other trigonometric results.
Karmarkar's algorithm solves linear programming problems
in polynomial time. These problems are represented by "n"
variables and "m" constraints. The previous method of
solving these problems consisted of problem representation
by an "x" sided solid with "y" vertices, where the solution
was approached by traversing from vertex to vertex.
Karmarkar's novel method approaches the solution by
cutting through the above solid in its traversal.
Consequently, complex optimization problems are solved
much faster using the Karmarkar algorithm.
His algorithm thus enables faster business and policy
decisions. Karmarkar's algorithm has stimulated the
development of several other interior point methods, some
of which are used in current codes for solving linear