**Brahmagupta (598–668 AD):**

The
great 7th Century Indian mathematician and astronomer Brahmagupta wrote
some important works on both mathematics and astronomy. He was from the
state of Rajasthan of northwest India (he is often referred to as
Bhillamalacarya, the teacher from Bhillamala), and later became the head
of the astronomical observatory at Ujjain in central India. Most of his
works are composed in elliptic verse, a common practice in Indian
mathematics at the time, and consequently have something of a poetic
ring to them.

It
seems likely that Brahmagupta's works, especially his most famous text,
the “Brahmasphutasiddhanta”, were brought by the 8th Century Abbasid
caliph Al-Mansur to his newly founded centre of learning at Baghdad on
the banks of the Tigris, providing an important link between Indian
mathematics and astronomy and the nascent upsurge in science and
mathematics in the Islamic world.

In
his work on arithmetic, Brahmagupta explained how to find the cube and
cube-root of an integer and gave rules facilitating the computation of
squares and square roots. He also gave rules for dealing with five types
of combinations of fractions. He gave 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.

**Brahmagupta’s rules for dealing with zero and negative numbers:**

Brahmagupta’s
genius, though, came in his treatment of the concept of (then
relatively new) the number zero. Although often also attributed to the
7th Century Indian mathematician Bhaskara I, his “Brahmasphutasiddhanta”
is probably the earliest known text to treat zero as a number in its
own right, rather than as simply a placeholder digit as was done by the
Babylonians, or as a symbol for a lack of quantity as was done by the
Greeks and Romans.

Brahmagupta
established the basic mathematical rules for dealing with zero (1 + 0 =
1; 1 - 0 = 1; and 1 x 0 = 0), although his understanding of division by
zero was incomplete (he thought that 1 ÷ 0 = 0). Almost 500 years
later, in the 12th Century, another Indian mathematician, Bhaskara II,
showed that the answer should be infinity, not zero (on the grounds that
1 can be divided into an infinite number of pieces of size zero), an
answer that was considered correct for centuries. However, this logic
does not explain why 2 ÷ 0, 7 ÷ 0, etc, should also be zero - the modern
view is that a number divided by zero is actually "undefined" (i.e. it
doesn't make sense).

Brahmagupta’s
view of numbers as abstract entities, rather than just for counting and
measuring, allowed him to make yet another huge conceptual leap which
would have profound consequence for future mathematics. Previously, the
sum 3 - 4, for example, was considered to be either meaningless or, at
best, just zero. Brahmagupta, however, realized that there could be such
a thing as a negative number, which he referred to as “debt” as a
opposed to “property”. He expounded on the rules for dealing with
negative numbers (e.g. a negative times a negative is a positive, a
negative times a positive is a negative, etc).

Furthermore,
he pointed out, quadratic equations (of the type x2 + 2 = 11, for
example) could in theory have two possible solutions, one of which could
be negative, because 32 = 9 and -32 = 9. In addition to his work on
solutions to general linear equations and quadratic equations,
Brahmagupta went yet further by considering systems of simultaneous
equations (set of equations containing multiple variables), and solving
quadratic equations with two unknowns, something which was not even
considered in the West until a thousand years later, when Fermat was
considering similar problems in 1657.

**Brahmagupta’s Theorem on cyclic quadrilaterals:**

Brahmagupta
even attempted to write down these rather abstract concepts, using the
initials of the names of colours to represent unknowns in his equations,
one of the earliest intimations of what we now know as algebra.

Brahmagupta
dedicated a substantial portion of his work to geometry and
trigonometry. He established √10 (3.162277) as a good practical
approximation for π (3.141593), and gave a formula, now known as
Brahmagupta's Formula, for the area of a cyclic quadrilateral, as well
as a celebrated theorem on the diagonals of a cyclic quadrilateral,
usually referred to as Brahmagupta's Theorem.

## 0 comments:

## Post a Comment