If x is a real number, then the bracket symbols x,
x, and {x} are used to denote the floor, ceiling, and
fractional part values, respectively, of x.
Square brackets [ ] and parentheses ( ) are placed at
the end points of an INTERVAL on the real number line
to indicate whether or not the end points of that interval are to be included.
See also EXPANDING BRACKETS; FLOOR/CEILING/
FRACTIONAL PART FUNCTIONS; ORDER OF OPERATION.
Brahmagupta (ca. 598–665) Indian Arithmetic, Geometry, Astronomy Born in Ujjain, India, scholar Brahmagupta is recognized as one of the important
mathematicians of the seventh century. His famous 628
text Brahmasphutasiddhanta (The opening of the universe) on the topic of astronomy includes such notable
mathematical results as his famous formula for the
AREA of a cyclic QUADRILATERAL, the integer solution to
certain algebraic equations, and methods of solution to
simultaneous equations. This work is also historically
significant as the first documented systematic use of
ZERO and negative quantities as valid numbers in
ARITHMETIC.
Brahmagupta was head of the astronomical observatory at Ujjain, the foremost mathematical center of
ancient India, and took an avid interest in the development of astronomical observation and calculation. The
first 10 of the 25 chapters of Brahmasphutasiddhanta
pertain solely to astronomy, discussing the longitude of
the planets, lunar and solar eclipses, and the timing of
planet alignments. Although rich in mathematical computation and technique, it is the remainder of the work
that offers an insight into Brahmagupta’s far-reaching
understanding of mathematics on an abstract level.
Brahmagupta goes on to describe the decimal PLACEVALUE SYSTEM used in India at his time for representing
numerals and the methods for doing arithmetic in this
system. (For instance, he outlines a method of “long
multiplication” essentially equivalent to the approach we
use today.) Brahmagupta permits zero as a valid number
in all of his computations, and in fact gives it the explicit
status of a number by defining it as the result of subtracting a quantity from itself. (Until then, zero acted as
nothing more than a placeholder to distinguish 203 from
23, for instance.) He also explains the arithmetical properties of zero—that adding zero to a number leaves that
number unchanged and multiplying any number by zero
produces zero, for instance. Brahmagupta also detailed
the arithmetic of negative numbers (which he called
“debt”) and suggested, for the first time, that they may
indeed be valid solutions to certain problems.
Brahmagupta next explores problems in ALGEBRA.
He develops some basic algebraic notations and then
presents a series of methods for solving a variety of linear and quadratic equations. For instance, he devised
an ingenious technique for finding integer solutions to
equations of the form ax
2 + c = y
2
. (For example, Brahmagupta correctly asserted that x = 226,153,980 and y
= 1,766,319,049 are the smallest positive integer solutions to 61x
2 + 1 = y
2
.) Brahmagupta also presents the
famous SUMS OF POWERS formulae:
as well as algorithms for computing square roots.
Unfortunately, as was the practice of writing at the
time, Brahmagupta never gave any word of explanation as to how his solutions or formulae were found.
No proofs were ever offered.
In the final sections of Brahmasphutasiddhanta,
Brahmagupta presents his famous formula for the area
of a cyclic quadrilateral solely in terms of the lengths
of its sides. Curiously, Brahmagupta does not state
that the formula is true only for quadrilaterals
inscribed in a CIRCLE.
In a second work, Khandakhadyaka, written in
665, Brahmagupta discusses further topics in astronomy. Of particular interest to mathematicians, Brahmagupta presents here an ingenious method for
computing values of sines.
Brahmagupta’s methods and discoveries were
extremely influential. Virtually every text that discusses Indian astronomy describes or uses some aspect
of his work.
See also BRAHMAGUPTA’S FORMULA.
Brahmagupta’s formula Seventh-century Indian mathematician and astronomer BRAHMAGUPTA derived a formula for the AREA of a QUADRILATERAL inscribed in a
1 2
1
2
1 2
1 2 1
6
1 2
1
4
2
2
2
3
3
3
2
2
+ + + =
+
+ + +
=
+
+
+ + +
=
+
L
L
L
n
n n
n
n n
n
n
n n
(
)
(
)(
)
(
)
Brahmagupta’s formula 51
x, and {x} are used to denote the floor, ceiling, and
fractional part values, respectively, of x.
Square brackets [ ] and parentheses ( ) are placed at
the end points of an INTERVAL on the real number line
to indicate whether or not the end points of that interval are to be included.
See also EXPANDING BRACKETS; FLOOR/CEILING/
FRACTIONAL PART FUNCTIONS; ORDER OF OPERATION.
Brahmagupta (ca. 598–665) Indian Arithmetic, Geometry, Astronomy Born in Ujjain, India, scholar Brahmagupta is recognized as one of the important
mathematicians of the seventh century. His famous 628
text Brahmasphutasiddhanta (The opening of the universe) on the topic of astronomy includes such notable
mathematical results as his famous formula for the
AREA of a cyclic QUADRILATERAL, the integer solution to
certain algebraic equations, and methods of solution to
simultaneous equations. This work is also historically
significant as the first documented systematic use of
ZERO and negative quantities as valid numbers in
ARITHMETIC.
Brahmagupta was head of the astronomical observatory at Ujjain, the foremost mathematical center of
ancient India, and took an avid interest in the development of astronomical observation and calculation. The
first 10 of the 25 chapters of Brahmasphutasiddhanta
pertain solely to astronomy, discussing the longitude of
the planets, lunar and solar eclipses, and the timing of
planet alignments. Although rich in mathematical computation and technique, it is the remainder of the work
that offers an insight into Brahmagupta’s far-reaching
understanding of mathematics on an abstract level.
Brahmagupta goes on to describe the decimal PLACEVALUE SYSTEM used in India at his time for representing
numerals and the methods for doing arithmetic in this
system. (For instance, he outlines a method of “long
multiplication” essentially equivalent to the approach we
use today.) Brahmagupta permits zero as a valid number
in all of his computations, and in fact gives it the explicit
status of a number by defining it as the result of subtracting a quantity from itself. (Until then, zero acted as
nothing more than a placeholder to distinguish 203 from
23, for instance.) He also explains the arithmetical properties of zero—that adding zero to a number leaves that
number unchanged and multiplying any number by zero
produces zero, for instance. Brahmagupta also detailed
the arithmetic of negative numbers (which he called
“debt”) and suggested, for the first time, that they may
indeed be valid solutions to certain problems.
Brahmagupta next explores problems in ALGEBRA.
He develops some basic algebraic notations and then
presents a series of methods for solving a variety of linear and quadratic equations. For instance, he devised
an ingenious technique for finding integer solutions to
equations of the form ax
2 + c = y
2
. (For example, Brahmagupta correctly asserted that x = 226,153,980 and y
= 1,766,319,049 are the smallest positive integer solutions to 61x
2 + 1 = y
2
.) Brahmagupta also presents the
famous SUMS OF POWERS formulae:
as well as algorithms for computing square roots.
Unfortunately, as was the practice of writing at the
time, Brahmagupta never gave any word of explanation as to how his solutions or formulae were found.
No proofs were ever offered.
In the final sections of Brahmasphutasiddhanta,
Brahmagupta presents his famous formula for the area
of a cyclic quadrilateral solely in terms of the lengths
of its sides. Curiously, Brahmagupta does not state
that the formula is true only for quadrilaterals
inscribed in a CIRCLE.
In a second work, Khandakhadyaka, written in
665, Brahmagupta discusses further topics in astronomy. Of particular interest to mathematicians, Brahmagupta presents here an ingenious method for
computing values of sines.
Brahmagupta’s methods and discoveries were
extremely influential. Virtually every text that discusses Indian astronomy describes or uses some aspect
of his work.
See also BRAHMAGUPTA’S FORMULA.
Brahmagupta’s formula Seventh-century Indian mathematician and astronomer BRAHMAGUPTA derived a formula for the AREA of a QUADRILATERAL inscribed in a
1 2
1
2
1 2
1 2 1
6
1 2
1
4
2
2
2
3
3
3
2
2
+ + + =
+
+ + +
=
+
+
+ + +
=
+
L
L
L
n
n n
n
n n
n
n
n n
(
)
(
)(
)
(
)
Brahmagupta’s formula 51