Nārāyaṇa Paṇḍita () (1340–1400) was an Indian

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, mathematical structure, structure, space, Mathematica ...

. Plofker writes that his texts were the most significant Sanskrit mathematics treatises after those of Bhaskara II, other than the Kerala school. He wrote the '' Ganita Kaumudi'' (lit. "Moonlight of mathematics") in 1356 about mathematical operations. The work anticipated many developments in

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...

Life and Works

About his life, the most that is known is that: Narayana Pandit wrote two works, an arithmetical treatise called ''Ganita Kaumudi'' and an

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

ic treatise called ''Bijaganita Vatamsa''. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavati, titled ''Karmapradipika'' (or ''Karma-Paddhati''). Although the ''Karmapradipika'' contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and

magic square In mathematics, especially History of mathematics, historical and recreational mathematics, a square array of numbers, usually positive integers, is called a magic square if the sums of the numbers in each row, each column, and both main diago ...

s. Narayana's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order

indeterminate equation In mathematics, particularly in number theory, an indeterminate system has fewer equations than unknowns but an additional a set of constraints on the unknowns, such as restrictions that the values be integers. In modern times indeterminate equati ...

''nq''² + 1 = ''p''² (

Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...

), solutions of indeterminate higher-order equations, mathematical operations with

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

, several

geometrical Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

rules, methods of integer factorization, and a discussion of magic squares and similar figures. Narayana has also made contributions to the topic of

cyclic quadrilateral In geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral (four-sided polygon) whose vertex (geometry), vertices all lie on a single circle, making the sides Chord (geometry), chords of the circle. This circle is called ...

s. Narayana is also credited with developing a method for systematic generation of all permutations of a given sequence.

Narayana's cows sequence

In his ''Ganita Kaumudi'' Narayana proposed the following problem on a herd of cows and calves: Translated into the modern mathematical language of recurrence sequences: : for , with initial values :. The first few terms are 1, 1, 1, 2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88,... . The limit ratio between consecutive terms is the

supergolden ratio In mathematics, the supergolden ratio is a geometrical aspect ratio, proportion, given by the unique real polynomial root, solution of the equation Its decimal expansion begins with . The name ''supergolden ratio'' is by analogy with the golde ...

. The definition of the sequence and the supergolden ratio are closely related to the definitions of the

Fibonacci sequence In mathematics, the Fibonacci sequence is a Integer sequence, sequence in which each element is the sum of the two elements that precede it. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted . Many w ...

and the

Golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \fr ...

References

1340 births 1400 deaths Indian Hindus 14th-century Indian mathematicians {{asia-mathematician-stub

Life and Works

Narayana's cows sequence

See also

References