A Legend of Mathematics-Bhaskaracharya
Abstract
Bhaskara II or Bhaskarachrya was an Indian mathematician and astronomer.
His mathematical works Lilawati and Bijaganita are considered to be
unparalleled and a memorial to his profound intelligence. His arithmetic
text Lilawati is divided into 13 chapters and covers many branches of
mathematics, arithmetic, algebra, geometry, and a little trigonometry and
mensuration and methods to solve indeterminate equations, and combinations.
His Bijaganita (Algebra) was a work in twelve chapters. It was the first
text to recognize that a positive number has two square roots (a positive
and negative square root). His work Bijaganita is e ectively a treatise on
algebra in his treatise Siddhant Shiromani he writes on planetary
positions, eclipses, cosmography, mathematical techniques and astronomical
equipment. Birth of Bhaskaracharya Bhaskara II or Bhaskarachrya was born
near Bijjada Bida (in present day Bijapur district, Karnataka state, South
India) into the Deshastha Brahmin family. His father Mahesvara was as an
astrologer, who taught him mathematics, which he later passed on to his son
Loksamudra. Loksamudras son helped to set up a school in 1207 for the study
of Bhskaras writings. Introduction Bhaskaracharya wrote Siddhanta Shiromani
in 1150 AD when he was 36 years old. It is divided into four parts,
Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact, each part can
be considered as separate book. This is a mammoth work containing about
1450 verses. The numbers of verses in Lilawati are 278, in Beejaganit there
are 213, in Ganitadhyaya there are 451 and in Goladhyaya there are 501
verses. Siddhanta Shiromani has surpassed all the ancient books on
astronomy in India. It consists of simple methods of calculations from
Arithmetic to Astronomy.
After Bhaskaracharya nobody could write excellent books on
mathematics and astronomy in lucid language in India. Bhaskaracharya used
to give no proofs of any theorem.64 J. of Ramanujan Society of Math. and
Math. Sc. Lilawati is an excellent example of how a di cult subject like
mathematics can be written in poetic language. Lilawati has been translated
in many languages throughout the world. Till 1857, for about 700 years,
mathematics was taught in India from Bhaskaracharyas Lilawati and
Beejaganit. No other textbook has enjoyed such long lifespan.
Bhaskaracharya was the first to discover gravity, 500 years before Sir
Isaac Newton. He was the champion among mathematicians of ancient and
medieval India. In the Surya Siddhant he makes a note on the force of
gravity: Objects fall on earth due to a force of attraction by the earth.
Therefore, the earth, planets, constellations, moon, and sun are held in
orbit due to this attraction. Bhaskaracharyas contributions to mathematics
Lilawati and Beejaganit together consist of about 500 verses.
A few important highlights of Bhaskaracharyas mathematics are as
follows: 1. Terms for numbers Bhaskaracharya has given the terms for
numbers in multiples of ten and he says that these terms were coined by
ancients for the sake of positional values. Bhaskar s terms for numbers are
as follows:
eka (1), dasha (10), shata (100), sahastra (1000), ayuta (10,000), laksha
(100,000), prayuta (1,000,000=million), koti (107), arbuda (108),
abja(109=billion), kharva (1010), nikharva (1011), mahapadma
(1012=trillion), shanku (1013), jaladhi (1014), antya(1015=quadrillion),
Madhya (1016) and parardha (1017).
2. Kuttak Kuttak means to crush to ne particles or
to pulverize. Kuttak is nothing but the modern indeterminate equation of
first order. There are many kinds of Kuttaks.
As for example- In the equation, ax + b = cy a and b are
known positive integers and the values of x and y are to be found in
integers. As a particular example, he considered 100x + 90 = 63y
Bhaskaracharya gives the solution of this example as, x =1881144207 and y
=30130230330 It is not easy to nd solutions of these equations but
Bhaskaracharya has given a generalized solution to get multiple answers.
3. Chakrawaal Chakrawaal is the
indeterminate equation of second order in modern mathematics. This type of
equation is also called Pells equation, though Pell had never solved the
equation. Much before Pell, the equation was solved by an ancient and
eminent Indian mathematician, Brahmagupta (628 AD). The solution is given
in A Legend of Mathematics-Bhaskaracharya 65 his Brahma-sphuta-siddhanta.
Bhaskara modified the method and gave a general solution of this equation.
For example, he considered the equation 61x2 +1 = y2 and gave the values of
x = 22615398 and y = 1766319049 There is an interesting history behind this
very equation. The Famous French mathematician Pierre de Fermat (1601-1664)
asked his friend Bessy to solve this very equation. Bessy used to solve the
problems in his head like present day Shakuntala devi. Bessy failed to
solve the problem. After about 100 years another famous French
mathematician solved this problem. But his method is lengthy and could nd a
particular solution only, while Bhaskara gave the solution for ve cases. In
his book History of mathematics, Carl Boyer says about this equation, In
connection with the Pells equation ax2 + 1 = y2, Bhaskara gave particular
solutions for ve cases, a = 8, 11, 32, 61, and 67. For example for 61x2 + 1
= y2 he gave the solutions, x = 226153980 and y = 1766319049, this is an
impressive feat in calculations and its veri cations alone will tax the e
orts of the reader Henceforth the so called Pells equation should be
recognized as Brahmagupta-Bhaskaracharya equation.
4. Simple mathematical methods Bhaskara has given simple methods
to find the squares, square roots, cube, and cube roots of big numbers. He
has proved the Pythagoras theorem in only two lines. The famous Pascal
Triangle was Bhaskaras Khandameru. Bhaskara has given problems on that
number triangle. Pascal was born 500 years after Bhaskara. Several problems
on permutations and combinations are given in Lilavati. He has called the
method ankapaash. Bhaskara has given an approximate value of Pie as 22/7
and a more accurate value as 3.1416. He knew the concept of infinity and
called it as khahar rashi, which means Anant. In the last, we can say that
his work is outstanding for its systemization, improved methods and the new
topics that he has introduced. Furthermore, the Lilawati contained
excellent recreational problems and it is thought that Bhaskaras intention
may have been that a student of Lilawati should concern himself with the
mechanical application of the method.
K Rajaram IRS 24525
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