NAME : BIPIN SAIGAL

TOPIC : BHASKER I
SUBMITTED TO SWATI MAAM

Bhāskara IBhāskara (c.

600 – c. 680) (commonly called Bhaskara I to avoid confusion
with the 12th-century mathematician Bhāskara II) was a 7th-
century mathematician and astronomer, who was the first to
write numbers in the Hindu decimal system with a circle for the
zero, and who gave a unique and remarkable rational
approximation of the sine function in his commentary on
Aryabhata's work.This commentary, Āryabhaṭīyabhāṣya, written
in 629 CE, is among the oldest known prose works in Sanskrit
on mathematics and astronomy. He also wrote two astronomical
works in the line of Aryabhata's school, the Mahābhāskarīya and
the Laghubhāskarīya.

On 7 June 1979 the Indian Space Research Organisation

launched Bhaskara I honouring the mathematician.
 BIOGRAPHY
Little is known about
Bhāskara's life. He was probably an astronomer.He was born in
India in the 7th century.There are references to places in India in
Bhaskara's writings. For example, he mentions Valabhi (today
Vala), the capital of the Maitraka dynasty in the 7th century, and
Sivarajapura, which were both in Saurastra which today is the
Gujarat state of India on the west coast of the continent. Also
mentioned are Bharuch (or Broach) in southern Gujarat and
Thanesar in the eastern Punjab which was ruled by Harsa for 41
years from 606. Harsa was the pre-eminent ruler in north India
through the first half of Bhaskara I's life. A reasonable guess
would be that Bhaskara was born in Saurastra and later moved
to Asmaka.

His astronomical education was given by his father. Bhaskara is

considered the most important scholar of Aryabhata's
astronomical school. He and Brahmagupta are two of the most
renowned Indian mathematicians who made considerable
contributions to the study of fractions.
 Representation of numbers
Bhaskara's probably most important mathematical contribution
concerns the representation of numbers in a positional system.
The first positional representations had been known to Indian
astronomers approximately 500 years prior to this work.
However, these numbers, prior to Bhaskara, were written not in
figures but in words or allegories and were organized in verses.
For instance, the number 1 was given as moon, since it exists
only once; the number 2 was represented by wings, twins, or
eyes since they always occur in pairs; the number 5 was given
by the senses. Similar to our current decimal system, these
words were aligned such that each number assigns the factor of
the power of ten correspondings to its position, only in reverse
order: the higher powers were right from the lower ones.

His system is truly positional since the same words representing,

can also be used to represent the values 40 or 400 Quite
remarkably, he often explains a number given in this system,
using the formula ankair api ("in figures this reads"), by
repeating it written with the first nine Brahmi numerals, using a
small circle for the zero . Contrary to his word system, however,
the figures are written in descending values from left to right,
exactly as we do it today. Therefore, at least since 629, the
decimal system is definitely known to the Indian scientists.
Presumably, Bhaskara did not invent it, but he was the first
having no compunctions to use the Brahmi numerals in a
scientific contribution in Sanskrit.
 Further contributions

Mathematics:
Bhaskara wrote three astronomical
contributions. In 629 he annotated the Aryabhatiya, written in
verses, about mathematical astronomy. The Comments referred
exactly to the 33 verses dealing with mathematics. There

His work Mahabhaskariya divides into eight chapters about

mathematical astronomy. In chapter 7, he gives a remarkable
approximation formula for sin x.

Astronomy
The Mahabhaskariya consists of eight chapters dealing with
mathematical astronomy. The book deals with topics such as:
the longitudes of the planets; association of the planets with each
other and also with the bright stars; the lunar crescent; solar and
lunar eclipses; and rising and setting of the planet
Works of Bhaskara I
Bhaskara i is famous for the following works:

Zero, positional arithmetic, the approximation of sine

Introduction
12th-century mathematician Bhaskara

Bhāskara i (c. 600 – c. 680) was a 7th-century Indian

mathematician and astronomer. He is referred to as Bhaskara i in
order to differentiate from the 12th-century mathematician
Bhaskara. Bhaskara-i is considered to be one of the three pearls
of Indian Astronomy and Mathematics along with Brahmagupta
and Madhava Samgramagrama.

Who is Bhāskara i?
Bhāskaracharya was a famous mathematician but not much is
known about his early life except what has been inferred from
his writings. Many believe that he must have been working in a
school of mathematicians in Asmaka which was probably in the
Nizamabad District of Andhra Pradesh. There are other
references to places in India in Bhaskara's writings. There are
some allusions to Valabhi (today Vala), the capital of the
Maitraka dynasty in the 7th century, and Sivarajapura, which
were both in Saurashtra, which today is the state of Gujarat.
There is yet another school of thought which believes that he
was born in Bori, in Parbhani district of Maharashtra. By and
large, it is believed that Bhaskara was born in Saurashtra and
later moved to Asmaka. He was tutored in astronomy by his
father.

It is believed his father taught him astronomy. Bhaskara i is

considered to be a follower of Aryabhata. He is considered to be
the most important scholar of Aryabhata's astronomical school.

Works of Bhaskara i

Zero, positional arithmetic, approximation of sine

One of the most important mathematical contributions is related
to the representation of numbers in a positional system. The first
positional representations were known to Indian astronomers
about 500 years ago before Bhaskaracharya, but the numbers
were not written in figures, but in words, symbols or pictorial
representations. For example, the number 1 was given as the
moon, since there is only one moon. The number 2 was
represented anything in pairs; the number 5 could relate to the
five senses and so on…

He explains a number given in this system, using the formula

ankair api, ("in figures, this reads") by repeating it written with
the first nine Brahmi numerals, using a small circle for the zero.
Brahmi numerals system, dating from 3rd century B.C is an
ancient system for writing numerals and are the direct graphic
ancestors of the modern Indian and Hindu-Arabic numerals.
Since 629, the decimal system has been known to the Indian
scientists. Though Bhaskara i did not invent it, he was the first to
use the Brahmi numerals in a scientific contribution in Sanskrit

Bhaskara I's sine approximation formula

Bhaskara i knew the approximation to the sine functions that
yields close to 99% accuracy, using a function that is simply a
ratio of two quadratic functions.

The formula is given in verses 17 – 19, Chapter VII,

Mahabhaskariya of Bhaskara I. He stated the formula in stylised
verse. Below is briefly stated the rule for finding the bhujaphala
( result from the bThe Aryabhatiyabhashya (629)
The Aryabhatiyabhashya is Bhaskara I’s commentary on the
Aryabhatiya. The Aryabhatiya is a treatise on astronomy written
in Sanskrit. It is said to be the only known surviving work of the
5th-century Indian mathematician Aryabhata. It is estimated that
the book was written around 510 B.C.

Bhaskara I wrote the Aryabhatiyabhasya in 629

Bhaskara I’s comments revolve around the 33 verses in

Aryabhatiya which is about mathematical astronomy. He also
expounds on the problems of indeterminate equations and
trigonometric formulas. While discussing Aryabhatiya, he
discussed cyclic quadrilaterals. He was the first mathematician
to discuss quadrilaterals whose four sides are not equal with
none of the opposite sides parallel. Bhaskara i explains in detail
Aryabhata’s method of solving linear equations with illustrative
examples.
He stressed on the need for providing mathematical rules.ase
sine)and the kotiphala, etc.) The result obtained by multiplying
the R sine of the koṭi due to the planet's kendra by the tabulated
epicycle and dividing the product by 80 without making use of
the Rsine-differences 225, etc.

Subtract the degrees of a bhuja (or koti) from the degrees of a

half-circle (that is, 180 degrees). Then multiply the remainder by
the degrees of the bhuja or koti and put down the result at two
places. At one place subtract the result from 40500. By one-
fourth of the remainder (thus obtained), divide the result at the
other place as multiplied by the 'anthyaphala (that is, the
epicyclic radius). Thus is obtained the entire bahuphala (or,
kotiphala) for the sun, moon or the star-planets. So also are
obtained the direct and inverse Rsines. It is not known how
Bhaskara I arrived at his approximation formula though many
historians of mathematics have marveled at the accuracy of the
formula. The formula is simple and enables one to compute
reasonably accurate values of trigonometric sines without using
any geometry whatsoever.

The Mahabhaskariya (“Great Book of Bhaskara i”):

The Mahabhaskariya is a work on Indian mathematical

astronomy consisting of eight chapters dealing with
mathematical astronomy. The book deals with topics such as the
longitudes of the planets; association of the planets with each
other, conjunctions among the plant and the stars; the lunar
crescent; solar and lunar eclipses; and rising and setting of the
planets. As referred to earlier, this treatise also includes chapters
which illustrate the sine approximation formula. Both the
treatises, Mahabhaskariya knew and Laghubhaskariya''), are
astronomical works in verse. It is interesting to note that Parts of
Mahabhaskariya were later translated into Arabic. The
Aryabhatiyabhashya (629)
The Aryabhatiyabhashya is Bhaskara I’s commentary on the
Aryabhatiya. The Aryabhatiya is a treatise on astronomy written
in Sanskrit. It is said to be the only known surviving work of the
5th-century Indian mathematician Aryabhata. It is estimated that
the book was written around 510 B.C.

Bhaskara I wrote the Aryabhatiyabhasya in 629

Bhaskara I’s comments revolve around the 33 verses in

Aryabhatiya which is about mathematical astronomy. He also
expounds on the problems of indeterminate equations and
trigonometric formulas. While discussing Aryabhatiya, he
discussed cyclic quadrilaterals. He was the first mathematician
to discuss quadrilaterals whose four sides are not equal with
none of the opposite sides parallel. Bhaskara i explains in detail
Aryabhata’s method of solving linear equations with illustrative
examples.

He stressed on the need for providing mathematical rules.

Summary :
It would not be wrong to say that the Bhaskara i has played a
pivotal role in the lasting influence of Aryabhata’s works.
Befitting a mathematician of his stature, the Indian Space
Research Organisation launched Bhaskara I honouring the
mathematician on 7 June 1979.

Mathematicians have agreed that Bhaskara i’s sine

approximation formula is fairly accurate for all practical
purposes. The formula, whether in its original form or modified
versions has been used by authors down the line. Keeping in
mind its origins centuries ago, it reflects a very high standard of
mathematics in India at that time. Truly wondrous to imagine
how the seeds of modern science were sown centuries ago.
PICTURES OF BHASKAR 1 :
 1

1
1 INTRODUCTION OF BHASKAR 1
2 MAHABHASKRIYA AND LAGHUBAHASKRIYA