2. Indian mathematicians have made a
number of contributions to mathematics
that have significantly influenced scientists
and mathematicians in the modern era.
These include place-value arithmetical
notation, the ruler, the concept of zero, and
most importantly, the Arabic-Hindu
numerals predominantly used today.
12. Aryabhata was born in Taregna, which is a small
town in Bihar, India, about 30 km from Patna
(then known as Pataliputra), the capital city of
Bihar State. Evidences justify his birth there. In
Taregna Aryabhata set up an Astronomical
Observatory in the Sun Temple 6th century.
There is no evidence that he was born outside
Patliputra and traveled to Magadha, the centre of
instruction, culture and knowledge for his studies
where he even set up a coaching institute. However,
early Buddhist texts describe Ashmakas as being
further south, in dakshinapath or the Deccan, while
other texts describe the Ashmakas as having
fought Alexander.
13. It is fairly certain that, at some point, he
went to Kusumapura for advanced
studies and that he lived there for some
time. A verse mentions that Aryabhatta
was the head of an institution at
Kusumapura, and, because the
university of Nalanda was in Patliputra
at the time and had an astronomical
observatory, it is speculated that
Aryabhata might have been the head of
the Nalanda university as well.
Aryabhata is also reputed to have set up
an observatory at the Sun temple
in Taregana, Bihar.
15. The place-value system, first seen in the
3rd century Bakhshali Manuscript, was
clearly in place in his work. While he did
not use a symbol for zero, the French
mathematician Georges Ifrah explains that
knowledge of zero was implicit in
Aryabhata's place-value system as a place
holder for the powers of ten
with null coefficients
However, Aryabhata did not use the Brahmi
numerals. Continuing the Sanskrit tradition
from Vedic times, he used letters of the
alphabet to denote numbers, expressing
quantities, such as the table of sine's in
a mnemonic form.
16. In Ganitapada 6, Aryabhata gives the area of a
triangle as
tribhujasya phalashariram samadalakoti
bhujardhasamvargah
that translates to: for a triangle, the result of a
perpendicular with the half-side is the area.
Aryabhata discussed the concept of sine in his
work by the name of ardha-jya. Literally, it
means "half-chord". For simplicity, people started
calling it jya. When Arabic writers translated his
works from Sanskrit into Arabic, they referred it
as jiba
17. In Aryabhatiya Aryabhata provided elegant results for the
summation of series of squares and cubes:
AND
18. Aryabhata worked on the approximation for
pi and may have come to the conclusion
that it is irrational. In the second part of
theAryabhatiyam , he writes:
caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā
sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add
62,000. By this rule the circumference of a circle
with a diameter of 20,000 can be approached."
This implies that the ratio of the circumference
to the diameter is ((4 + 100) × 8 + 62000)/20000
= 62832/20000 = 3.1416, which is accurate to
five significant figures.
19. The Aryabhatta numeration is a system
of numerals based on Sanskrit phonemes. It
was introducedin the early 6th century by
Āryabhaṭa, in the first chapter titled Gītika
Padam of his Aryabhatiya. It attributes a
numerical value to each syllable of the form
consonant vowel possible in Sanskrit
phonology, from ka = 1 up to hau = 10
20.
21. Āryabhaṭa's sine table is a set of twenty-
four of numbers given in the astronomical
treatise Āryabhaṭiya composed by the fifth
century Indian mathematician and
astronomer Āryabhaṭa (476–550 CE), for
the computation of the half-chords of
certain set of arcs of a circle. It is not a
table in the modern sense of a
mathematical table; that is, it is not a set of
numbers arranged into rows and columns.
22. The second section of Āryabhaṭiya titled Ganitapāda
contains a stanza indicating a method for the
computation of the sine table. There are several
ambiguities in correctly interpreting the meaning of this
verse. For example, the following is a translation of the
verse given by Katz wherein the words in square brackets
are insatz
"When the second half- chord partitioned is less than the
first half-chord, which is approximately equal to the
corresponding arc, by a certain amount, the remaining
sine-differences are less than the previous ones
