Aryabhata,Zero, Bhaskara

Aryabhata wrote many books on mathematics, astronomy etc but most of them are lost today.His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.Arabic translation of Aryabhata’s work is Al ntf or Al-nanf and it claims that it is a translation of Aryabhatiya like other Ancient Indian texts -translated mostly by Persian scholars during golden age and advent of Islam.

Place Value System and ZERO
----------------------------------

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 Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.


Approximation of π
------------------------

Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes :

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.||

Translation : “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 calculates to 3.1416 close to the actual value Pi (3.14159).Aryabhata used the word āsanna (approaching / approximating), to mean that not only is this an approximation but that the value is incommensurable (or irrational).This is quite a sophisticated insight, because the irrationality of pi(π) was proved only in 1761 by Johann Heinrich Lambert.After Aryabhatiya was translated into Arabic (during 820 CE) this approximation was mentioned in Al-Khwarizmi‘s book on algebra.

Contributions in Trigonometry

-------------------------------

In Ganitapada 6, Aryabhata gives the area of a triangle as :

tribhujasya phalashariram samadalakoti bhujardhasamvargah ||

Translation : “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, which literally means “half-chord (half-wave)“. For simplicity, people started calling it jya.When Arabic writers translated his works from Sanskrit into Arabic, they referred it as "jiba".However, in Arabic writings, vowels are omitted, and it was abbreviated as jb.
Later writers substituted it with "jaib", meaning “pocket” or “fold (in a garment)“.Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic "jaib" with its Latin counterpart, sinus, which means “cove” or “bay“; thence comes the English SINE.Alphabetic code has been used by him to define a set of increments. If we use Aryabhata’s table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher