Aryabhata: School Projects

Famous Indian mathematician-Astronomer Aryabhata: School projects

Mathematics project for class 10: Biography of Aryabhata

Aryabhata name:- While there is a tendency to misspell his name as “Aryabhatta” by analogy with other names having the “Bhatta” suffix, his name is properly spelled aryabhata every astronomical text spells his name thus, including brahmagupta’s references to him “In more than a hundred places by name”. Furthermore, in most instances “Aryabhatta” would not fit the metre either.

Aryabhata mentions in the aryabhatiya that it was composed 3,600 years into the kali yuga, when he was 23 years old. This corresponds to 499 ce, and implies that he was born in 476. Aryabhata provides no information about his place of birth. The only information comes from bhāskara I, who describes aryabhata as āśmakīya, “One belonging to the aśmaka country.” during the buddha’s time, a branch of the aśmaka people settled in the region between the narmada and godavari rivers in central india; aryabhata is believed to have been born there.

It is fairly certain that, at some point, he went to kusumapura for advanced studies and lived there for some time. Both hindu and buddhist tradition, as well as bhāskara I (ce 629), identify kusumapura as pāṭaliputra, modern patna. A verse mentions that aryabhata was the head of an institution (kulapa) at kusumapura, and, because the university of nalanda was in pataliputra 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.

Major works:- 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. The arya-siddhanta, a lost work on astronomical computations, is known through the writings of aryabhata’s contemporary, varahamihira, and later mathematicians and commentators, including brahmagupta and bhaskara I. This work appears to be based on the older surya siddhanta and uses the midnight-day reckoning, as opposed to sunrise in aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhaya-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.
9. 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 argues 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 pi:- aryabhata worked on the approximation for pi (pi), and may have come to the conclusion that pi is irrational. In the second part of the aryabhatiyam (gaṇitapāda 10), 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. After aryabhatiya was translated into arabic (c. 820 ce) this approximation was mentioned in al-khwarizmi’s book on algebra.

Trigonometry:- aryabhata discussed the concept of sine in his work by the name of ardha-jya, which literally 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. 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)”. (in arabic, jiba is a meaningless word.) 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 word sine.

Algebra:- in aryabhatiya, aryabhata provided elegant results for the summation of series of squares and cubes:- 1^2 + 2^2 + cdots + n^2 = {n(n + 1)(2n + 1) over 6} and 1^3 + 2^3 + cdots + n^3 = (1 + 2 + cdots + n)^2 (see squared triangular number)