ARYABHATTA

Aryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta.

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,[1] including Brahmagupta's references to him "in more than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta" does not fit the metre either.[1]

[edit] Birth

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 CE.[1]

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." While aśmaka was originally situated in the northwest of India, it is widely attested that, during the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers, in the South Gujarat–North Maharashtra region of central India. Aryabhata is believed to have been born there.[1][3] However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north.[3]

Work

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there for some time.[4] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pā ṭ aliputra , modern Patna.[1] 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.[1] Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.[5]

Kerala hypothesis

It has also been suggested that aśmaka (Sanskrit for "stone") might be the region in Kerala that is now known as Koṭuṅṅallūr, based on the belief that it was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance").[1] It is also claimed that the fact that several commentaries on the Aryabhatiya have come from Kerala suggest that it was Aryabhata's main place of life and activity. But K. V. Sarma, the authority on Kerala's astronomical tradition,[6] disagrees and cites many commentaries that have come from outside Kerala and the Aryasiddhanta's being completely unknown in Kerala.[1] In recent (2007) papers, K. Chandra Hari uses a discrepancy in Aryabhata's astronomical values to deduce that he carried out his calculations from a place in Kerala at the same meridian as Ujjayini, possibly Chamravattam (10°N51, 75°E45) in central Kerala. He further hypothesizes that Asmaka was the Jain country surrounding Shravanabelagola, taking its name from the stone monoliths there.[6][7][8]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.[9]

[edit] Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. 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.[3]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.[3]

Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha(ca. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)

3. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.

4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, ca. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE).

Mathematics

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[10]

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.[11]

[edit] Pi as irrational

Aryabhata worked on the approximation for Pi (π), and may have come to the conclusion

that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno 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."[12]

This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert).[13]

After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[3]

Mensuration and trigonometry

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."[14]

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. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jiab, meaning "cove" or "bay." (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 jiab with its Latin counterpart, sinus, which means "cove" or "bay". And after that, the sinus became sine in English.[15]

TEENAGERS Age limits

There is not exact age limit of teen years. Specialists have different opinions about it. Some of them think that it is rather large period from 11 till 18; others think it is 12-15, 13-15, 14-16, and 13-17. There are specialists who divide teen’s years into periods: early teen, middle teen, late teen. It looks like the most common opinion is to think about teenager’s years as the period between 13 and 17 years. But the period limit is individual and can begin and end at different ages for different people.

Physical development

At this period teenagers have big changes physically. Their body proportions are becoming like that of adults. The growth is usually rapid, especially in height. Skeleton grows quicker than muscles and at this period teenagers can feel very awkward. Many teenagers have problems with hearth activity that can not fully serve the growth. That is the reason of rising of blood pressure, head aches, getting tired quickly. This can lead to frequent changes in the mood, general emotional instability. That is why at the beginning of teen period parents should take care that children have enough sleep and eat healthy food. Many teenagers have problems with their skin, especially on faces. Owing to rapid growth, teenagers start to pay close attention to their appearance. All drawbacks are often exaggerated but for some it can result in low self esteem and becoming reserved, if their real appearance is far from their ideal picture of themselves.

Feeling of being adult

Rapid growth and physical development determines one of the most important characteristics of teenagers. It is feeling of already being adult. A teenager understands that he is not a child anymore and he does not want to be treated like a child. He wants to be accepted according to his new status but this status exists often only in his mind. Parents and other adults are often not ready to accept this. The change was so quick, only yesterday a child was obedient and today he tells he is grown. For a large number of teenagers adulthood is connected with some symbols, like smoking, drinking, coming home whenever they want to, being out of control, dating, using make up, and wearing fashionable or expensive clothes. They try to do some of this to prove them and show others that they are not children. Trying to seem adult is one of

important reasons why teenagers start smoking or drinking. If society does not accept a teenager as a grown individual and he is continued to be treated like a child, he starts to fight for his rights and this might results in serious conflicts with parents and teachers.

Relation with parents As it was told, a teenager feels grown and his relations with parents change. He refuses to obey demands that earlier he did, he wants to have less control and more independence, and he wants to be equal. But it is not always easy for parents to change the relations, because they are used too much control and coordination, and because they do not see reasons to change something (a teenager is still a student and still dependant on them). Besides, teenagers often do not show any proof of more mature behavior.

It is very important if parents choose democratic style of relations. At this age parents should give more freedom and independence, less control. Teen years are a very important period for children to learn norms of social behavior and adults should help them with it. The time of communication should increase. Parents should choose such a style of relations in which they can be as older and wiser friends, not to command but explain and guide. Relations should be cooperation based on respect of each other and where parents serve as example. Teenagers should not only be given more independence and rights but more duties.

If there is no good contact and understanding between parents and teenagers, it will result in constant conflicts. This, in this turn, can have bad consequences for a teenager; they will go to the street in search of people who will? Understand? Them and become a victim of drug sellers.

Study

 

At this age all brain functions are rapidly developing too, that is why the period is very good for study and acquiring practical skills. But schools stops being the only place of getting knowledge. As it was told, at this age communication with friends comes to the first place, and teenager begins to be interested in many other things, besides study, volume of information received form other sources increases.

That is why lessons should be very interesting and motivating. Only with this condition learning and school performance will be very successful. Boring lessons is the reason of discipline problems, missing classes and loss of interest and motivation.

Teachers should be careful in communication with teenagers. Opinion of class is very important and they do not want to look funny in classmates' eyes. All critics should be done in friendly manner and not touch human dignity. Praise of achievements and progress can play a good role too; it will increase self opinion of a teenager and help him to become more interested in the subject. If a teenager feels humiliated, he will break discipline to restore the respect of the class. It is up to teachers to take care that there are no isolated students. Grouping is in nature of teenager and if he is not accepted by his class, he might search for company of other isolated students at school and it is not always good for all, because such groups usually form gangs and bring a lot of trouble making their revenge.

Relations with friends

Necessity to be with others is characteristics features of teenagers. Communication with friends takes a very important place in life, for many it becomes even more important than communication with parents or study. If at younger age communication with friends is often based on school study, at teen years it becomes much wider. Teenagers are connected with other on hobbies, common interests or just talking. Personal qualities of friends start to become more important than progress at school. Teenagers consider relations with friends as their private matter and do not like interference. If parents are not tactful discussing friends, it can cause protest and conflicts. At this age it is very important for teenagers to be accepted by others. Friendship is one of the ways to satisfy this need. A teenager needs someone who will understand him, someone whom he can tell about his problems and concerns and get help. Friends become example for teenagers and have a lot of influence on them, more influence than teachers or parents. That is why choice of friends is very important. It is very common for teenagers to want to be "like others", like their friends. And it often becomes a problem for parents when children demand different things that friends have and they do not. Communicating with friends teenagers learn vital skills of living and interacting in society and get experience in building personal relations.

Parents should start to worry if their children do not have friends and try to help to find a reason of it and correct situation.

Development of individuality

Teen years are a very important period in development of individuality. It influences the whole future life of children. Teenagers start to learn themselves, their inner world, strong and weak personal qualities. Sometimes it might happen that teenagers have a big gap between their ideal picture of themselves and a real one. It can lead to depression and low self esteem. Many children at this age begin to try many different hobbies and start to be interested in many things. Then they might give up interests and try new ones. All this is attempted to find them in life and express individuality. Sometimes they try to express individuality through appearance, like crazy clothes, dyeing hair in crazy color, piercing and so on. Personal instability is characteristics of teenagers. They support one point of view but in some time they will support the opposite point of view. All this shows that teenagers are in search. It is very important to have the right to personal opinion. At teen age children start to form their system of values, which, in most cases stays for the rest of their life. They begin to think about what is good and what is evil. That is why it is very important to have good contact with adults who can show the right direction.