Place values system and Zero interduce to world by Aryabhata

Mathematics

Place value system and zero

The palce values system, first seen in the 3rd-century Bakhshali 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  Palce – Values System as a place holder for the powers of ten with Null -Coefficient

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 sine’s in a mnemonic form.

Place values Chart

 

 Pi ( π ) values

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

Caturadhikam satama stagunam devasa stistatha sahasranam

Ayutadayavi siambhasyasanno vittaparinahah


“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 figure.

OR

Aryabhatta

chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAm AyutadvayaviShkambhasyAsanno vr^ttapariNahaH.
[gaNita pAda, 10] Aryabhatiyam (499 CE)
“Add 4 to 100, multiply by 8 and add to 62,000. This is approximately the circumference of a circle whose diameter is 20,000.”
i.e.
correct to four places.

 

It is speculated that Aryabhata used the word ‘asana’ (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”.

After Aryabhatiya was translated into Arabic  (c. 820 CE) this approximation was mentioned in Al-Khwarizmi’s  book on algebra.

he value of pi is being used in India from ancient times. It gives us an insight about how evolved our past was. There is a shloka, a hymn to Lord Krishna or Shiva, which gives the value of pi upto 31 decimal places.

Its amazing that our forefathers used an encryption technique to easily remember it. What is more astonishing is that they needed pi upto 31 places.

Importance of Pi
Pi deals with circles and circles are very important in many fields. Pi is a very important number in the fields of :

  • Geometry and trigonometry,
  • Complex number and analysis,
  • Number theory,
  • Nrobability and statistic physics,
  • Engineering and geology.
  • Computers and many more
  • Katapayadi Encryption
  • gopiibhaagya madhuvraataH shruMgashodadhi saMdhigaH .
    khalajiivitakhaataava galahaalaa rasaMdharaH
  • This shloka, a hymn to Lord Krishna or Shiva, gives the value of pi upto 31 decimal places.
Pi using Encryption
  • • Katapayadi system is used to encode numbers in many shlokas

    ga – 3 pii – 1 bhaa – 4 gya – 1 ma – 5 dhu – 9 ra – 2 ta -6 shru – 5 ga – 3 sho – 5 da – 8 dhi – 9 sa – 7 dha – 9 ga – 3 kha – 2 la – 3 jii – 8 vi – 4 ta – 6 kha – 2 ta – 6 va – 4 ga – 3 la – 3 ra – 2 sa – 7 dha – 9 ra – 2

    pi = 3.1415926535897932384626433832792

 

to be continue …….2

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Ancient Mathematician & Astrologer Aryabhata

Aryabhatt

Aryabhata born December 476 CE Ashmaka is old name of Patna (Bihar) India was the first in the line of great mathematician- astrologer from the classical age of  Indian Mathematician and astronomer . His works include the Aryabhata (499 CE, when he was 23 years old) and the Arya-SidinataThe works of Aryabhata dealt with mainly mathematics and astronomy. 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 , plan trigonometry, and spherical trigonometry . It also contains continued fractions , quadratics 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. 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.

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, Abu Rayhan Al Biruni.

Aryabhatiya Main article: 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 Bhaskar  calls it Ashmakatantra . 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 (formula) 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  (c. 1st century BCE). There is also a table of sins (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 menstruation  (ketra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic ,simultaneous , and indeterminate equations
  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 elliptical equator , 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, c. 600 CE) and by Nilkantha Somayaji  in his Aryabhatiya Bhasya, (1465 CE).

To be continue…….1