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.
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.”
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