AEGAEUM JOURNAL ISSN NO: 0776-3808 Ancient Indian Method of Calculating Sixth Cube Root Dr. Puja Kumari Shreewastav 1 & Dr. K. B. Singh 2 1Department of Mathematics, B. R. A. Bihar University, Muzaffarpur 2Department of Physics, Samastipur College Samastipur, L. N. M. U. Darbhanga Abstract: In this paper firstly I have tried my best to make attempts to ancient Indian method of finding cube root of a perfect cubic number to 6th root of any integer. Keywords: Cube root, Integer, Vadic Mathematics, Ancient Hindu Mathematics. 1. Introduction Mathematics has been an integral part of intellectual training and heritage for at least twenty five hundred years. The Hindus apparently were attracted by the arithmetical and computational aspects of Mathematics, rather than by the geometrical and rational features of the subject. They delighted more in the tricks that could be played with numbers than in the thoughts, the mind could produce so that neither Euclidean Geometry nor Aristotelian logic made strong. Hindu Mathematicians treated rational and irrational quantities, curvilinear and rectilinear magnitudes indiscriminately. They saw no essential unlikeness between rectilinear and curvilinear figures for each could be measured in terms of numbers, arithmetic and mensuration rather that Geometry and considerations of congruence were fundamental. It is generally held that numerical symbols were included after writing had been in use for sometime and that in the early stages the numbers were written out in full, in words. The writings on the inscriptions of Ashoka show that in his time the use of numerical symbols in India was quite common. The variations in the forms of the numerical signs suggest that the symbols had been in use for a long time. Most of the inscriptions of Ashoka and the following period, some written in a script which has been called Brahmi whilst someone in a different script known as Kharosthi. The Kharosthi lipi is a script meant for clerks and men of business. The Brahmi inscriptions are found distributed all over India. The Brahmi script, thus, was the national script of ancient Hindu. The third and most important fact of the Hindu numeral notations is the decimal place-value notation. In this system there are only ten symbols those called “Aka” for the numbers one to nine and the zero symbol, ordinarily called Sunya. A large number of scripts differing from each other are in use in different parts of India today, although all the Hindu scripts are derived from a common source, the Brahmi script. The need of the invention of arithmetical figures was felt to meet a certain kind of want like that of the language. To remove the want, figures corresponding to the alphabet of a language came into being. The invention of the alphabet marked a milestone in the path of progress of civilization. 2. Originity of the Number Symbol in the Indus Valley Civilization The vedic Civilization was considered the oldest civilization of our Country till 1920 A.D. But, it has been proved wrong. Two old great cities have been discovered after 1920 A.D. They are Mohen-jo-daro and Harappa. The excavations at Mohen-jo-daro and Harappa had unearthed many documents viz. Seals, inscriptions etc. dating from 3500 B.C. 1 Here, the civilization is known as Indus Valley civilization. In Indus Valley Civilization weights were used in multiples of twos and also in decimal System of numeration. There is also an idea of Geometrical progression in the weights used. Thus, we have got an indication that the invention of Numeral Symbols had also taken place in India during Indus Valley Civilization which prepared the back ground for the discovery of zero. 3. Divine Links to the Invention of Zero Sunya or zero or null or void or nothing – a mark of absence of everything, identifies itself with blackness, the Colour of Goddess Kali or Lord Krsna, the absence of all Volume 8, Issue 8, 2020 http://aegaeum.com/ Page No: 1753 AEGAEUM JOURNAL ISSN NO: 0776-3808 colours; with Nirguna Brahma, the absence of all qualities. The Blackness in the colour of Goddess Kali or no quality in Brahma represents the fact that She (or He) is not guided by any of the qualities, Sattva, Raja or Tama but at the Same time it symbolises Her (or His) pervading nature and connotates the conception that She (or He) is the source of all qualities, energy, power and strength which sublimate every living and non-living thing in the world and which guide not only this world but the whole Universe and Cosmos. Zerohas been used as a Symbol for one-sensed soul, two-sensed soul, etc denoted by 0, 00, etc. This has been symbolically represented in the Jain work, Dhavala of Virasena.The great Mathematician Mahaviracarya has used the word ‘Pudgala’ for an atom. The meaning of this word is also ‘Soul’ ir ‘epithet of Shiva’. Atom presents the idea of infinitesimal. Thus, Mahaviracarya has presented ‘Shiva’ or ‘Soul’ in the form of Zero as in infinitesimal. In Buddhism or Buddhist Philosphy, Sunyata (Zeroism) plays a very important role. T.R.V. Murti in his treatise “The central philosophy of Buddhism” has given the following remarks while referring to Nagarjuna’s interpretation of Buddist philosophy. 4. An Extension of Ancient Indian Method of Calculating Cube Root An attempt has been made to give a chronological description of methods for obtaining the real root of an integer given by the Ancient Hindu Mathematician. Bh āskara II, in his book Lil āvati gave the Method to obtain the cube root of a perfect cubic number. It is obvious that ancient Indian mathematician knew the cube root technique Bh āskara’s method is indeed slightly different from his predecessors. 4.1 6 th Root Method Note that: Sixth powers of numbers from 1 to 9 are respectively 1, 64, 729, 4096, 15625, 44656, 117649, 262144 and 531441. We explain the ‘method by an ‘example’. Example 1: Find 6 th root of 47045881. Step I : Put a vertical line on the unit place and five dots on the fine digits and again proceed in the same manner. (Total no. of vertical lines = Total digits in its 6 th root). Step II : The unit place of the given number is 1. Hence the unit place of its root is 9. Step III : Leave first 6 digits in the given number and get the remaining. Leave 045881 and we have 47. Step IV : Subtract the greatest possible 6 th power of any number from 47. Now the greatest possible 6th power of 1 can be only subtracted from 47. Now I would occupy the tens place in its 6 th root. Step V : 6th root of 47045881 is 19. Note that: Similarly we can find out the others root. Proof By the application of Binomial theorem, we can find out the n th root by writing down the expression of (a+b) n for n = 2, 3, 4, 5, 6, ………….. , n in descending form of a or b. n n n n n-1 n n-2 2 n n-3 3 n n-r 2 n n (a + b) = coa + c1a b+ c2a b + c3a b +…. cra b +….+ cnb Let n= 2 2 2 2 2 2-1 2 2-2 2 (a + b) = coa + c1a b + c2a h = a2 + 2ab + b 2 a2 + 2ab + b 2 a2 a20 2a ) 2ab ( b 2ab 0 b2 b20 x Volume 8, Issue 8, 2020 http://aegaeum.com/ Page No: 1754 AEGAEUM JOURNAL ISSN NO: 0776-3808 Hence the root of a 2 + 2ab + b 2 is a + b. Let n = 3 ͧ Μ Μͯͥ Μͯͦ ͦ Μͯͧ ͧ (a + b) = nΑΝa + nΑͥa b + nΑͦa b + nΑͧa b ͧ ͧͯͥ ͧͯͦ ͦ ͧͯͧ ͧ = 3ΑΝa + 3Αͥa b + 3Αͦa b + 3Αͧa b = a 3 + 3a 2b + 3ab 2 + b 3 a3 a3 + 3a 2b + 3ab 2 + b 3 a30 3a 2 ) 3a 2b ( b 3a 2b 0 3ab 2 3ab 2 3ab 20 b3 b3 b30 x Hence (a 3 + 3a 2b + 3ab 2 + b 3)1 = a+b Let n = 4 ͨ Μ Μͯͥ Μͯͦ ͦ Μͯͧ ͧ Μͯͨ ͨ (a + b) = nΑΝa + nΑͥa b + nΑͦa b + nΑͧa b + nΑͨa b ͨ ͨͯͥ ͨͯͦ ͦ ͨͯͧ ͧ ͨͯͨ ͨ = 4ΑΝa + 4Αͥa b + 4Αͦa b + 4Αͧa b + 4Αͨa b = a 4 + 4a 3b + 6a 2b2 + 4ab 3 + b 4 a4 + 4a 3b + 6a 2b2 + 4ab 3 + b 4 a4 a40 4a 3 ) 4a 3b ( b 4a 3b 0 6a 2b2 6a 2b2 6a 2b20 4ab 3 4ab 3 4ab 30 b4 b4 b4 x u 4 3 2 2 3 4 Hence (a + 4a b + 6a b + 4ab + b )x = a + b. Let n =5 ͩ Μ Μͯͥ Μͯͦ ͦ Μͯͧ ͧ Μͯͨ ͨ Μͯͩ ͩ (a + b) = nΑΝa + nΑͥa b + nΑͦa b + nΑͧa b + nΑͨa b + nΑͩa b ͩ ͩͯͥ ͩͯͦ ͦ ͩͯͧ ͧ ͩͯͨ ͨ ͩͯͩ ͩ = 5ΑΝa + 5Αͥa b + 5Αͦa b + 5Αͧa b + 5Αͨa b + 5Αͩa b = a 5 + 5a 4b + 10a 3b2 + 10a 2b3 + 5ab 4 + b 5 a5 + 5a 4b + 10a 3b2 + 10a 2b3 + 5ab 4 + b 5 a5 a50 5a 4 ) 5a 4b ( b 5a 4b 0 10a 3b2 10a 3b2 10a 3b20 10a 2b3 10a 2b3 10a 2b30 5ab 4 5ab 4 5ab 4 b5 b5 b5 x u 5 4 3 2 2 3 4 5 Hence (a + 5a b + 10a b + 10a b + 5ab + b )y = a +b. Let n =6 ͪ Μ Μͯͥ Μͯͦ ͦ Μͯͧ ͧ Μͯͨ ͨ Μͯͩ ͩ (a + b) = nΑΝa + nΑͥa b + nΑͦa b + nΑͧa b + nΑͨa b + nΑͩa b Μͯͪ ͪ + nΑͪa b Volume 8, Issue 8, 2020 http://aegaeum.com/ Page No: 1755 AEGAEUM JOURNAL ISSN NO: 0776-3808 ͪ ͪͯͥ ͪͯͦ ͦ ͪͯͧ ͧ ͪͯͨ ͨ ͪͯͩ ͩ = 6ΑΝa + 6Αͥa b + 6Αͦa b + 6Αͧa b + 6Αͨa b + 6Αͩa b ͪͯͪ ͪ + 6Αͪa b = a 6 + 6a 5b + 15a 4b2 + 20a 3b3 + 15a 2b4 + 6ab 5 + b 6 a6 + 6a 5b + 15a 4b2 + 20a 3b3 + 15a 2b4 + 6ab 5 + b 6 a6 a60 6a 5 ) 6a 5b ( b 6a 5b 0 15a 4b2 15a 4b2 15a 4b20 20a 3b3 20a 3b3 20a 3b30 15a 2b4 15a 2b4 15a 2b40 6ab 5 6ab 5 6ab 50 b5 b6 b5 x u 6 5 4 2 3 3 2 4 5 6 Hence (a + 6a b + 15a b + 20a b + 15a b + 6ab + b )y = a +b.