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Vigesimal System of Numeration As Prevailed in the Bārelā Tribe Dipak

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Vigesimal System of Numeration As Prevailed in the Bārelā Tribe Dipak Vigesimal System of Numeration as Prevailed in the Bārelā Tribe Dipak Jadhav Lecturer in Mathematics Govt. Boys Higher Secondary School, Anjad Distt. Barwani 451556 (M. P.) India [email protected] Abstract Bārelā is a tribe. It resides in the western part of India. In Bārelā system of numeration other numbers stand in relation to twenty. This paper finds that no other vigesimal system whether it was in India or from abroad has been influential in shaping Bārelā system of numeration if their corresponding terms are taken into consideration. Gunnīsa, uganīsa, and oganīsa are the alternative terms used by Bārelās for 19. This paper also offers an insight into the formation of each of them. Key Words: Bārelā, Vigesimal system of numeration, Vīsa, Vīsu. Introduction Bārelā is an Indian tribe. Most of its population lives in Barwani District of State of Madhya Pradesh in India. The surrounding region, where as well Bārelās dwell, to this District encompasses the neighboring Khargone District of Madhya Pradesh and the three neighboring Districts, namely, Jalgoan, Dhule, and Nandurbar, of State of Maharashtra. The term used to identify Bārelā in Maharashtra is Pāvarā.1 The entire region of their dwelling, which is in the western part of India, falls in the forest of Satpura range. The dialect which Bārelās speak is called Bārelī. Their system of numeration is based on twenty. It is observed to be mostly practised in Pati and Barwani Blocks of Barwani District. Bārelās dealing with retailers in its terms can be seen in weekly haat bazars2 conducted in this region (Parihar et el 2000; Thakur 2005, p. 7). Since it is practised among Bārelā community, 1Bārelā colleagues of the present author including Mr. Subhash Solanki and Mr. Balram Pawar from Pansemal Block of Barwani District has apprised him so. 2 A haat bazaar, simply called haat or hat or hāṭa (if transliterated), is an open-air market. It serves as a trading venue, conducted on a regular basis, for local people in rural or forest areas. Journal of Mathematics and Culture 61 December 2019 13(1) ISSN-1558-5336 an identifiable cultural group, it is an ethnomathematics. See definition of ethnomathematics suggested by D‟Ambrosio (D‟Ambrosio, 1985). Very older Bārelās still remember an accurate vigesimal system. Group leader Vikas Parihar and his four co-workers of Grade XI from J. N. Govt. Model Higher Secondary (Residential) School, Barwani (now known as Ekalavya Model Residential School, Barwani) undertook a project on “Number- Language System of Bārelās: A Mathematical Study” under the guidance of the present author in the academic year 2000-2001. Vikas Parihar, after their project had been selected first at District level and then at State level, presented its report in Hindi at National level under focal theme „Indigenous scientific knowledge: For a better tomorrow‟ in National Children Science Congress, held during December 27-31, 2000 at Kolkata (Parihar et el, 2000). The project represented State of Madhya Pradesh in Indian Science Congress, held during January 3-7, 2001 at New Delhi (Anonymous, 2001, p. 27). It was selected for participation in the “Exchange of Young Researchers” program taken up by National Council for Science and Technology Communication, Department of Science and Technology, Government of India. Under this program a fellowship to visit various science and research institutions in Germany was awarded to Vikas and his co-worker Nilesh. They visited Germany in April, 2001.3 The project-report was submitted to Honorable Governor of Madhya Pradesh in Bhopal on May 04, 2001 when Vikas and Nilesh paid a courtesy visit to Governor House.4 The project was selected for featuring in the Television Show Ignited Minds. Vikas Parihar accompanied by his guide teacher participated in its shooting conducted 3See letter D.O.No. CO/FP/O11/2000 dated January 12, 2001 issued by Ministry of Science and Technology, Government of India. 4 See Hindi Daily Newspaper Naiduniya dated May 05, 2001. Journal of Mathematics and Culture 62 December 2019 13(1) ISSN-1558-5336 during November 3-6, 2004 at Noida.5 The related episode of Ignited Minds was televised on DD-National on December 17, 2005. The study in the above project was confined to the numbers from 1 to 100. It was aimed at understanding number-language system of Bārelās in mathematical terms, documenting it, and approximately defining a Bārelā number. In the next section of this paper, we shall see how the young researchers studied this peculiar system of numeration and looked at its underlying characteristics and laws. We will also see what the interesting features they found are. This paper is the outcome of a variety of side trips. It is mainly aimed at making the essence of the above project-report available in English to the wider class of scholars in the field of Ethnomathematics. This will be done in the next section. It is also aimed at discovering the differences and similarities between Bārelā system and various other vigesimal systems of numeration prevailed in the other parts of India and abroad. It will also discuss various aspects of Bārelā system related to History of Mathematics. Bārelā Number System The young researchers collected the terms spoken by Bārelās for numbers from 1 to 100 from their Bārelā fellows of the residential school and divided them into two categories. The first is category of „generally spoken‟ and the second is that of „sometimes spoken‟. See Table 1. Table 1 is slightly enriched in information if compared with theirs. 5See letters, written by Pulse Media Pvt. Ltd. separately to the principal of the school and to the present author, received on October 18, 2004. Journal of Mathematics and Culture 63 December 2019 13(1) ISSN-1558-5336 From Table 1 they notice that ne stands for “plus”, koma or kama means “minus” or “less”, and no term is used for “multiplied by”. They also observe that Bārelā numbers are formed using five laws. Each law contains one or two of the three operations, namely, addition, subtraction, and multiplication. If two operations are contained in the law, one of them must be the operation of multiplication. Below are shown those five laws using arrow. The Bārelā number can be read along the direction of and above the arrow. What is below the arrow shows the working of the law. Let us denote 20 by v , which is the first letter not only of vīsa (twenty) or vīsu (twenty) but also of vigesimal. Figure 1 shows how law of addition works. vīsu ne duī 22 v + 2 Figure 1 Figure 2 displays how law of subtraction works. vīsu koma tīna 17 v - 3 Figure 2 Figure 3 shows how law of multiplication works. Journal of Mathematics and Culture 64 December 2019 13(1) ISSN-1558-5336 cyāra vīsu 80 4 × v Figure 3 Each of Figure 4 and Figure 5 shows how law of compound operation works. In Figure 4 it contains the operations of addition and multiplication while in Figure 5 it contains the operations of subtraction and multiplication. tīna vīsu ne duī 62 3 × v + 2 Figure 4 vīsu duī koma āṭha 32 v × 2 - 8 Figure 5 Each of Figure 6 and Figure 7 shows how law of flexibility works. duī vīsu ne suḷe 56 2 × v + 16 Figure 6 Journal of Mathematics and Culture 65 December 2019 13(1) ISSN-1558-5336 vīsu tīna koma cyāra 56 v × 3 - 4 Figure 7 56 is formed in Figure 6 using the operations of addition and multiplication while it is formed in Figure 7 using the operations of subtraction and multiplication. This law provides facility to the Bārelās in numeration. Observing Table 1, the young researchers find that there are cycles in Bārelā number system. They call those cycles vigesimal cycles (vīsu cakras). Each of them ends in multiples of twenty. There are five cycles in their study; first from 1 to 20, second from 21 to 40, third from 41 to 60, fourth from 61 to 80, and fifth from 81 to 100. See the third cycle in Figure 8. The operation of multiplication with that of subtraction is generally employed in the second half of each vigesimal cycle while the operation of multiplication with that of addition in its first half except for the first cycle. 2v + 5 First half 3v 2v + 10 Second half 3v - 5 Figure 8 Further, they find that Bārelā number has two parts. One is vigesimal part (vīsu bhāga) and the other is non-vigesimal part (avīsu bhāga). See Figure 9. Journal of Mathematics and Culture 66 December 2019 13(1) ISSN-1558-5336 43 = 2v + 3 vigesimal part non-vigesimal part Figure 9 If z is a Bārelā number, z x y x, y where x v , 2v , 3v , 4v , 5v and y 10, -9, ..., 9, 10 . Assuming X-axis to be vigesimal axis (vīsu akṣa) and Y-axis to be non-vigesimal axis (avīsu akṣa), they demonstrate the entire column of „generally spoken‟ from Table 1 as shown in Figure 10. Non-vigesimal axis 10 cycle 5 First half of second First half of third cycle First half of fourth cycle First half of fifth cycle First half of first cycle Vigesimal axis v 2v 3v 4v 5v 0 h f2e hal cycle cycle a con f of d cycle thir fifth lf d fourth fou -5 o cyc rth f le cyc Second halfof Second halfof Second halfof cycle first Second halfof fi Second halfof second cycle le -10 ft Figure 10 h From Figure 10 they infer that in Bārelā number system is found an upward move in the first half of each vigesimalc cycle and the move is maintained in its second half as well with a downward yjump just after the first half.
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