Solution of Repeating Non-Terminating Problem in Division

Abstract

This article discusses the drawbacks of decimal theory in terms of repeating and non-terminating problems that may not give complete real results. This problem of decimal theory has persisted from its discovery to the present day. For this problem a theory has been developed in this research which proves to be better for repeating and never ending problem which is called L-Sign theory. The data which does not give hundred present absolute real result by decimal division method like 10/3, 5/11, 22/7[approximation of ?(Pi)] and many more like these can be divided by this new L-Sign method to get absolute real result, the result obtained will be absolute real value of the given data. Mathematical operations can be easily performed with other numbers of the obtained results. Apart from this, the reality of these results has been checked by different methods.

Introduction

Conclusion

In this paper, an L-Sign method has been developed to solve repeating and non-terminating problems and reach its absolute real value, which proves to be an important discovery in the history of mathematics after the failure of the decimal division method. Through this L-Sign method, we can easily solve such problems which could not reach their real value like ? and which were impossible to bring hundred percent real value in mathematical operations. In this paper compatibility of L-Sign numbers with normal numbers has also been proved by addition, subtraction, multiplication and division.

References

[1] A., Volkov Calculation of ? in ancient China : from Liu Hui to Zu Chongzhi, Historia Sci. (2) 4 (2) (1994), 139-157 [2] Ahmad, A., On the ? of Aryabhatta I, Ganita Bharati 3 (3-4) (1981), 83-85. [3] Archimedes. \"Measurement of a Circle.\" From Pi: A Source Book. [4] Beckman, Petr. The History of Pi. The Golem Press. Boulder, Colorado, 1971. [5] Berggren, Lennart, and Jonathon and Peter Borwein. Pi: A Source Book. Springer-Verlag. New York, 1997. [6] Bruins, E. M., with roots towards Aryabhatt’s ?- value, Ganita Bharati 5 (1-4) (1983), 1-7. [7] C Pereira da Silva, A brief history of the number ? (Portuguese), Bol. Soc. Paran. Mat. (2) 7 (1) (1986), 1-8. [8] Cajori, Florian. A History of Mathematics. MacMillan and Co. London, 1926 [9] Cohen, G. L. and A G Shannon, John Ward\'s method for the calculation of pi, Historia Mathematica 8 (2) (1981), 133-144 [10] Florian Cajori, A History of Mathematics, second edition, p.143, New York: The Macmillan Company, 1919. [11] Greenberg, Marvin Jay (2008), Euclidean and Non-Euclidean Geometries (Fourth ed.), W H Freeman, pp. 520–528, ISBN 0-7167-9948-0 [12] Heath, Thomas (1981). History of Greek Mathematics. Courier Dover Publications. [13] Hobson, E. W., Squaring the circle (London, 1953). [14] I Tweddle, John Machin and Robert Simson on inverse-tangent series for ?, Archive for History of Exact Sciences 42 (1) (1991), 1-14. [15] Jami, C., Une histoire chinoise du \'nombre ?\', Archive for History of Exact Sciences 38 (1) (1988), 39-50. [16] K Nakamura, On the sprout and setback of the concept of mathematical \"proof\" in the Edo period in Japan : regarding the method of calculating number ?, Historia Sci. (2) 3 (3) (1994), 185-199 [17] L Badger, Lazzarini\'s lucky approximation of ?, Math. Mag. 67 (2) (1994), 83-91. [18] M D Stern, A remarkable approximation to ?, Math. Gaz. 69 (449) (1985), 218-219. [19] N T Gridgeman, Geometric probability and the number ?, Scripta Math. 25 (1960), 183-195. [20] O\'Connor, John J. and Robertson, Edmund F. (2000). [21] P Beckmann, A history of ? (Boulder, Colo., 1971). [22] P E Trier, Pi revisited, Bull. Inst. Math. Appl. 25 (3-4) (1989), 74-77.

Copyright

Copyright © 2023 Lokesh Dewangan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.