A NEW CONCEPT TO PROVE, √(- 1)= -1 IN BOTH GEOMETRIC AND ALGEBRAIC METHODS WITHOUT USING THE CONCEPT OF IMAGINARY NUMBERS

Journal > Journal > Journal Vol – 18 No -9, September 2023 > A NEW CONCEPT TO PROVE, √(- 1)= -1 IN BOTH GEOMETRIC AND ALGEBRAIC METHODS WITHOUT USING THE CONCEPT OF IMAGINARY NUMBERS

September 17, 2023

Journal Vol – 18 No -9, September 2023

Authors:

Prabir Chandra Bhattacharyya,

DOI NO:

https://doi.org/10.26782/jmcms.2023.09.00003

Keywords:

Cartesian Coordinate System,Dynamics of Numbers,Extended form of Pythagoras Theorem,Imaginary Number,Quadratic Equation,Three Types of Rectangular Bhattacharyya’s coordinate systems,

Abstract

A particular branch of mathematics is coordinate geometry where geometry is studied with the help of algebra. According to the new concept of three types of Rectangular Bhattacharyya’s coordinate systems, plane coordinate geometry consists of four axes. In type – I, Rectangular Bhattacharyya’s coordinate system, the four axes are all neutral straight lines having no direction; in the type – II coordinate system the four axes are all count up straight lines and, in the type – III coordinate system all the axes are countdown straight lines. The author has considered all four axes to be positive in type II and type III coordinate systems. Ultimately, the author has established relations among the three types of coordinate systems and used the extended form of Pythagoras Theorem to prove √(- 1)= -1. In this paper, algebra is studied with the help of geometry. The equation, x2 + 1 = 0, means x2 = – 1 and therefore, the value of √(- 1)= -1, has been proved by the author with the help of geometry by using the new concept of the three types of coordinate systems without using the concept of the imaginary axis. Also, the author has given an alternative method of proof of √(- 1)= -1 algebraically by using the concept of the theory of dynamics numbers. The square root of any negative number can be determined in a similar way. This is the basic significance of that paper. This significance can be widely used in Mathematics, Science, and Technology and also, in Artificial Intelligence (AI), and Crypto-system

Refference:

I. Ahmad Raza, : “Research Work of Mathematics in Algebra. New Inventory Work in Quadratic Equation (P.E. Degree 2), Cubic Equation (P.E. Degree 3) & Reducible Equation General Term”. European Journal of Mathematics and Statistics. pp. 310-315. Vol 4, Issue 1, January 2023. 10.24018/ejmath.2023.4.1.159
II. Ashwannie Harripersaud, : “The Quadratic Equation Concept”. American Journal of Mathematics and Statistics. 2021; 11(3): 67-71. 10.5923/j.ajms.20211103.03
III. B. B. Dutta, (1929). : “The Bhakshali Mathematics”. Calcutta, West Bengal: Bulletin of the Calcutta Mathematical Society.
IV. B. B. Datta, & A. N. Singh, (1938). : “History of Hindu Mathematics, A source book”. Mumbai, Maharashtra: Asia Publishing House.
V. Gandz, S. (1937). : “The origin and development of the quadratic equations in Babylonian, Greek, and Early Arabic algebra”. History of Science Society, 3, 405-557.
VI. H. Lee Price, Frank R. Bernhart, : “Pythagorean Triples and a New Pythagorean Theorem”. arXiv:math/0701554 [math.HO]. 10.48550/arXiv.math/0701554
VII. Katz, V., J. (1998). : “A history of mathematics (2nd edition)”. pp. 226-227. Harlow, England: Addison Wesley Longman Inc.
VIII. L. Nurul , H., D., (2017). : “Five New Ways to Prove a Pythagorean Theorem”. International Journal of Advanced Engineering Research and Science. Volume 4, issue 7, pp.132-137. 10.22161/ijaers.4.7.21
IX. M. Janani et al, : “Multivariate Crypto System Based on a Quadratic Equation to Eliminate in Outliers using Homomorphic Encryption Scheme”. Homomorphic Encryption for Financial Cryptography. pp 277–302. 01 August 2023. Springer.
X. M. Sandoval-Hernandez et al, : “The Quadratic Equation and its Numerical Roots”. International Journal of Engineering Research & Technology (IJERT). Vol. 10 Issue 06, June-2021 pp. 301-305. 10.17577/IJERTV10IS060100
XI. Prabir Chandra Bhattacharyya, : “AN INTRODUCTION TO RECTANGULAR BHATTACHARYYA’S CO-ORDINATES: A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-16, No.-11, November (2021). pp 76-86. 10.26782/jmcms.2021.11.00008
XII. Prabir Chandra Bhattacharyya, : “AN INTRODUCTION TO THEORY OF DYNAMICS OF NUMBERS: A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-17, No.-1, January (2022). pp 37-53. 10.26782/jmcms.2022.01.00003
XIII. Prabir Chandra Bhattacharyya, : ‘A NOVEL CONCEPT IN THEORY OF QUADRATIC EQUATION’. J. Mech. Cont. & Math. Sci., Vol.-17, No.-3, March (2022) pp 41-63. 10.26782/jmcms.2022.03.00006
XIV. Prabir Chandra Bhattacharyya. : “A NOVEL METHOD TO FIND THE EQUATION OF CIRCLES”. J. Mech. Cont. & Math. Sci., Vol.-17, No.-6, June (2022). pp 31-56. 10.26782/jmcms.2022.06.00004
XV. Prabir Chandra Bhattacharyya, : “AN OPENING OF A NEW HORIZON IN THE THEORY OF QUADRATIC EQUATION: PURE AND PSEUDO QUADRATIC EQUATION – A NEW CONCEPT”. J. Mech. Cont. & Math. Sci., Vol.-17, No.-11, November (2022). pp 1-25. 10.26782/jmcms.2022.11.00001
XVI. Prabir Chandra Bhattacharyya, : “A NOVEL CONCEPT FOR FINDING THE FUNDAMENTAL RELATIONS BETWEEN STREAM FUNCTION AND VELOCITY POTENTIAL IN REAL NUMBERS IN TWO-DIMENSIONAL FLUID MOTIONS”. J. Mech. Cont. & Math. Sci., Vol.-18, No.-02, February (2023) pp 1-19. 10.26782/jmcms.2023.02.00001
XVII. Prabir Chandra Bhattacharyya, : “A NEW CONCEPT OF THE EXTENDED FORM OF PYTHAGORAS THEOREM”. J. Mech. Cont. & Math. Sci., Vol.-18, No.-04, April (2023) pp 46-56. 10.26782/jmcms.2023.04.00004
XVIII. Salman Mahmud, : “14 New Methods to Prove the Pythagorean Theorem by using Similar Triangles”. International Journal of Scientific and Innovative Mathematical Research (IJSIMR). vol. 8, no. 2, pp. 22-28, 2020. 10.20431/2347-3142.0802003
XIX. S. Mahmud, (2019). : “Calculating the area of the Trapezium by Using the Length of the Non Parallel Sides: A New Formula for Calculating the area of Trapezium”. International Journal of Scientific and Innovative Mathematical Research. volume 7, issue 4, pp. 25-27. 10.20431/2347- 3142.0704004
XX. Smith, D. (1953). : “History of mathematics”, Vol. 2. Pp. 293. New York: Dover
XXI. Smith, D. (1953). : “History of mathematics”, Vol. 2. pp. 443. New York: Dover.
XXII. T. A. Sarasvati Amma, : “Geometry in Ancient and Medieval India”. Pp. – 17. Motilal Banarasidass Publishers Pvt. Ltd. Delhi.
XXIII. T. A. Sarasvati Amma, : “Geometry in Ancient and Medieval India”. pp. – 219. Motilal Banarasidass Publishers Pvt. Ltd. Delhi.
XXIV. Thapar, R., (2000). : “Cultural pasts: Essays in early Indian History”, New Delhi: Oxford University Press.

View Download