#### Authors:

Prabir Chandra Bhattacharyya,#### DOI NO:

https://doi.org/10.26782/jmcms.2022.11.00001#### Keywords:

Dimension of Numbers,Dynamics of Numbers,,Quadratic Equation,Rectangular Bhattacharra’s Coordinates,significance of roots of a Quadratic Equation,#### Abstract

In this paper, the author has opened a new horizon in the theory of quadratic equations. The author proved that the value of x which satisfies the quadratic equation cannot be the only criteria to designate as the root or roots of an equation. The author has developed a new mathematical concept of the dimension of a number. By introducing the concept of the dimension of number the author structured the general form of a quadratic equation into two forms: 1) Pure quadratic equation and 2) Pseudo quadratic equation. First of all the author defined the pure and pseudo quadratic equations. In the case of pure quadratic equation ax^2+bx+c=0 , the root of the equation will be a two-dimensional number having one root only while in the case of pseudo quadratic equation ax^2+bx+c=0, the root of the equation will be a one-dimensional number having two roots only. The author proved that all pseudo quadratic equation is factorizable but all factorizable quadratic equation is not a pseudo quadratic equation. The author begs to differ from the conventional theorem: "A quadratic equation has two and only two roots." By introducing the concept that any quadratic surd is a two-dimensional number, the author developed a new theorem: “In a quadratic equation with rational coefficients, irrational roots cannot occur in conjugate pairs” and proved it. Any form of quadratic equation ax^2+bx+c=0, can be solved by the application of the ‘Theory of Dynamics of Numbers’ even if the discriminant b^2-4ac<0 in real number only without introducing the concept of an imaginary number. Therefore, the question of imaginary roots does not arise in the method of solution of any quadratic equation#### Refference:

I. Bhattacharyya, Prabir Chandra. : “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

II. Bhattacharyya, Prabir Chandra. : “A NOVEL CONCEPT IN THEORY OF QUADRATIC EQUATION”. J. Mech. Cont. & Math. Sci., Vol.-17, No.-3, March (2022) pp 41-63

III. Bhattacharyya, Prabir Chandra. : “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.

IV. Boyer, C. B. & Merzbach, U. C. (2011). A history of mathematics. New York: John Wiley & Sons.

V. Cajori, F., (1919). A History of Mathematics 2nd ed., New York: The Macmillan Company.

VI. Dutta, B.B. ( 1929). The Bhakshali Mathematics, Calcutta, West Bengal: Bulletin of the Calcutta Mathematical Society.

VII. Datta, B. B., & Singh, A. N. (1938). History of Hindu Mathematics, A source book. Mumbai, Maharashtra: Asia Publishing House.

VIII. 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.

IX. Gandz, S. (1940). Studies in Babylonian mathematics III: Isoperimetric problems and the origin of the quadratic equations. Isis, 3(1), 103-115.

X. Hardy G. H. and Wright E. M. “An Introduction to the Theory of Numbers”. Sixth Edition. P. 52.

XI. Katz, V. J. (1997), Algebra and its teaching: An historical survey. Journal of Mathematical Behavior, 16(l), 25-36.

XII. Katz, V., J. (1998). A history of mathematics (2nd edition). Harlow, England: Addison Wesley Longman Inc.

XIII. Katz Victor, (2007). The Mathematics of Egypt, Mesopotamia, China, India and Islam: A source book 1st ed., New Jersey, USA: Princeton University Press.

XIV. Kennedy, P. A., Warshauer, M. L. & Curtin, E. (1991). Factoring by grouping: Making the connection. Mathematics and Computer Education, 25(2), 118-123.

XV. Ling, W. & Needham, J., (1955). Horner’s method in Chinese Mathematics: Its root in the root extraction procedures of the Han Dynasty, England: T’oung Pao.

XVI. Nataraj, M. S., & Thomas, M. O. J. (2006). Expansion of binomials and factorisation of quadratic expressions: Exploring a vedic method. Australian Senior Mathematics Journal, 20(2), 8-17.

XVII. Rosen, Frederic (Ed. and Trans). (1831). The algebra of Mohumed Ben Muss. London: Oriental Translation Fund; reprinted Hildesheim: Olms, 1986, and Fuat Sezgin, Ed., Islamic Mathematics and Astronomy, Vol. 1. Frankfurt am Main: Institute for the History of Arabic-Islamic Science 1997.

XVIII. Smith, D. (1951). History of mathematics, Vol. 1. New York: Dover. Smith, D. (1953). History of mathematics, Vol. 2. New York: Dover. Stols, H. G. (2004).

XIX. Smith, D. (1953). History of mathematics, Vol. 2. New York: Dover.

XX. Thapar, R., (2000). Cultural pasts: Essays in early Indian History, New Delhi: Oxford University Press.

XXI. Yong, L. L. (1970). The geometrical basis of the ancient Chinese square-root method. The History of Science Society, 61(1), 92-102.

XXII. http://en. wikipedia.org/wiki/Shridhara

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