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

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