New Form of Mathematics Excluding Imaginary Numbers

How to cite this paper: Gode, D.B. (2014) Complex Number Theory without Imaginary Number (i). Open Access Library Journal, 1: e856. http://dx.doi.org/10.4236/oalib.1100856 Abstract In this paper new method is introduced for complex numbers; this method does not include imaginary number " i " but produces the same results that occur in Addition, Subtraction, Multiplication & Division of complex numbers, also proof of Eluer Formula e iθ and De Moivre theorem without using imaginary number " i ". Furthermore placing the light on the square root of a negative number, the square root of a negative number is equal to the square root of the same positive real number but with an angle of 90 degree to real line. The intention is that there's nothing mystical about imaginary number " i ". The square root of minus one is just as real as any other number. Complex numbers exists without imaginary number " i ". This paper is limited to the results which are already established.

JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES

In this paper, the author proved that the product of two negatively directed numbers is a negatively directed number. This article is the outcome of previously published (2021-2024) ten (10) articles of this author. It is true that the negative of a negative number is a positive number. It has been done by applying the inversion process to a negative number. The process of inversion does not satisfy the basic concept of multiplication. Multiplication is defined as the adding of a number concerning another number repeatedly. So, the process of inversion does not comply with the fundamental concept of multiplication. According to the Theory of Dynamics of Numbers there exist three types of numbers: (1) Neutral or discrete numbers (2) Positively directed numbers (3) Negatively directed numbers. In general, we use four types of operators: addition (+), subtraction (-), multiplication (x), and division (÷) in mathematical calculations. Besides these, we use the negative sign (-) as an inversion operator. The positive sign (+) and negative sign (-) also represent the direction of neutral or discrete numbers. In this paper, the author introduced Fermat's Last Theorem: x n + y n = z n for n = 2 in Bhattacharyya's Theorem to prove that the product of two negatively directed numbers is a negatively directed number using the concept of the Theory of Dynamics of Numbers. In this paper, the author framed new laws of multiplication and inversion. Also, the author has given a comparative study between the conventional method of multiplication and the present concept of multiplication citing some practical examples. The author has become successful in finding the root of a quadratic equation in real numbers even if the discriminant, b 2-4ac < 0 without using the concept of the imaginary number and also in determining the radius of a circle even if g 2 + f 2 < c, in real number without using the concept of complex numbers. With one example the author proved that this theorem is applicable in 'Calculus' also.

Pure and Applied Mathematics Journal, 2021

Most mathematician, have accepted that a constant divided by zero is undefined. However, accepting this situation is an unsatisfactory solution to the problem as division by zero has arisen frequently enough in mathematics and science to warrant some serious consideration. The aim of this paper was to propose and prove the existence of a new number set in which division by zero is well defined. To do this, the paper first uses set theory to develop the idea of unstructured numbers and uses this new number to create a new number set called “Semi-structured Complex Number set” (Ś). It was then shown that a semi-structured complex number is a three-dimensional number which can be represented in the xyz-space with the x-axis being the real axis, the y-axis the imaginary axis and the z-axis the unstructured axis. A unit of rotation p was defined that enabled rotation of a point along the xy-, xz- and yz- planes. The field axioms were then used to show that the set is a “complete ordered ...

JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES

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, x 2 + 1 = 0, means x 2 =-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.

Angelaki, 2020

The operations that produce beings outside nature can themselves be quite ordinary, quotidian, even rational. We can extend our world by amalgamating “monsters” – beings against and therefore outside nature – not by taming them or by exhibiting them in cages, but by enlarging our social, technical, political, epistemic practices to include them. This chapter focuses on specific practices in twentieth- and twenty-first-century mathematics of articulating, barring, taming, and operating with what mathematicians widely call mathematical monsters (Wagner; Kucharski; Feferman; Cooper; Poincaré). We can regard these practices as examples of what more generically can be called problematization, a powerful mode of propositional thought shared by mathematicians and speculative philosophers.

In various fields of science and philosophy, including mathematics and theoretical physics, accuracy and precision in the foundational principles are the basis for progress and a deeper understanding of truth. One of the fundamental principles that has been repeatedly emphasized throughout the history of science is that "If the first brick is laid crookedly, the entire building will be crooked." This proverb serves as a warning that carelessness in the foundational principles can derail the entire course of a science, ultimately leading to erroneous and misleading conclusions. Regarding imaginary numbers and other abstract concepts used in mathematics and theoretical physics, what may initially seem like useful conceptual tools may, if misused or inaccurately defined and applied, lead to increasing confusion at the lower levels of science. These confusions not only lead to scientific mistakes, but they can also misguide the search for truth. This article aims to show how the proper and accurate application of basic concepts in philosophy and ontology can not only lay the foundation for more precise and reliable theories in various sciences, but also prevent fundamental errors in theoretical physics, guiding the quest for truth on a correct and stable path. In this context, we will critique and examine concepts such as imaginary numbers and their impact on scientific deviations, and we will explore how an emphasis on clarity in foundational concepts can free science from endless confusions and deviations.

IAVM, 2023

In this paper, I aim to summarize a brief history of negative numbers in the context of ancient mathematics. I will explore the methods in Vedic mathematics where negative numbers are used constructively. Finally, I will suggest the methodologies which will help students to deal with negative numbers effectively.

The American Mathematical Monthly, 2007

Journal of the Learning Sciences, 1994