Complex Number Theory without Imaginary Number (i)

Historically, complex numbers were introduced to solve algebraic equations. It was observed that the algebraic equation x 2 + 1 = 0 possesses no real solutions and the notation ± √ −1 was used to represent the roots. It was the year 1545 when the mathematicians Giro-lamo Cardano 1 , Ferro Tartaglia 2 and Rafel Bombeli 3 first used the square root of negative numbers to represent roots of equations. In 1797 a Norwegian Surveyor Wessel 4 reported to the Danish Academy of Science on the geometric interpretation of complex numbers of the form z = a + ib. Later in 1806 Robert Argand 5 also produced an essay on the geometric interpretation of complex numbers. The Swiss mathematician Leonard Euler 6 was the first to introduce the symbol i to represent the imaginary component of a number. In 1831 Carl Fredrich Gauss 7 formalized the work of Wessel and Argand and introduced numbers z = x + iy where i is a symbol used to represent a pure imaginary number having the property that i 2 = −1. Numbers of the form z = x + iy, where x and y are real numbers, were called complex numbers and many mathematicians have contributed to the development of the theory of complex numbers and functions associated with these numbers. In particular, much of the early work in the complex domain was done by the mathematicians Cauchy 8 , Weierstrass 9 and Riemann 10. For the last two hundred years many applications of complex numbers and complex functions have been developed in the areas of science and engineering. For example, the study areas of mappings, integration, differentiation, solutions of algebraic equations, solutions of ordinary differential equations, solutions of partial differential equations , summation of series, sequences, fractals and potential theory are just a few of the many application areas where one can find complex variable theory employed. You will find examples from many of these application areas presented throughout this textbook.

Maths for Chemists

Contents 1.1 Sequences 1.2 Finite Series 1.3 Infinite Series 1.4 Power Series as Representations of Functions 2.1 The Imaginary Number i 2.2 The General Form of Complex Numbers 2.3 Manipulation of Complex Numbers 2.4 The Argand Diagram 2.5 The Polar Form for Complex Numbers 3 6 7 11 29 29 30 32 35 3.1 Origins: the Solution of Simultaneous Linear Equations 46 3.2 Expanding Determinants 48 3.3 Properties of Determinants 51 3.4 Strategies for Expanding Determinants where n > 3 52 4.1 Introduction: Some Definitions 4.2 Rules for Combining Matrices 4.3 Origins of Matrices 56 57 60 V vi Contents ~~ 4.4 Operations on Matrices 4.5 The Determinant of a Product of Square Matrices 4.6 Special Matrices 4.7 Matrices with Special Properties 4.8 Solving Sets of Linear Equations 4.9 Molecular Symmetry and Group Theory

interst

The imaginary number “i” is an amphibian as it links two real numbers and renders the ordered pair of real numbers complex: z=x+iy. Its discovery has made rapid stride in mathematics and its application. The equation e^iπ+1=0 is one of the most profound equations. The solution to quadratic equation x^2+1=0 are ±i. Geometric algebra, pioneered by Hestenes, Baylis and others, has given a better place for i. It can be considered as the analogue of scalar I. We discuss briefly about polar vectors and axial (pseudo) vectors, scalars and pseudo scalars.

Review of complex numbers. A complex number is any expression of the form x + iy where x and y are real numbers. x is called the real part and y is called the imaginary part of the complex number x + iy. The complex number x − iy is said to be complex conjugate of the number x + iy.

JOURNAL OF MECHANICS OF CONTINUA AND MATHEMATICAL SCIENCES

In this paper, the author stated and proved Bhattacharyya's Theorem : √-(𝑥 2 + 𝑦 2) =-√ 𝑥 2 + 𝑦 2. With the help of this theorem, the author finds the square root of any negative number introducing Fermat's last theorem without using the concept of complex numbers. The author has introduced Fermat's Last Theorem in Bhattacharyya's Theorem to find the square root of any negative number in real numbers in a very simple way. Indeed it is a new invention in mathematics in this era.

If the expression í µí±– = −1 has a solution that is not a real number, what might be that? So the question is a philosophical and foundational one, simply to wonder about what we are dealing with. Concern about the foundations of mathematics therefore means that the problem of imaginary numbers is in fact a problem and cannot be dismissed as superfluous or trivial. What is the essence of í µí±– = −1 ? We discover the very nature of í µí±– = −1 to be í µí±– = −1 = −∞. From Euler formula í µí±’í µí±¥í µí±2í µí¼‹í µí±– = 1 , we deduce the imaginary phase of the zero, that is 0 = 2í µí¼‹í µí±–. Thus we tame the zero. We develop new axioms to tame infinity, zero and the imaginary number. Hence we develop a new arithmetic, solves the problem of indeterminants. Eventually, many paradoxes were resolved if infinity had taken to be both even and odd simultaneously. .

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.

The American Mathematical Monthly, 2007

2024

The Canonical Order of Operations introduces a unified framework to resolve ambiguities in traditional arithmetic, particularly in handling negative values, roots, and exponents. By eliminating inconsistencies that necessitate the use of imaginary numbers and redefining the treatment of operations involving negatives, this framework preserves the integrity of real numbers without resorting to abstract constructs. Rooted in centuries of mathematical tradition, the Canonical Order replaces symbols such as the radical () with explicit fractional exponents, making these concepts more accessible and intuitive for learners. This paper explores the historical context, defines the Canonical Order, and demonstrates its potential to simplify mathematical education and computation while maintaining logical consistency.