The common sense on the division by zero with the long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on differential coefficients we have a great missing since tan(π/2) = 0. Our mathematics is also wrong in elementary mathematics on the division by zero. In this book in a new and definite sense, we will show and give various applications of the division by zero 0/0 = 1/0 = z/0 = 0. In particular, we will introduce several fundamental concepts in calculus, Euclidean geometry, analytic geometry, complex analysis and differential equations. We will see new properties on the Laurent expansion, singularity, derivative, extension of solutions of differential equations beyond analytical and isolated singularities, and reduction problems of differential equations. On Euclidean geometry and analytic geometry, we will find new fields by the concept of the division by zero. We will collect many concrete properties in mathematical sciences from the viewpoint of the division by zero. We will know that the division by zero is our elementary and fundamental mathematics.
Components of the Book:
- Copyright
- Introduction
- Preface
- Acknowledgements
- Contents
- Chapter 1
Introduction
- Chapter 2
Introduction and definitions of general fractions
- Chapter 3
Stereographic projection
- Chapter 4
Mirror image with respect to a circle
- Chapter 5
Division by zero calculus
- Chapter 6
Triangles, trigonometric functions and harmonic mean
- Chapter 7
Derivatives of functions
- Chapter 8
Differential equations
- Chapter 9
Euclidean spaces and division by zero
- Chapter 10
Applications to Wasan geometry
- Chapter 11
Introduction of formulas log 0 = log1 = 0
- Chapter 12
Divergence series and integrals from the viewpoint of the division by zero calculus
- Chapter 13
Basic meanings of values at isolated singular points of analytic functions
- Chapter 14
Division by zero calculus on multidimensional spaces
- Chapter 15
Division by zero calculus in physics
- Chapter 16
What is the zero?
- Chapter 17
Conclusion
- References
- Index
Readership:
Students, academics, teachers and other people attending or interested in mathematics and operations
SABUROU SAITOH
Saburou Saitoh is Professor Emeritus of Gunma University with 74 years old and whose major is the theory of reproducing kernels with many applications in analysis and he published 7 books with co-authors, in particular, with Professor H. Sawano, Theory of Reproducing Kernels and Applications, Developments in Mathematics 44, Springer (2016). He published over 170 original papers.He was appointed as Emeritus Professor of Gunma University in 2009. After that, he got a five years Researcher Posit.