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Foundations Of Transcomplex Numbers:

An extension of the complex number system to four dimensions

By E. Pérez

Cover page of the E-Book Foundations of Transcomplex Numbers: An extension of the complex number system to four dimensions.ISBN 978-0-9802037-0-7
Original copy deposited at U.S. Copyright Office.

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Not a collection of sample chapters, its a full PDF book, with the added benefit for the visually impaired that PDF texts can be appropriately sized on screen for easy readability.

Foundations of transcomplex numbers is a mathematics book about a novel way of extending the complex numbers system to four-coordinate variables, maintaining the usual operations attributed to the complex numbers.

The ISBN for the E-Book is 978-0-9802037-0-7

The book is typeset in high quality Latex format for superb equations and text readability, formatted for 8½ x 11 letter size printing. With 194 pages, generously illustrated with 51 illustrations,  and many cross-references like:

  • Table of Contents
  • Bibliography
  • Index
  • Index of Definitions
  • Index of Figures
  • Index of Nomenclature
  • Index of Theorems

Seven chapters divided as follows:

  • Ordered Pairs. The whole theory of transcomplex functions is based on the ordered pair concept: from the two-dimension plane up to the four-dimension space.
  • Complex Numbers. The complex numbers system is derived from the ordered pairs concept.
  • Transcomplex Numbers. Here starts the extension of the complex numbers into ordered pairs of complex numbers, arriving at the concept of transcomplexs.
  • The Coordinate System S4. This chapter is devoted to deriving a suitable coordinate system to plot transcomplex functions.
  • Transcomplex Functions. Functions of complex variables evolve to make space for functions of four-entries ordered pairs.
  • Transcomplex Surfaces. A radical and totally new perception of surfaces generated by complex variables.
  • Theorem Proofs. This chapter collects all the proofs of the theorems stated along the book.

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Some sample pages from the book

Multiplication of two ordered pairs
Fig. 1-2. Multiplication of ordered pairs
The norm of an ordered pair
Fig. 1-4. The norm of an ordered pair
The norm of a complex number
Fig. 2-4. The norm of a complex number
The argument of a complex number
Fig. 2-5. The argument of a complex number
The 4 subspaces of the hyperspace S4
Fig. 4-10. The 4 subspaces of the hyperspace S4
The cubicle number 4 of the hyperspace S4
Fig. 4-15. The cubicle number 4 of the hyperspace S4
The real planes of the real cubicle
Fig. 4-4. The real planes of the real cubicle.
The complex trigonometric function
Fig. 5-12. The the complex
trigonometric function W = sin Z
The essence of the transcomplex maps
Fig. 5-16. The essence of the transcomplex maps