# Foundations Of Transcomplex Numbers:

## An extension of the complex number system to four dimensions

### By E. Pérez

ISBN 978-0-9802037-0-7

Original copy deposited at U.S. Copyright Office.

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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 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 S**^{4}. 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

Fig. 1-2. Multiplication of ordered pairs

Fig. 1-4. The norm of an ordered pair

Fig. 2-4. The norm of a complex number

Fig. 2-5. The argument of a complex number

Fig. 4-10. The 4 subspaces of the hyperspace S^{4}

Fig. 4-15. The cubicle number 4 of the hyperspace S^{4}

Fig. 4-4. The real planes of the real cubicle.

Fig. 5-12. The the complex

trigonometric function *W *= sin *Z*

Fig. 5-16. The essence of the transcomplex maps