ORDER OF THE COMPLEX NUMBERS AND ITS CONSEQUENCES

by Antonio González Carlomán

Introduction

From the formula  Euler developed the power of the complex numbers. Nowadays, in the self

construction of basic Mathematics ( Mathematical logic, Set theory, Relations and structures and the Numerical system

[''Mathematical Atlas'' by Reinhard and Soeder, Publisher Alianza]) a didactic lack is presented because the formula from

Euler can not be included in its development to build the power of the complex numbers.

In the essay that is resumed here, this didactic lack is avoided by means of an order of the complex numbers (modular

order) from which, by means of sequences of powers of complex number base and rational number index of denominator

power of two, the power of complex numbers with real numbers as index is defined. And by means of an inverse process

to the one followed by Euler, the trigonometry is very easily developed. And finally, with the support of the

trigonometry already built, the pi number  can be defined.

1.Quadrants

Being  the set of the complex numbers, the subset of the complex numbers of modulus one and the set of the

positive real numbers; if  ,we call  the subset of the elements of  of modulus .

And over  we define the following subsets:

Which we call quadrants of    (If .

Obviously  and  would be the quadrants of .

From now on, we agree: If  , then  and .

1.1 Partition in

The family of quadrants  form a partition in .

1.2. Canonical form of the complex numbers of modulus one.

Any  can be uniquely written in the form

in wich  and   and .

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