  Since the beginning of the quest of hypercomplex numbers in the late
eighteenth century, many hypercomplex number systems have been proposed but
none of them succeeded in extending the concept of complex numbers to higher
dimensions. This paper provides a definitive solution to this problem by
defining the truly hypercomplex numbers of dimension N greater than or equal to
3. The secret lies in the definition of the multiplicative law and its
properties. This law is based on spherical and hyperspherical coordinates.
These numbers which I call spherical and hyperspherical hypercomplex numbers
define Abelian groups over addition and multiplication. Nevertheless, the
multiplicative law generally does not distribute over addition, thus the set of
these numbers equipped with addition and multiplication does not form a
mathematical field. However, such numbers are expected to have a tremendous
utility in mathematics and in science in general.
