  This paper aims to build a new understanding of the nonstandard mathematical
analysis. The main contribution of this paper is the construction of a new set
of numbers, $\mathbb{R}^{\mathbb{Z}_< }$, which includes infinities and
infinitesimals. The construction of this new set is done na\"ively in the sense
that it does not require any heavy mathematical machinery, and so it will be
much less problematic in a long term. Despite its na\"ivety character, the set
$\mathbb{R}^{\mathbb{Z}_< }$ is still a robust and rewarding set to work in. We
further develop some analysis and topological properties of it, where not only
we recover most of the basic theories that we have classically, but we also
introduce some new enthralling notions in them. The computability issue of this
set is also explored. The works presented here can be seen as a contribution to
bridge constructive analysis and nonstandard analysis, which has been
extensively (and intensively) discussed in the past few years.
