This research has been done with the aim of building a new understanding of the nonstandard mathematical analysis. In order to achieve this goal, we first explore the two existing models of numbers: reals (R) and hyperreals ( R), and also the main feature of the latter one which is the transfer principle – which, as Goldblatt said, is where the strength of nonstandard analysis lies. However, here we analyse some serious problems with that transfer principle and moreover, propose an idea to solve it: combining the two languages of R and RZ< into one language. Nevertheless, there is one obvious big problem with this idea, which is the occurrence of contradiction.

Two possible ways are proposed in resolving this contradiction issue. One of them, which we favour in this research, is by having a subsystem in our new theory. This idea was based on Chunk and Permeate strategy proposed by Brown and Priest in 2003. In the process of doing that, we then turn to the primary contribution of this thesis: the construction of a new set of numbers, RZ<, which also include infinities and infinitesimals in it. The construction of this new set is done naïvely (in comparison to other sets) 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 of its naïve way of construction, it has been demonstrated in this thesis that the set RZ< is still a robust and rewarding set to work in. We further develop some analysis and topological properties of RZ<, where not only we recover most of the basic theories that we have classically, but we also introduce some new enthralling notions in them. Lastly, we also deal with the computability aspect of our set RZ<. We define the set RZ< c , a set of all computable numbers in RZ<, and show that its standard arithmetic operations (functions) are computable. We use a concrete implementation of these ideas in the programming language Python, whose syntax should be intuitively understandable even by those not familiar with it.