How to add infinitesimal to the real number system My question may seem foolish, but I don't know that why it is hard to add infinitesimal to the real number system.
For example, if I add to the real number system two special symbols to denote infinitesimal and infinitely large quantity, and give an order relation and operations on the new system, then what problems do occur?
(Of course such a number system may not satisfy the Dedekind completeness, but I hear existing real number systems with infinitesimal also don't.)

To complement sos440's fine answer, I would mention that what you are proposing is in fact close to what is known in the literature as "dual numbers"; see Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?  Here you have basically a single (up to a multiple) infinitesimal $\epsilon$ such that $\epsilon^2=0$.  The resulting number system is helpful in certain applications in physics.  However, to be able to extend some standard functions (beyond polynomials) to the new number system, one needs a more elaborate approach.
