Summary

We now have three kinds of computation available to us. First, there are computations involving hyperreal numbers. In Example 2, the two steps giving

are computations of this kind. The computations of this first kind are justified by the Transfer Principle.

Second, we have computations which involve standard parts. In Example 2, the three steps giving

are of this kind. This second kind of computation depends on Theorem 3.

Third there are computations with ordinary real numbers. Sometimes the real numbers will appear as standard parts. In Example 2, the last two steps which give

are computations with ordinary real numbers.

Usually, in computing the standard part of a hyperreal number, we use the first kind of computation, then the second kind, and then the third kind, in that order. We shall give two more somewhat different examples and pick out these three stages in the computations.