Hyperreal Numbers

Notice that each infinitesimal number is finite. Before going through the whole list of rules, let us take a close look at two of them.

If b is infinitesimal and a is finite, then the product a · s is infinitesimal. For example, ½ε, -6ε, 1000ε, (5 - ε)ε are infinitesimal. This can be seen intuitively from Figure 1.5.2; an infinitely thin rectangle of length a has infinitesimal area.

If ε is positive infinitesimal, then 1/ε is positive infinite. From experience we know that reciprocals of small numbers are large, so we intuitively expect 1/ε to be positive infinite. We can use the Transfer Principle to prove 1/ε is positive infinite. Let r be any positive real number. Since e is positive infinitesimal, 0 < ε < 1/r. Applying the Transfer Principle, 1/ε > r > 0. Therefore, 1/ε is positive infinite.