Summary

Let us summarize our intuitive description of the hyperreal numbers from Section 1.4. The real line is a subset of the hyperreal line; that is, each real number belongs to the set of hyperreal numbers. Surrounding each real number r, we introduce a collection of hyperreal numbers infinitely close to r. The hyperreal numbers infinitely close to zero are called infinitesimals. The reciprocals of nonzero infinitesimals are infinite hyperreal numbers. The collection of all hyperreal numbers satisfies the same algebraic laws as the real numbers. In this section we describe the hyperreal numbers more precisely and develop a facility for computation with them.

This entire calculus course is developed from three basic principles relating the real and hyperreal numbers: the Extension Principle, the Transfer Principle, and the Standard Part Principle. The first two principles are presented in this section, and the third principle is in the next section.

We begin with the Extension Principle, which gives us new numbers called hyperreal numbers and extends all real functions to these numbers. The Extension Principle will deal with hyperreal functions as well as real functions. Our discussion of real functions in Section 1.2 can readily be carried over to hyperreal functions. Recall that for each real number a, a real function/of one variable either associates another real number b = f(a) or is undefined. Now, for each hyperreal number H, a hyperreal function F of one variable either associates another hyperreal number K = F(H) or is undefined. For each pair of hyperreal numbers H and J, a hyperreal function G of two variables either associates another hyperreal number K = G(H, J) or is undefined. Hyperreal functions of three or more variables are defined in a similar way.