The Hyperreal Line

First we shall give an intuitive picture of the hyperreal numbers and show how they can be used to find the slope of a curve. The set of all hyperreal numbers is denoted by R*. Every real number is a member of R*, but R* has other elements too. The infinitesimals in R* are of three kinds: positive, negative, and the real number 0. The symbols Δx, Δy,... and the Greek letters ε (epsilon) and δ (delta) will be used for infinitesimals. If a and b are hyperreal numbers whose difference a - b is infinitesimal, we say that a is infinitely close to b. For example, if Δx is infinitesimal then x₀ + Δx is infinitely close to x₀. If ε is positive infinitesimal, then -ε will be a negative infinitesimal. 1/ε will be an infinite positive number, that is, it will be greater than any real number. On the other hand, -1/ε will be an infinite negative number, i.e., a number less than every real number. Hyperreal numbers which are not infinite numbers are called finite numbers. Figure 1.4.3 shows a drawing of the hyperreal line. The circles represent "infinitesimal microscopes" which are powerful enough to show an infinitely small portion of the hyperreal line. The set R of real numbers is scattered among the finite numbers. About each real number c is a portion of the hyperreal line composed of the numbers infinitely close to c (shown under an infinitesimal microscope for c = 0 and c = 100). The numbers infinitely close to 0 are the infinitesimals.

In Figure 1.4.3 the finite and infinite parts of the hyperreal line were separated from each other by a dotted line. Another way to represent the infinite parts of the hyperreal line is with an "infinite telescope" as in Figure 1.4.4. The field of view of an infinite telescope has the same scale as the finite portion of the hyperreal line, while the field of view of an infinitesimal microscope contains an infinitely small portion of the hyperreal line blown up.