Standard Parts

In this section we shall develop a method that will enable us to compute the slope of a curve by means of infinitesimals. We shall use the method to find slopes of curves in Chapter 2 and to find areas in Chapter 4. The key step will be to find the standard part of a given hyperreal number, that is, the real number that is infinitely close to it.

Here are three simple remarks.

(1) If ε is infinitesimal, then b ≈ b + ε. This is true because the difference, b - (b + ε) = - ε, is infinitesimal.

(2) b is infinitesimal if and only if b ≈ 0. The formula b ≈ 0 will be used as a short way of writing "b is infinitesimal."

(3)  If b and c are real and b is infinitely close to c, then b equals c. b - c is real and infinitesimal, hence zero; so b = c.

The relation x between hyperreal numbers behaves somewhat like equality, but, of course, is not the same as equality. Here are three basic properties of ≈.