The ebook Elementary Calculus is based on material originally written by H.J. Keisler. For more information please read the copyright pages.


Algebraic Manipulation of Hyperreal Numbers

We have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus it is helpful to imagine a line in physical space as a hyperreal line. The hyperreal line is, like the real line, a useful mathematical model for a line in physical space.

The hyperreal numbers can be algebraically manipulated just like the real numbers. Let us try to use them to find slopes of curves. We begin with the parabola

y = x2

Consider a real point (x0, y0) on the curve y = x2. Let Δx be either a positive or a negative infinitesimal (but not zero), and let Δy be the corresponding change in y. Then the slope at (x0, y0) is defined in the following way:

[slope at (x0, y0)] = [the real number infinitely close to 01_real_and_hyperreal_numbers-85.gif]

We compute 01_real_and_hyperreal_numbers-86.gif as before:

01_real_and_hyperreal_numbers-87.gif

This is a hyperreal number, not a real number. Since Δx is infinitesimal, the hyperreal number 2x0 + Δx is infinitely close to the real number 2x0. We conclude that

[slope at (x0, y0)] = 2x0.

The process can be illustrated by the picture in Figure 1.4.5, with the infinitesimal changes Δx and Δy shown under a microscope.

01_real_and_hyperreal_numbers-88.gif

Figure 1.4.5


Last Update: 2010-11-25