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 ]

We compute as before:

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.