I. The Extension Principle

(a)     The real numbers form a subset of the hyperreal numbers, and the order relation x < y for the real numbers is a subset of the order relation for the hyperreal numbers.

(b)     There is a hyperreal number that is greater than zero but less than every positive real number.

(c)     For every real function f of one or more variables we are given a corresponding hyperreal function f* of the same number of variables, f* is called the natural extension of f.

Part (a) of the Extension Principle says that the real line is a part of the hyperreal line. To explain part (b) of the Extension Principle, we give a careful definition of an infinitesimal.

With this definition, part (b) of the Extension Principle says that there is at least one positive infinitesimal. We shall see later that there are infinitely many positive infinitesimals. A positive infinitesimal is a hyperreal number but cannot be a real number, so part (b) ensures that there are hyperreal numbers that are not real numbers.

Part (c) of the Extension Principle allows us to apply real functions to hyperreal numbers. Since the addition function + is a real function of two variables, its natural extension + * is a hyperreal function of two variables. If x and y are hyperreal numbers, the sum of x and y is the number x + * y formed by using the natural extension of +. Similarly, the product of x and y is the number x ·* y formed by using the natural extension of the product function ·. To make things easier to read, we shall drop the asterisks and write simply x + y and x · y for the sum and product of two hyperreal numbers x and y. Using the natural extensions of the sum and product functions, we will be able to develop algebra for hyperreal numbers. Part (c) of the Extension Principle also allows us to work with expressions such as cos (x) or sin (x + cos (y)), which involve one or more real functions. We call such expressions real expressions. These expressions can be used even when x and y are hyperreal numbers instead of real numbers. For example, when x and y are hyperreal, sin (x + cos (y)) will mean sin* (x + cos* (y)), where sin* and cos* are the natural extensions of sin and cos. The asterisks are dropped as before.

We now state the Transfer Principle, which allows us to carry out computations with the hyperreal numbers in the same way as we do for real numbers. Intuitively, the Transfer Principle says that the natural extension of each real function has the same properties as the original function.