II. Transfer Principle

Every real statement that holds for one or more particular real functions holds for the hyperreal natural extensions of these functions.

Here are seven examples that illustrate what we mean by a real statement. In general, by a real statement we mean a combination of equations or inequalities about real expressions, and statements specifying whether a real expression is defined or undefined. A real statement will involve real variables and particular real functions.

1. Closure law for addition: for any x and y, the sum x + y is defined.
2. Commutative law for addition: x + y = y + x.
3. A rule for order: If 0 < x < y, then 0 < 1/y < 1/x.
4. Division by zero is never allowed: x/0 is undefined.
5. An algebraic identity: (x - y)2 = x2 - 2xy + y2.
6. A trigonometric identity: sin2 x + cos2 x = 1.
7. A rule for logarithms: If x > 0 and y > 0, then log10 (xy) = log10 x + log10 y.

Each example has two variables, x and y, and holds true whenever x and y are real numbers. The Transfer Principle tells us that each example also holds whenever x and y are hyperreal numbers. For instance, by Example (3), x/0 is undefined, even for hyperreal x. By Example (6), sin2 x + cos2 x = 1, even for hyperreal x.

Notice that the first five examples involve only the sum, difference, product, and quotient functions. However, the last two examples are real statements involving the transcendental functions sin, cos, and log10. The Transfer Principle extends all the familiar rules of trigonometry, exponents, and logarithms to the hyperreal numbers.

In calculus we frequently make a computation involving one or more unknown real numbers. The Transfer Principle allows us to compute in exactly the same way with hyperreal numbers. It "transfers" facts about the real numbers to facts about the hyperreal numbers. In particular, the Transfer Principle implies that a real function and its natural extension always give the same value when applied to a real number. This is why we are usually able to drop the asterisks when computing with hyperreal numbers.

A real statement is often used to define a new real function from old real functions. By the Transfer Principle, whenever a real statement defines a real function, the same real statement also defines the hyperreal natural extension function. Here are three more examples.

1. The square root function is defined by the real statement y = y√x if, and only if, y2 = x and y ≥ 0.
2. The absolute value function is defined by the real statement y = |x| if, and only if, y =
3. The common logarithm function is defined by the real statement y = log10 x if, and only if, 10y = x.

In each case, the hyperreal natural extension is the function defined by the given real statement when x and y vary over the hyperreal numbers. For example, the hyperreal natural extension of the square root function, √*, is defined by Example (8) when x and y are hyperreal.

Last Update: 2006-11-15