1.5: Infinitesimal, Finite, and Infinite Numbers

Let us summarize our intuitive description of the hyperreal numbers from Section 1.4. The real line is a subset of the hyperreal line; that is, each real number belongs to the set of hyperreal numbers. Surrounding each real number \(r\), we introduce a collection of hyperreal numbers infinitely close to \(r\). The hyperreal numbers infinitely close to zero are called infinitesimals. The reciprocals of nonzero infinitesimals are infinite hyperreal numbers. The collection of all hyperreal numbers satisfies the same algebraic laws as the real numbers. In this section we describe the hyperreal numbers more precisely and develop a facility for computation with them.

This entire calculus course is developed from three basic principles relating the real and hyperreal numbers: the Extension Principle, the Transfer Principle, and the Standard Part Principle. The first two principles are presented in this section, and the third principle is in the next section.

We begin with the Extension Principle, which gives us new numbers called hyperreal numbers and extends all real functions to these numbers. The Extension Principle will deal with hyperreal functions as well as real functions. Our discussion of real functions in Section 1.2 can readily be carried over to hyperreal functions. Recall that for each real number \(a\), a real function \(f\) of one variable either associates another real number \(b = f(a)\) or is undefined. Now, for each hyperreal number \(H\), a hyperreal function \(F\) of one variable either associates another hyperreal number \(K = F(H)\) or is undefined. For each pair of hyperreal numbers \(H\) and \(J\), a hyperreal function \(G\) of two variables either associates another hyperreal number \(K = G(H, J)\) or is undefined. Hyperreal functions of three or more variables are defined in a similar way.

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.

Definition:

A hyperreal number \(b\) is said to be:

positive infinitesimal if \(b\) is positive but less than every positive real number.

negative infinitesimal if \(b\) is negative but greater than every negative real number.

infinitesimal if \(b\) is either positive infinitesimal, negative infinitesimal, or zero.

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 \(x \cdot^{*} y\) formed by using the natural extension of the product function \(\cdot\). To make things easier to read, we shall drop the asterisks and write simply \(x + y\) and \(x \cdot y\) 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.

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: \((x - y)^{2} = x^{2} - 2xy + y^{2}\).

(6) A trigonometric identity: \(\sin^{2} x + \cos^{2} x = 1\).

(7) A rule for logarithms: If \(x > 0\) and \(y > 0\), then \(\log_{10} (xy) = log_{10} x + log_{10} y\).

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 (4), \(x/0\) is undefined, even for hyperreal \(x\). By Example (6), \(\sin^{2} x + \cos^{2} x = 1\), 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 \(\log_{10}\). 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.

(8) The square root function is defined by the real statement \(y = \sqrt{x}\) if, and only if, \(y^{2} = x\) and \(y \geq 0\).

(9) The absolute value function is defined by the real statement \(y = |x|\) if, and only if, \(y = p\).

(10) The common logarithm function is defined by the real statement \(y = \log_{10} x\) if, and only if, \(10^{y} = x\).

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, \(\sqrt{\}^{*}\),is defined by Example (8) when \(x\) and \(y\) are hyperreal.

An important use of the Transfer Principle is to carry out computations with infinitesimals. For example, a computation with infinitesimals was used in the slope calculation in Section 1.4. The Extension Principle tells us that there is at least one positive infinitesimal hyperreal number, say \(\varepsilon\). Starting from \(\varepsilon\), we can use the Transfer Principle to construct infinitely many other positive infinitesimals. For example, \(\varepsilon^{2}\) is a positive infinitesimal that is smaller than \(\varepsilon\), \(0 < \varepsilon^{2} < \varepsilon\). (This follows from the Transfer Principle because \(0 < x^{2} < x\) for all real \(x\) between 0 and 1.) Here are several positive infinitesimals, listed in increasing order: \[\varepsilon^{3}, \varepsilon^{2}, \varepsilon/100, \varepsilon, 75\varepsilon, \sqrt{\varepsilon}, \varepsilon + \sqrt{\varepsilon}. \nonumber\]

We can also construct negative infinitesimals, such as \(-\varepsilon\) and \(-\varepsilon^{2}\) , and other hyperreal numbers such as \(1 + \sqrt{\varepsilon}\), \((10 - \varepsilon)^{2}\), and \(1/\varepsilon\).

We shall now give a list of rules for deciding whether a given hyperreal number is infinitesimal, finite, or infinite. All these rules follow from the Transfer Principle alone. First, look at Figure \(\PageIndex{1}\), illustrating the hyperreal line.

Figure \(\PageIndex{1}\): The hyperreal line.

Definition:

A hyperreal number \(b\) is said to be:

finite if \(b\) is between two real numbers.
positive infinite if \(b\) is greater than every real number.
negative infinite if \(b\) is less than every real number.

Notice that each infinitesimal number is finite. Before going through the whole list of rules, let us take a close look at two of them.

If \(\varepsilon\) is infinitesimal and \(a\) is finite, then the product \(a \cdot \varepsilon\) is infinitesimal. For example, \(\frac{1}{2} \varepsilon\), \(-6 \varepsilon\), \(1000 \varepsilon\), \((5 - \varepsilon) \varepsilon\) are infinitesimal. This can be seen intuitively from Figure \(\PageIndex{2}\); an infinitely thin rectangle of length \(a\) has infinitesimal area.

If \(\varepsilon\) is positive infinitesimal, then \(1/\varepsilon\) is positive infinite. From experience we know that reciprocals of small numbers are large, so we intuitively expect \(1/\varepsilon\) to be positive infinite. We can use the Transfer Principle to prove \(1/\varepsilon\) is positive infinite. Let \(r\) be any positive real number. Since \(\varepsilon\) is positive infinitesimal, \(0 < \varepsilon < 1/r\). Applying the Transfer Principle, \(1/\varepsilon > r > 0\). Therefore, \(1/\varepsilon\) is positive infinite.

Figure \(\PageIndex{2}\): A rectangle of length \(a\) and infinitesimal height \(\varepsilon\).

RULES FOR INFINITESIMAL, FINITE, AND INFINITE NUMBERS

Assume that \(\varepsilon\), \(\delta\) are infinitesimals; \(b\), \(c\) are hyperreal numbers that are finite but not infinitesimal; and \(H\), \(K\) are infinite hyperreal numbers.

1. Real numbers:
   The only infinitesimal real number is 0.
   Every real number is finite.
2. Negatives:
   \(\varepsilon\) is infinitesimal.
   \(-b\) is finite but not infinitesimal.
   \(-H\) is infinite.
3. Reciprocals:
   If \(\varepsilon \neq 0\), \(1/\varepsilon\) is infinite.
   \(1/b\) is finite but not infinitesimal.
   \(1/H\) is infinitesimal.
4. Sums:
   \(\varepsilon + \delta\) is infinitesimal.
   \(b + \varepsilon\) is finite but not infinitesimal.
   \(b + c\) is finite (possibly infinitesimal).
   \(H + \varepsilon\) and \(H + b\) are infinite.
5. Products:
   \(\delta \cdot \varepsilon\) and \(b \cdot \varepsilon\) are infinitesimal.
   \(b \cdot c\) is finite but not infinitesimal.
   \(H \cdot b\) and \(H \cdot K\) are infinite.
6. Quotients:
   \(\varepsilon/b\), \(\varepsilon/H\), and \(b/H\) are infinitesimal.
   \(b/c\) is finite but not infinitesimal.
   \(b/\varepsilon\), \(H/e\), and \(H/b\) are infinite, provided that \(\varepsilon \neq 0\).
7. Roots:
   If \(\varepsilon > 0\), \(\sqrt[n]{\varepsilon}\) is infinitesimal.
   If \(b > 0\), \(\sqrt[n]{b}\) is finite but not infinitesimal.
   If \(H > 0\), \(\sqrt[n]{H}\) is infinite.

Notice that we have given no rule for the following combinations:

\(\varepsilon/\delta\), the quotient of two infinitesimals.
\(H/K\), the quotient of two infinite numbers.
\(H\varepsilon\), the product of an infinite number and an infinitesimal.
\(H + K\), the sum of two infinite numbers.

Each of these can be either infinitesimal, finite but not infinitesimal, or infinite, depending on what \(\varepsilon\), \(\delta\), \(H\), and \(K\) are. For this reason, they are called indeterminate forms. 

Here are three very different quotients of infinitesimals.

\(\dfrac{\varepsilon^{2}}{\varepsilon}\) is infinitesimal (equal to \(\varepsilon\)).

\(\dfrac{\varepsilon}{\varepsilon}\) is finite but not infinitesimal (equal to 1).

\(\dfrac{\varepsilon}{\varepsilon^{2}}\) is infinite (equal to \(\dfrac{1}{\varepsilon}\)).

Table \(\PageIndex{1}\) below shows the three possibilities for each indeterminate form. Here are some examples which show how to use our rules.

Example \(\PageIndex{1}\)

Consider \((b - 3\varepsilon) / (c + 2\delta)\). \(\varepsilon\) is infinitesimal, so \(-3\varepsilon\) is infinitesimal, and \(b - 3\varepsilon\) is finite but not infinitesimal. Similarly, \(c + 2\delta\) is finite but not infinitesimal. Therefore the quotient \[\frac{b - 3\varepsilon}{c + 2 \delta} \nonumber\]is finite but not infinitesimal.

Table \(\PageIndex{1}\). Possibilities for indeterminate forms.

	Examples
indeterminate form	infinitesimal	finite (equal to 1)	infinite
\(\dfrac{\varepsilon}{\delta}\)	\(\dfrac{\varepsilon^{2}}{\varepsilon}\)	\(\dfrac{\varepsilon}{\varepsilon}\)	\(\dfrac{\varepsilon}{\varepsilon^{2}}\)
\(\dfrac{H}{K}\)	\(\dfrac{H}{H^{2}}\)	\(\dfrac{H}{H}\)	\(\dfrac{H^{2}}{H}\)
\(H \varepsilon\)	\(H \cdot \dfrac{1}{H^{2}}\)	\(H \cdot \dfrac{1}{H}\)	\(H^{2} \cdot \dfrac{1}{H}\)
\(H + K\)	\(H + (-H)\)	\((H + 1) + (-H)\)	\(H + H\)

The next three examples are quotients of infinitesimals.

Example \(\PageIndex{2}\)

The quotient \[\frac{5 \varepsilon^{4} - 8 \varepsilon^{3} + \varepsilon^{2}}{3 \varepsilon} \nonumber\]is infinitesimal, provided \(\varepsilon \neq 0\).
The given number is equal to \[\frac{5}{3} \varepsilon^{3} - \frac{8}{3} \varepsilon^{2} + \frac{1}{3} \varepsilon.\]

We see in turn that \(\varepsilon, \varepsilon^{2}, \varepsilon^{3}, \frac{1}{3} \varepsilon, -\frac{8}{3} \varepsilon^{2}, \frac{5}{3} \varepsilon^{3}\) are infinitesimal; hence the sum \((\PageIndex{1})\) is infinitesimal.

Example \(\PageIndex{3}\)

If \(\varepsilon \neq 0\), the quotient \[\frac{3 \varepsilon^{3} + \varepsilon^{2} - 6 \varepsilon}{2 \varepsilon^{2} + \varepsilon} \nonumber\]is finite but not infinitesimal.

Cancelling an \(\varepsilon\) from numerator and denominator, we get \[\frac{3 \varepsilon^{2} + \varepsilon - 6}{2 \varepsilon + 1}.\]

Since \(3\varepsilon^{2} + \varepsilon\) is infinitesimal while \(-6\) is finite but not infinitesimal, the numerator \[3 \varepsilon^{2} + \varepsilon - 6 \nonumber\]is finite but not infinitesimal. Similarly, the denominator \(2\varepsilon + 1\) is finite but not infinitesimal, and hence the quotient \((\PageIndex{2})\) is finite but not infinitesimal.

Example \(\PageIndex{4}\)

If \(\varepsilon \neq 0\), the quotient \[\frac{\varepsilon^{4} - \varepsilon^{3} + 2\varepsilon^{2}}{5 \varepsilon^{4} + \varepsilon} \nonumber\]is infinite.

We first note that the denominator \(\varepsilon^{4} + \varepsilon^{3}\) is not zero because it can be written as a product of nonzero factors, \[5\varepsilon^{4} + \varepsilon^{3} = \varepsilon \cdot \varepsilon \cdot \varepsilon \cdot (5 \varepsilon + 1).\nonumber\]

When we cancel \(\varepsilon^{2}\) from the numerator and denominator we get \[\frac{\varepsilon^{2} - \varepsilon + 2}{5 \varepsilon^{2} + \varepsilon}. \nonumber\]

We see in turn that:

\(\varepsilon^{2} - \varepsilon + 2\) is finite but not infinitesimal,

\(5\varepsilon^{2} + \varepsilon\) is infinitesimal,

\(\dfrac{\varepsilon^{2} - \varepsilon + 2}{5 \varepsilon^{2} + \varepsilon}\) is infinite.

Example \(\PageIndex{5}\)

\(\dfrac{2H^{2} + H}{H^{2} - H + 2}\) is finite but not infinitesimal.

In this example the trick is to multiply both numerator and denominator by \(1/H^{2}\). We get \[\frac{2 + 1/H}{1 - 1/H + 2/H^{2}}. \nonumber\]

Now \(1/H\) and \(1/H^{2}\) are infinitesimal. Therefore both the numerator and denominator are finite but not infinitesimal, and so is the quotient.

In the next theorem we list facts about the ordering of the hyperreals.

Theorem \(\PageIndex{1}\)

(i) Every hyperreal number which is between two infinitesimals is infinitesimal.
(ii) Every hyperreal number which is between two finite hyperrealnumbers is finite.
(iii) Every hyperreal number which is greater than some positive infinite number is positive infinite.
(iv) Every hyperrealnumber which is less than some negative infinite number is negative infinite.

All the proofs are easy. We prove (iii), which is especially useful. Assume \(H\) is positive infinite and \(H < K\). Then for any real number \(r\), \(r < H < K\). Therefore, \(r < K\) and \(K\) is positive infinite.

Example \(\PageIndex{6}\)

If \(H\) and \(K\) are positive infinite hyperreal numbers, then \(H + K\) is positive infinite. This is true because \(H + K\) is greater than \(H\).

Our last example concerns square roots.

Example \(\PageIndex{7}\)

If \(H\) is positive infinite then, surprisingly, \[\sqrt{H+1} - \sqrt{H-1} \nonumber\]is infinitesimal.

This is shown using an algebraic trick. \[\begin{align*} \sqrt{H+1} - \sqrt{H-1} &= \frac{\left(\sqrt{H+1} - \sqrt{H-1}\right) \left(\sqrt{H+1} + \sqrt{H-1}\right)}{\sqrt{H+1} + \sqrt{H-1}} \\[4pt] &= \frac{(H+1) - (H-1)}{\sqrt{H+1} + \sqrt{H-1}} = \frac{2}{\sqrt{H+1} + \sqrt{H-1}}. \end{align*}\]

The numbers \(H + 1\), \(H - 1\), and their square roots are positive infinite, and thus the sum \(\sqrt{H+1} + \sqrt{H-1}\) is positive infinite. Therefore the quotient \[\sqrt{H+1} - \sqrt{H-1} = \frac{2}{\sqrt{H+1} - \sqrt{H-1}}, \nonumber\]a finite number divided by an infinite number, is infinitesimal.

Problems for Section 1.5

In Problems 1-40, assume that: \(\varepsilon\), \(\delta\) are positive infinitesimal, \(H\), \(K\) are positive infinite. Determine whether the given expression is infinitesimal, finite but not infinitesimal, or infinite.

1.	\(76,000,000 \varepsilon\)	2.	\(3 \varepsilon + 4 \delta\)
3.	\(1 + 1/\varepsilon\)	4.	\(3 \varepsilon^{3} - 2\varepsilon^{2} + \varepsilon + 1\)
5.	\(1 / \sqrt{\varepsilon}\)	6.	\(\varepsilon/H\)
7.	\(H/1,000,000\)	8.	\((3 + \varepsilon)^{2} - 9\)
9.	\((3 + \varepsilon)(4 + \delta) - 12\)	10.	\(\dfrac{1 + \varepsilon + 3 \varepsilon^{2}}{2 - \varepsilon - 8\varepsilon^{3}}\)
11.	\(\dfrac{2 \varepsilon^{3} + \varepsilon^{4}}{4 \varepsilon - \varepsilon^{2} + \varepsilon^{3}}\)	12.	\(\dfrac{2 \varepsilon^{3} - \varepsilon^{4}}{4\varepsilon^{3} + \varepsilon^{4}}\)
13.	\(\dfrac{3 \varepsilon - 4 \varepsilon^{2}}{\varepsilon^{2} + 5 \varepsilon^{3}}\)	14.	\(\dfrac{\sqrt{\varepsilon} + \varepsilon}{\sqrt{\varepsilon} + 1}\)
15.	\(\dfrac{1}{\sqrt{\varepsilon} - \varepsilon}\)	16.	\(\dfrac{1}{\varepsilon} \cdot \varepsilon^{3}\)
17.	\(\dfrac{1}{\varepsilon} \cdot 5\varepsilon\)	18.	\(\dfrac{1}{\varepsilon} \cdot \varepsilon^{3}\)
19.	\(\dfrac{1}{\varepsilon} \left(\dfrac{1}{3 + \varepsilon} - \dfrac{1}{3}\right)\)	20.	\(\dfrac{2H + 1}{3H + 2}\)
21.	\(\dfrac{2H^{4} + 3H - 6}{4H^{3} + 5}\)	22.	\(\dfrac{H + 4 + \varepsilon}{H^{2} + 2\varepsilon}\)
23.	\(\dfrac{H + K}{HK}\)	24.	\(\dfrac{H-K}{H^{2} + K^{2}}\)
25.	\(H^{2} - H\)	26.	\(\sqrt{H+1} - \sqrt{H}\)
27.	\(\left(H + \dfrac{1}{H}\right)^{2} - \left(H - \dfrac{1}{H}\right)^{2}\)	28.	\(\left(H + \dfrac{\varepsilon}{H}\right)^{2} - \left(H - \dfrac{\varepsilon}{H}\right)^{2}\)
29.	\(\dfrac{\sqrt{4 + \varepsilon} - 2}{\varepsilon}\)	30.	\(\dfrac{1}{\varepsilon} \left(1 - \dfrac{1}{\sqrt{1 + \varepsilon}}\right)\)
31.	\(H \left(\sqrt{3 + \dfrac{1}{H}} - \sqrt{3}\right)\)	32.	\(\dfrac{\sqrt{H}}{\sqrt{H+1} + \sqrt{H + 2}}\)
33.	\(H (\sqrt{H + 2} - \sqrt{H})\)	34.	\(\dfrac{1 - \sqrt[3]{1 + \varepsilon}}{\varepsilon}\)
35.	\(\sqrt[3]{H} - \sqrt[3]{H+1}\)	36.	\(H - \sqrt{H+1} \sqrt{H+2}\)
37.	\(\dfrac{(3 + \varepsilon)(4 + \delta) - 12}{\varepsilon \delta}\)	38.	\(\dfrac{5 + \varepsilon}{7 + \delta} - \dfrac{5}{7}\)
39.	\(\dfrac{\varepsilon + \delta}{\sqrt{\varepsilon^{2} + \delta^{2}}}\) (Hint: Assume \(\varepsilon \geq \delta\) and divide through by \(\varepsilon\).)	40.	\(\dfrac{H + K}{\sqrt{H^{2} + K^{2}}}\)
41.	In (a)-(f) below, determine which of the two numbers is greater. \(\quad\) (a) \(\varepsilon\) or \(\varepsilon^{2}\)  \(\quad\) (b) \(\dfrac{1}{\varepsilon^{3}}\) or \(\dfrac{1}{\varepsilon^{4}}\) \(\quad\) (c) \(H\) or \(H^{2}\) \(\quad\) (d) \(\varepsilon\) or \(\sqrt{\varepsilon}\) \(\quad\) (e) \(H\) or \(\sqrt{H}\) \(\quad\) (f) \(\sqrt{H}\) or \(\sqrt[3]{H}\)
\(\square\) 42.	Let \(x\), \(y\) be positive hyperreal numbers. Can \(\dfrac{x}{y} + \dfrac{y}{x}\) be infinite? Finite? Infinitesimal?
\(\square\) 43.	Let \(a\) and \(b\) be real. When is \((3 \varepsilon^{2} - \varepsilon + a)/(4 \varepsilon^{2} + 2 \varepsilon + b)\) \(\quad\) (a) infinitesimal? \(\quad\) (b) finite but not infinitesimal? \(\quad\) (c) infinite?
\(\square\) 44.	Let \(a\) and \(b\) be real. When is \((aH^{2} - 2H + 5)/(bH^{2} + H - 2)\) \(\quad\) (a) infinitesimal? \(\quad\) (b) finite but not infinitesimal? \(\quad\) (c) infinite?