1.5 INFINITESIMAL,  FINITE,  AND  INFINITE  NUMBERS

Letus 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 hyperrealnumbers. Surrounding each real number r,we introduce a collection of hyperreal numbers infinitely close to r.The hyperreal numbers infinitely close to zero are calledinfinitesimals. The reciprocals of nonzero infinitesimals areinfinite hyperreal numbers. The collection of all hyperreal numberssatisfies the same algebraic laws as the real numbers. In thissection we describe the hyperreal numbers more precisely and developa facility for computation with them.

 Thisentire calculus course is developed from three basic principlesrelating the real and hyperrdal numbers: the Extension Principle, theTransfer Principle, and the Standard PartPrinciple. The first two principles are presented in this section,and the third principle is in the next section.

Webegin with the Extension Principle, which gives us new numbers calledhyperreal numbers and extends all real functions to these numbers.The Extension Principle will deal with hyperrealfunctions as well as real functions.Our discussion of real functions in Section 1.2 can readily becarried over to hyperreal functions. Recall that for each realnumber  α,a real function f ofone variable either associates another real number b = f(α)or is undefined. Now, for each hyperreal number H,a hyperrealfunction Fof onevariable either associates another hyperreal number K= F (H) oris undefined. For each pair of hyperreal numbers  H andJ,a hyperreal function Gof twovariables either associates another hyperreal number K= G (H, J)or is undefined. Hyperreal functions of three ormore variables are defined in a similar way.

I.  THE  EXTENSION  PRINCIPLE    

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

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

(c)   For every real function f of one or morevariables we are given a corre- sponding hyperreal function f* of thesame 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 ofthe hyperreal line. To explain part (b) of the Extension Principle,we give a careful definition of aninfinitesimal.

DEFINITION

               Ahyperreal number b is said to be:

               positiveinfinitesimal  if  bis positive but less than every positive real number. negativeinfinitesimal  if  bis negative but greater than every negative real number.

Infinitesimalif  b is either positiveinfinitesimal, negative infinitesimal. or zero.

Withthis definition, part (b) of the Extension Principle says that thereis at least one positive infinitesimal. We shall see later that thereare infinitely many positive infinitesimals. A positive infinitesimalis 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 tohyperreal numbers. Since the addition function + is a real functionof two variables, its natural extension +* isa hyperreal function of two variables. If xand yare hyperreal numbers, the sum of xand yis the number x+* yformed by using the natural extension of +.Similarly, the product of xand yis the number x∙* yformed by using the natural extension of theproduct function ∙. To make thingseasier to read, we shall drop the asterisks and write simply x+ y andx ∙ yfor the sum and product of two hyperreal numbersx andy.Using the natural extensions of the sum andproduct functions, we will be able to develop algebra for hyperrealnumbers. Part (c) of the Extension Principle also allows us to workwith expressions such as cos (x)or sin (x+ cos (y)),whichinvolve one or more real functions. We call such expressions realexpressions. These expressions can beused even when xand yare hyperreal numbers instead of real numbers.For example, when xand yare hyperreal, sin (x+cos (y))willmean sin* (x+ cos* (y)),wheresin* and cos* are the natural extensions of sin and cos. Theasterisks are dropped as before.

Wenow state the Transfer Principle, which allows us to carry outcompu-tations with the hyperreal numbers in the same way as we do forreal numbers. Intuitively, the Transfer Principle says that thenatural extension of  each real function has the same propertiesas the original function.

ll. TRANSFER  PRINCIPLE

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

Hereare seven examples that illustrate what we mean by a real statement.In general, by a real statement we mean a combination of equations orinequalities about teal expressions, and statements specifyingwhether a real expression is defined or undefined. A real statementwill involve real functions.

⑴ Closurelaw for addition : for any xand y,the sum x+ y isdefined.

⑵  Commutativelaw for addition : x+ y =y + x.

⑶  Arule for order: If 0 < x< y,then 0 < 1/y <1/x.

⑷  Divisionby zero is never allowed: x/0is undefined.

⑸  Analgebraic identity: x-y2= x2-2xy+y2

⑹  Atrigonometric identity：sin2+cos2x=1.

⑺  Arule for logarithms: If x> 0 and y> 0, then log10(xy)=log10 x+                      log10 y.

Eachexample has two variables, xand y,and holds true whenever x andy arereal numbers. The Transfer Principle tells us that each example also holds whenever xand yare hyperreal numbers. For instance, by Example(4), x/0is undefined, even for hyperreal x.By Example (6), sin2 x+cos2x=1,even lon hyperreal x.

Noticethat the first five examples involve only the sum, difference,product, and quotient functions. However, the last two examples arereal statements involving the transcendental functions sin, cos, andlog10 .The Transfer Principle extends all the familiarrules of trigonometry, exponents, and logarithms to the hyperrealnumbers.

Incalculus wefrequently make a computationinvolving one or moreunknown real numbers. The TransferPrinciple allows usto compute inexactly the same way with hyperreal numbers. It “transfers” factsabout the real numbers to facts about the hyperreal numbers. Inparticular, the Transfer Principle implies that a real function andits natural extension always give the same value when applied to areal number. This is why we are usually able to drop the asteriskswhen computing with hyperreal numbers.

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

⑻Thesquare root function is defined by the real statement y=xif, and only if, y2= x andy ≥ 0.

⑼Theabsolute value function is defined by the real statement y=x if, and onlyif,  y=x2.

⑽Thecommon logarithm function is defined by the real statement y = log10x if, and onlyif, 10y=x.

Ineach case, the hyperreal natural extension is the function defined bythe given real statement when xand yvary over the hyperreal numbers. Forexample, the hyperreal natural extension of the square root function,*, is defined by Example ⑻ when xand yare hyperreal.

Animportant use of the Transfer Principle is to carry out computationswith infinitesimals. For example, acomputation with infinitesimals was used in the slope calculation inSection 1.4. The Extension Principle tells us that there is at leastone positive infinitesimal hyperreal number, say ε.  Starting from ε,we can use the Transfer Principle to constructinfinitely many other positive infinitesimals.For example, ε2 isa positive infinitesimal that is smaller than ε,0 < ε2<ε. (Thisfollows from the Transfer Principle because 0 < x2<x forall real xbetween 0 and 1.) Here are several positiveinfinitesimals, listde in increasing order:

                          ε3,ε2ε/100,ε,75ε,ε,ε + ε.

Wecan also construct negative infinitesimals, such as -εand -ε2, and otherhyper-real numbers such as 1+ε,10-ε2,and 1/ε.

Weshall now give a list of rules for deciding whether a given hyperrealnumber is infinitesimal, finite, or infinite. All these rules followfrom the Transfer Principle alone. First, look at Figure 1.5.1,illustrating the hyperreal line. 

DEFINITION

Ahyperrdal number b is said to be:

Finiteif b is between two real number.

positiveinfinite if b is greater thanevery real number.

negativeinfinite if b is less thanevery real number.

Noticethat each infinitesimal number is finite. Beforegoing through the whole list of rules, let us take a close look attwo lf them.

If ε isinfinitesimal and a is finite, then the product a ∙εis infinitesimal.For example, 12ε, -6ε,1000ε,(5-ε)εare infinitesimal. This can be seen intuitivelyfrom Figure 1.5.2; an infinitely thinrectangle of length a has infinitesimal area.

Ifε ispositive infinitesimal, then 1/εis positive infinite. Fromexperience we know that reciprocals of small numbers are large, so weintuitively expect 1/ε to be positive infinite. We can use the TransferPrinciple to prove 1/εis positive infinite. Let rbe any positive real number. Sinceε ispositive infinitesimal, 0 < ε< 1/r.Applying the Transfer Principle, 1/ε> r>0. Therefore, 1/εis positive infinite.

RULESFOR INFINITESIMAL, FINITE, AND INFINITE  NUMBERS     Assume that   ε,δ  areinfinitesimals; b, c are hyperreal numbers that are finite but notinfinitesimal; and H, K are infinite hyperreal nubers.

(i)Realnumbers:

        Theonly infinitesimal real number is 0.

Everyreal number is finite.

(ii)Negatives:

                -εis infinitesimal.

−bis finite but not infinitesimal.

−His infinite

(iii)Reciprocals:

If ε ≠0, 1/εis infinite.

1/bis finite but not infinitesimal.

1/His infinitesimal.

(iv)Sums:

ε+ δis infinitesimal.

b+ε is finite but notinfinitesimal.

b+ c is finite (possiblyinfinitesimal).

H+ ε and H + b are infinite

(v)Products:

δ·ε andb ·ε areinfinitesimal.

b·c isfinite but not infinitesimal.

H·b and H·K are infinite

(vi)Quotients:

ε/b,ε/H, and b/Hare infinitesimal.

b/cis finite but not infinitesimal.

b/ε,H/ε, and H/b are infinite,provided that ε≠0.

(vii)Roots:

If ε,>0, nεis infinitesimal.

If b > 0, nbis finite but not infinitesimal.

IfH >0, nHis infinite

Noticethat we have given no rule for the following combinations:

                 ε/δ, thequotient of two infinitesimals.

  H/K,the quotient of two infinite numbers.

                 Hε,the product of an infinite number and aninfinitesimal.

                 H+ K, the sum of two infinite numbers.

Eachof these canbe eitherinfinitesimal, finitebut notinfinitesimal, orinfinite, depending on what ε,δ,, H,and Kare. For this reason, they are calledindeterminate forms.

Hereare three very different quotients of infinitesimals.

                  ε2εis infinitesimal(equal to ε ).

                  εε is finite but not infinitesimal(equal to 1).

                 εε2 is infinite equalto1ε  .

Table1.5.1 on the following page shows thethree possibilities for each indeterminate form. Here are someexamples which show how to use our rules.

EXAMPLE1    Considerb-3ε/c+2δ. ε isinfinitesimal, so-3ε is infinitesimal, and                                      b-3εis finite but not infinitesimal.Similarly, c+2δ  isfinite but not infinitesimal.      Therefore the quotient

b-3εc+2δ

Isfinite but not infinitesimal.

Thenext three examples are quotients of infinitesimals.

EXAMPLE2    The quotient

5ε4-8ε3+ε23ε

isinfinitesimal, provided  ε ≠ 0.

Thegiven number is equal to

⑴                   
53ε3-83ε2+13ε.

We see  in turn that ε,  ε2,ε3,  13ε,- 83ε2,53ε3 are infinitesimal; hence the sum⑴ isinfinitesimal.

EXAMPLE 3   If ε ≠ 0, the quotient

3ε3+ε2-6ε2ε2+ε

Is finite but not infinitesimal.

Cancellingan ε  fromnumerator and denominator, we get

⑵
3ε2+ε-62ε+1

Since3ε2 +ε  is infinitesimal while -6is finite but not infinitesimal, the numerator

3ε2+ε-6

Is finite but not infinitesimal. Similarly, the denominator 2ε+1,and hence the quotient ⑵  is finite but not infinitesimal.

EXAMPLE 4  If ε ≠ 0  , the quotient

ε4-ε3+2ε25ε4+ε3

isinfinite.

We first note that the denominator  5ε4+ε3is not zero because it can be written as aproduct  of  nonzero factors,

5ε4+ε3=ε·ε·ε·(5ε+1).

When we cancel ε2  fromthe numerator and denominator we get

ε2-ε+25ε2+ε

Wesee in turn that:

ε2-ε+2 is finite but not infinitesimal,

5ε2+ε is infinitesimal,

ε2-ε+25ε2+ε is infinite.

EXAMPLE 5    2H2+HH2-H+2 is finite but not infinitesimal.

Inthis example the trick is to multiply both numerator and denominatorby  1/ H2.We get

2+1/H1-1/H+2/H2

Now 1/H  and 1/H2  areinfinitesimal. Therefore both the numerator and denominator arefinite but not infinitesimal, and so is the quotient.

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

THEOREM 1

(i)Everyhyperreal number which is between two infinitesimals isinfinitesi-mal.

(ii)Everyhyperreal number which is between two finite hyperreal numbers isfinite.

(iii)Everyhyperreal number which is greater than some positive infinite numberis positive infinite.

(iv)Everyhyperreal number which is less than some negative infinite number isnegative infinite .

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

EXAMPLE 6   If Hand K are positive infinite hyperrealnumbers, then H + K ispositive infinite. This is true because H+ K is greater than H.

Ourlast example concerns square roots.