第1.5节 无穷小，有限超实数与无穷大

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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 tor.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.

Thisentire calculus course is developed from three basic principlesrelating the real and hyperreal numbers: the Extension Principle, theTransfer Principle, and the Standard Part Principle. The first twoprinciples are presented in this section, and the third principle isin 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 hyperreal functions as well asreal functions. Our discussion of real functions in Section 1.2 canreadily be carried over to hyperreal functions. Recall that for eachreal number a,a real function fofone variable either associates another real number b=f(a) oris undefined. Now, for each hyperreal number H,a hyperreal function Fofone variable either associates another hyperreal number K=F(H)oris undefined. For each pair of hyperreal numbers HandJ, ahyperreal function G oftwo variables either associates another hyperreal number K=G(H,J)oris undefined. Hyperreal functions of three or more variables aredefined in a similar way.

1.THEEXTENSION PRINCIPLE

(a)Thereal numbers form a subset of the hyperreal numbers, and the orderrelation x<y for

therealnumbers is a subset of the order relation for the hyperreal numbers.

(b)Thereis a hyperreal number that is greater than zero but less than everypositive real

number.

(c)Forevery real function f of one or more variables we are given acorresponding hyperreal

functionf*of the same number of variables. f* is called the natural extensionof 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 an infinitesimal.

DEFINITION

Ahyperreal number b is said to be:

Positiveinfinitesimalif b is positive but less than every positive real number, negativeinfinitesimal if b is negative but greater than every negative realnumber.

Infinitesimalifb is either positive infinitesimal, 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 function oftwo variables, its natural extension +* is a hyperreal function oftwo variables. If x andy arehyperreal numbers, the sum of xandy isthe number x+*yformedby using the natural extension of +. Similarly, the product ofx andy isthe number x.*y formedby using the natural extension of the product function ·. To makethings easier to read, we shall drop the asterisks and write simply x+y andx·yforthe sum and product of two hyperreal numbers x andy. Usingthe natural extensions of the sum and product functions, we will beable to develop algebra for hyperreal numbers. Part (c) of theExtension Principle also allows us to work with expressions such ascos(x)or sin(x+cos(y)),which involve one or more real functions. We call such expressionsreal expressions.These expressions can be used even when x andy arehyperreal numbers instead of real numbers. For example, when xandy arehyperreal, sin(x+cos(y))will mean sin*(x+cos*(y)),where sin* 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 outcomputations 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 properties asthe original function.

‖.TRANSFER PRINCIPLE

Everyreal statement that holds for one or more particular real functionsholds for the

hyperrealnaturalextensions of these functions.

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

(1)Closurelaw for addition: for any xandy,the sum x+yisdefined.

(2)Commutativelaw for addition : x+y=y+x

(3)Arule for order : If 0<x<y,then 0<1/y<1/x.

(4)Divisionby zero is never allowed : x/0is undefined.

(5)Analgebraic identity : (x-y)²=x²-2xy+y².

(6)Atrigonometric identity : sin²x+cos² x=1.

(7)Arule for logarithms : If x>0and y>0,then log10(xy)= log 10x +log 10y.

Eachexample has two variables, x andy, andholds true whenever xandy arereal numbers. The Transfer Principle tells us that each example alsoholds whenever xandy arehyperreal numbers. For instance, by Example (4), x/0is undefined, even for hyperreal x.By Example (6),

sin²x+cos ²x =1,even 1or 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 familiar rules oftrigonometry, exponents, and logarithms to the hyperreal numbers.

Incalculus we frequently make a computation involving one or moreunknown real numbers. The Transfer Principle allows us to 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.

Areal statement is often used to define a new real function from oldreal functions. 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.

(8)Thesquare root function is defined by the real statementy=____if,and only if, y²=xandy≥0.

(9)Theabsolute value function is defined by the real statement y=|x|if,and only if, y=____.

(10)Thecommon logarithm function is defined by the real statement

y=log10x if,and only if, 10y=x.

Ineach case, the hyperreal natural extension is the function defined bythe given real statement when xandy varyover the hyperreal numbers. For example, the hyperreal naturalextension of the square function ___*,is defined by Example (8) when x and y are hyperreal.

Animportant use of the Transfer Principle is to carry out computationswith infinitesimal. For example, a computation with infinitesimalswas used in the slope calculation in Section 1.4. The ExtensionPrinciple tells us that there is at least one positive infinitesimalhyperreal number, say ε.Starting from ε,we can use the Transfer Principle to construct infinitely many otherpositive infinitesimals. For example, ε²isa positive infinitesimal that is smaller than ε, 0<ε²<ε.(thisfollows from the Transfer Principle because 0<x²<x forall real xbetween0 and 1.) Here are several positive infinitesimal, listed inincreasing order:

______________________

Wecan also construct negative infinitesimals, such as -εand ε²,and other hyperreal numbers such as 1+_____,(10- ε)²,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

Ahyperreal number b is said to be:

finiteifb is between two real numbers.

positiveinfiniteif b is greater than every real number.

negativeinfiniteif b is less than every real number.

Noticethat each infinitesimal number is finite. Before going through thewhole list of rules, let us take a close look at two of them.

Ifε is infinitesimal and a is finite, then the product a·εis infinitesimal. For example, ________areinfinitesimal. This can be seen intuitively from Figure 1.5.2; aninfinitely thin rectangle of length a has infinitesimal area.

Ifε is positive infinitesimal, then 1/ε is positiveinfinite. From experience we know that reciprocals of small numbersare large, so we intuitively expect 1/ε to be positiveinfinite. We can use the Transfer Principle to prove 1/ε ispositive infinite. Let r be any positive real number. Since εis positive infinitesimal, 0<ε<1/r. Applying the TransferPrinciple, 1/ε >r >0. Therefore, 1/ε ispositive infinite.

Figure1.5.2

RULESFOR INFINITESIMAL, FINITE, AND INFINITE NUMBERS

Assumethat ε, δ are infinitesimal; b, care hyperreal numbers that are finite but not infinitesimal; and H, Kare infinite hyperreal numbers.

(i)Real numbers:

Theonlyinfinitesimal real number is 0.

Everyrealnumber is finite.

(ii)Negatives:

-εis infinitesimal.

-bisfinite but not infinitesimal.

-Hisinfinite.

(iii)Reciprocals:

Ifε ≠ 0, 1/εis infinite.

1/bis finite but not infinitesimal.

1/His infinitesimal.

(iv)Sums:

ε+δ is infinitesimal.

b+ ε is finite but not infinitesimal.

b+ c is finite (possible infinitesimal).

H+εand H + b are infinite.

(v)Products:

δ·εand b·ε are infinitesimal.

b·cisfinite but not infinitesimal.

H·bandH·K are infinite.

(vi)Quotients:

ε/b,ε/H, and b/ H are infinitesimal.

b/cis finite but not infinitesimal.

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

(vii)Roots:

Ifε >0,____ is infinitesimal.

Ifb >0, ____ isinfinitesimal but not infinitesimal.

IfH >0, ____ is infinite.

Noticethat we have given no rule for the following combinations:

ε/δ,the quotient of two infinitesimals.

H/K,the quotient of two infinite numbers.

Hε,the product of an infinite number and an infinitesimal.

H+ K,the sum of two infinite numbers.

Eachof these can be either infinitesimal, finite but not infinitesimal,or infinite, depending on what ε,δ ,H, andK are.For this reason, they are called indeterminateforms.

Hereare three very different quotients of infinitesimals.

___isinfinitesimal (equal to ε).

______isfinite but not infinitesimal (equal to 1).

______isinfinite ( equal to ___).

Table1.5.1 on the following page shows the three possibilities for eachindeterminate form. Here are some examples which show how to use ourrules.

EXAMPLE1 Consider(b-3ε)/(c+2δ).ε isinfinitesimal, so -3εisinfinitesimal and b-3εisfinite but not infinitesimal. Similarly, c+2δisfinite but not infinitesimal. Therefore the quotient

b-3ε

c+2δ

isfinite but not infinitesimal.

Thenext three examples are quotients of infinitesimals.

EXAMPLE2 The quotient

5ε4-8ε3+ε2

3ε

isinfinitesimal, provided ε≠0.

Thegiven number is equal to

(1)_____________________

Weseein turn that ______________areinfinitesimal; hence the sum(1) is infinitesimal.

EXAMPLE3 Ifε≠ 0,the quotient

3ε3+ε2-6ε

2ε2+ε

isfinite but not infinitesimal.

Cancellingan ε fromnumerator and denominator, we get

(2)

3ε3+ε-6

2ε+1

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

3ε3+ε-6

isfinite but not infinitesimal. Similarly, the denominator 2ε+1, and hence

thequotient(2) is finite but not infinitesimal.

EXAMPLE4 If ε≠0,the quotient

ε4_ ε3+2ε2

5ε4+ε3

isinfinite.

Wefirst note that the denominator 5ε4+ε3isnot zero because it can be written as a product of nonzero factors,

5ε4+ε3=ε·ε·ε·(5ε+1).

Whenwe cancel ε2fromthe numerator and denominator we get

ε2_ ε+2

5ε2+ε

Wesee in turn that :

ε2_ ε+2isfinite but not infinitesimal,

5ε2+εisinfinitesimal.,

ε2_ ε+2

5ε2+εisinfinite.

EXAMPLE5 __________isfinite but not infinitesimal.

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

2+1/H

1-1/H+2/H2

Now1/H and1/H2areinfinitesimal. Therefore both the numerator and denominator arefinite but not infinitesimal, and so is the quotient.

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

THEOREM1

(i)Every hyperreal number which is between twoinfinitesimal is infinitesimal.

(ii)Every hyperreal number which is between two finite hyperreal numbersis finite.

(iii)Every hyperreal number which is greater than some positive infinitenumber is positive infinite.

(iv)Every hyperreal number which is less than some negative infinitenumber is negative 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 andK ispositive infinite.

EXAMPLE6 If H andK arepositive infinite hyperreal numbers, then H+K ispositive infinite.

Thisistrue because H+Kisgreater than H.

Ourlast example concerns square roots.

EXAMPLE7 IfH ispositive infinite then, surprisingly,

____________________

isinfinitesimal.

Thisisshown using an algebraic trick.

Thenumbers H +1, H -1,and their square roots are positive infinite, and thus the sum_______ispositive infinite. Therefore the quotient

afinite number divided by an infinite number, is infinitesimal.

PROBLEMSFOR SECTION 1.5

InProblems 1-40, assume that: ε, δarepositive infinitesimal, H, Karepositive infinite.

Determinewhether the given expression is infinitesimal, finite but notinfinitesimal, or infinite.

176,000,000ε23ε +4δ

31+1/ ε 43ε3-2ε²+ε+1

5________6ε/H

7H/1,000,0008 (3+ε)²-9

9(3+ε)(4+δ)-1210 _________

11____________12_________

13___________14_________

15__________ 16__________

17__________18__________

19__________20__________

21__________22__________

23__________24__________

25H²-H 26_________

27__________28__________

29__________30__________

31__________32__________

33__________34__________

35___________36__________

37___________38___________

39____________40___________

(Hint:Assume ε ≥δ

anddividethrough by ε.)

41In (a) - (f) below, determine which of the two numbers is greater.

(a)ε orε²(b)___or_____(c)H orH²

(d)ε or____(d)H or_____(b)___or_____

□42Let x, y bepositive hyperreal numbers. Can _______beinfinite? Finite ? Infinitesimal?

□43Let a andb bereal. When is (3ε²-ε+a)/ (4ε²+2ε+b)

(a)infinitesimal?

(b)finitebutnot infinitesimal?

(c)infinite?

□44Let a andb bereal. When is (aH²-2H+5)/ (bH²+H-2)

(a)infinitesimal?

(b)finitebutnot infinitesimal?

(c)infinite?