1.6 STANDARD  PARTS

In this section we shall develop a method that will enable us to compute the slope of a curve by means of infinitesimals. We shall use the method to find slopes of curves in Chapter 2 and to find areas in Chapter 4. The key step will be to find the standard part of a given hyperreal number, that is, the real number that is infinitely close to it.

 

DEFINITION

       Two hyperreal numbers b and c are said to be infinitely close to each other, in symbols

       b ≈ c, if their difference b-c is infinitesimal. b≠c means that b is not infinitely close to c.

 

Here are three simple remarks.

 

(1)If ε is infinitesimal, then b≈b+ε. This is true because the difference, b-(b+ε)= -ε,

is infinitesimal.

(2) b is infinitesimal if and only if b≈0. The formula b≈0 will be used as a short way

of writing “b is infinitesimal.”

(3) if b and c are real and b is infinitely close to c, then b equals c.

     b-c is real and infinitesimal, hence zero; so b=c.

 

The relation ≈ between hyperreal numbers behaves somewhat like equality, but, of course, is not the same as equality. Here are three basic properties of ≈.

 

THEOREM 1

Let a, b and c be hyperreal numbers.

(i) a≈a.

(ii) if a ≈ b, then b ≈ a.

(iii) if a ≈ b, and b ≈ c, then a ≈c.

 

These properties are useful when we wish to show that two numbers are infinitely close to each other.

 

The reason for (i) is that a-a is an infinitesimal, namely zero. For(ii), we note that if a -b is an infinitesimal ε, then b-a = -ε, which is also infinitesimal. Finally, (iii) is true because a-cis the sum of two infinitesimals, namely a-band b-c.

 

THEOREM 2

        Assume a ≈ b, then

         (i) if a is infinitesimal, so is b.

         (ii) if a is finite, so is b.

         (iii) if a is infinite, so is b.

 

The real numbers are sometimes called “standard” numbers, while the hyperreal numbers that are not real are called “nonstandard” numbers. For this reason, the real number that is infinitely close to bis called the “standard part” of b. An infinite number cannot have a standard part, because it can’t be infinitely close to a finite number (Theorem 2). Our third principle (stated next) on hyperreal numbers is that every finite numbers has a standard part.

 

|||.  STANDARD  PART  PRINCIPLE

 

Every finite hyperreal number is infinitely close to exactly one real number.

 

DEFINITION

      Let b be a finite hyperreal number. The standard part of b, denoted by st(b), is the real number which is infinitely close to b. Infinite hyperreal numbers do not have standard parts.

 

Here are some facts that follow at once from the definition.

Let b be a finite hyperreal number.

(1) st(b)is a real number.

(2) b=st(b) +ε for some infinitesimal ε.

(3) If b is real, then b=st(b).

 

Our next aim is to develop some skill in computing standard parts. This will be one of the basic methods throughout the Calculus course. The next theorem is the principal tool.

 

THEOREM  3

       Let a and b be finite hyperreal numbers. Then

       (i)  st(-a) = -st(a).

       (ii)  st(a+b) = st(a)+st(b).

       (iii)  st(a-b) = st(a) -st(b).

       (iv)  st(ab) = st(a)·st(b).

       (v)  if st(b) ≠ 0, then st(a/b)=st(a)/st(b).

       (vi)  st(an)=(st(a))n.

       (vii)  if a ≥ 0, then st(____)=________.

       (viii)  if a ≤ b, then st(a)≤ st(b).

 

This theorem gives formulas for the standard parts of the simplest expressions.

All of the rules in Theorem 3 follow from our three principles for hyperreal numbers. As an illustration, let us prove the formula (iv) for st(ab). Let rbe the standard part of a and sthe standard part of b, so that

                           a=r+ε,    b=s+δ

Where ε and δ are infinitesimal. Then

                         ab = (r +ε) (s+ δ)

                            = rs +r δ +sε +εδ ≈rs.

 

Therefore      st(ab) = rs = st(a) ·st(b)

 

Often the symbols Δx, Δy, etc. Are used for infinitesimals. In the following examples we use the rules in Theorem 3 as a starting point for computing standard parts of more complicated expressions.

 

EXAMPLE 1 When Δxis an infinitesimal and xis real, compute the standard part of

                     3x²+3 x Δx+ (Δx)².

      Using the rules in Theorem 3, we can write

                    st(3x²+ 3xΔx+(Δx)²)= st(3x² ) + st(3xΔx) + st((Δx)²)

                                  = 3x+st(3x) ·st(Δx) + st(Δx)²

                                  =3x²+ 3x·0 + 0² =3x².

 

EXAMPLE 2  If st(c)=4 and c ≠ 4, find

                         ____________________
         We note that the denominator has standard part 0,

                   st(c²-16)= st(c²) -16 = 4²-16=0

However, since c ≠ 4 the fraction is defined, and it can be simplified by factoring

the numerator and denominator,

                     

 

 

Then

 

 

We now have three kinds of computation available to us. First, there are computations involving hyperreal numbers. In Example 2, the two steps giving

 

                     ___________________

are computations of this kind. The computations of this first kind are justified by the Transfer Principle.

 

Second, we have computations which involve standard parts. In Example 2,the three steps giving

         

 

 

 

are of this kind. This second kind of computation depends on Theorem 3.

 

Third there are computations with ordinary real numbers. Sometimes the real numbers will appear as standard parts. In Example 2, the last two steps which give

                           

 

are computations with ordinary real numbers.

 

Usually, in computing the standard part of a hyperreal number, we use the first kind

of computation, the the second kind, and then the third kind, in that order. We shall give two more somewhat different examples and pick out these three stages in the computations.

 

EXAMPLE 3 If His a positive infinite hyperreal number, compute the standard part of

            

 

 

         In this example both the numerator and denominator are infinite, and we have to use

         the first type of computation to get the equation into a different form before we can

         take standard parts.

 

First stage

 

 

 

Second stage   H-1 and H-2  are infinitesimal, so

          

 

 

 

 

 Third stage     ______________________.

 

EXAMPLE  4  If εis infinitesimal but not zero, find the standard part of

 

                     

 

      Both the numerator and denominator are nonzero infinitesimals.

First stage   We multiply both numerator and denominator by 5+___________.

           

         

 

 

 

 

Second stage   ___________________________

             ___________________________

Third stage     st(b)= -5 - ____ = -10.

 

EXAMPLE  5  Remember that infinite hyperreal numbers do not have standard parts.

               Consider the infinite hyperreal number

                      3+ε  

                      4ε+ε²                     

 

Where ε is a nonzero infinitesimal. The numerator and denominator have standard parts

                 st(3+ ε)=3,    st(4ε +ε²)=0.

However, the quotient has no standard part. In other words,

                __________________ is undefined.

 

PROBLEMS  FOR  SECTION 1.6

Compute the standard parts of the following.

1   2+ε+3ε²,              εinfinitesimal

2   b+2ε - ε²,        st(b)=5,   εinfinitesimal

3   ____________,        εinfinitesimal

4   y4+2y2Δy+Δy3,         yreal, Δyinfinitesimal

5   (x²+3xΔx+ Δx² )6.      xreal, Δxinfinitesimal

6    _________________ xpositive real, Δxinfinitesimal  

7    ________________,   ε=0infinitesimal

8    ________________,   ε≠ 0infinitesimal

9   ________________,    ε≠ 0infinitesimal

10  (2+ε+δ) ( 3-εδ ),       ε, δ infinitesimal

11  ________________,  st(a) =3, ε, δ infinitesimal

12  ________________,  Hinfinite

13 ________________,  Hinfinite

14  ________________, Hinfinite

15  ________________,  Hinfinite, ε infinitesimal

16  ________________,  Hinfinite

17  ________________,   st(b)= 2, st(c)= -1

18   ________________,   st(b)= 3, st(c)= 2

19   ________________,   x, yreal , ε≠0 infinitesimal

20  ________________,   xreal , Δx≠0 infinitesimal

21   ________________,   xreal , Δx≠0 infinitesimal

22    ________________, a≠0 real, ε ≠ 0 infinitesimal

23    ________________,  b≠ 5 and st(b)= 5

24    ________________,  a≠ 4 and st(a)= 4

25    ________________,  c≠ 7 and st(c)= 7

26    ________________,  st(c)= 5

27    ________________,  a≠ -3 and st(a)= 3

28    ________________,  b≠ 2 and st(b)= 2

29    ________________,  c≠ -3 and st(c)= -3

30    ________________,  ε ≠ 0 and infinitesimal

31   ________________,   ε≠ 0 and infinitesimal

32    ________________,   Hpositive infinite

33    ________________,   Hpositive infinite

34    ________________,   Hpositive infinite

 

In the following problems let a, b, a1, b1be hyperreal numbers with a≈a1,b≈b1.

 

□35 show that a+b≈ a1+b1.

    Hinit: put a1=a+ε, b1=b+δ, and compute the difference (a1+b1) - (a+b).

□36 Show that if a, b are finite, then ab≈ a1b1.

□37 Show that if a=b=H,  a1=b1=H +1/H, then ab≈ a1b1. (Hpositive infinite).

 

EXTRA  PROBLEMS  FOR  CHAPTER 1

1   Find the distance between the points P(2,7) and Q(1, -4).

2   Find the slope of the line through the points P(2,-6) and Q(3,4).

3   Find the slope of the line through P(3,5) and Q(6,0).

4   Find the equation of the line through P(4,4) and Q(5,9).

5   Find the equation of the line through P(4,5) with slope m= -2.

6   Find the velocity and equation of motion of a particle which moves with constant velocity

and has positions y=2 at t=0, y=5 at t=2.

7   Find the equation of the circle with radius____and center at(1,3).

8   Find the equation of the circle that has center(1,0)and passes through the point(0,1).

 

Let εbe positive infinitesimal. Determine whether the following are infinitesimal, finite but not infinitesimal, or infinite.

 

9     (4 ε+ 5)(2ε+6)                       10    (4ε+5) (ε²-ε)

11    1/ε-2/ε²                              12    1- ________

 

Let Hbe positive infinite. Determine whether the following are infinitesimal, finite but not infinitesimal, or infinite.

13    (H-2) (2H+5)                       14  _________

15    __________                        16 ____________

 

Compute the standard parts in Problems 17-22.

17   (b + 2ε )(3b - 4ε ),  st(b) = 4,ε infinitesimal

18    __________, εinfinitesimal

19     _________, εinfinitesimal

20     _________, 0 ≠ Δx infinitesimal

21     _______, H, Kpositive infinite

22     _______H, Hpositive infinite

23     If f(x) = ______, find f(x+Δx)- f(x).

24     what is the domain of the function f(x) = _________ ?

□25   Show that if a < b, then (a+b)/2 is between aand b; that is , a<(a+b)/2 .

□26   Show that every open interval has infinitely many points.

□27   The union of two sets X and Y, X ∪ Y,is the set of all x such that xis either in X or Yor

       both. Prove that the union of two bounded sets is bounded.

  28   The intersection of X and Y, X ∩ Y, is the set of all x such that xis in both Xand Y.

       Prove that the intersection of two closed intervals is either empty or is a closed interval.

□29   Prove that the intersection of two open intervals is either empty or is an open interval.

□30   Prove that two (real) straight lines with different slopes intersect.

□31   prove that if His infinite, then 1/His infinitesimal.

□32   prove that if H is infinite and bis finite, then H + b is infinite.

□33   Prove that if εis positive infinitesimal, so is ______.

□34   Prove that if a,bare not infinitesimal and a≈b, then 1/a≈1/b.

□35   Prove that if a is finite, then st(|a|) = [st(a)|.

□36   Suppose ais finite , ris real, and st(a)<</span> r.Prove that a<</span> r.

□37   Suppose a and b are finite hyperreal numbers with st(a)Prove that there is a real

       number r with a < r < b.

□38  suppose that f is a real function.

      Show that the set of real solutions of the equation f(x)=0 is bounded if and only if every

      hyperreal solution of f*(x) =0 is finite.