Pocket Electronic book: What Is Infinity approaching?

It now seems logical to introduce infinity and extend the real system. So, what is called "infinitely close". What's new in the hyper-real system?

In the electronic edition of Basic Calculus, chapter sixth, page 35th, J.keisler introduces the following definitions:

DEFINITION

Two hyperreal numbers B and C are said to is infinitely close to all other (mutual infinity) in symbols b≈c,if their difference b -C is infinitesimal, b┒≈c means that B isn't infinitely close to C.

From the definition, if the difference between the two super real numbers is infinitesimal, it is said that the two "infinitely close", this definition is very consistent with people's intuitive ideas. The following facts are obvious: if ε is infinitesimal, then there is a formula: b≈b+ε; if B is infinitesimal, then the necessary and sufficient condition is b≈0, that is, B is infinitesimal, can be expressed in b≈0, and if B and C are real and b≈c, then there must be b=c. From this, the infinity approach to "≈" is similar to the equal sign "=", but it is different. These three things should be kept in mind.

In the electronic edition of Basic Calculus, chapter sixth, page 36th, j.keisler the following definitions:

DEFINITION

Let B is a hyperreal numben. The standard part of b,denoted by St (b), are the real number which are infinitely close to B.infinite hyperreal do no t have standard part.

This definition is critical and must be thoroughly understood. It means that every finite super real number has a so-called "standard part" that is infinitely close to the super real number and is recorded as St (b).

Assuming A and B are super real numbers, the following equations are set up:

(1) St (a) =-st (a)

(2) St (a+b) =st (a) +st (b)

(3) St (A-b) =st (a)-st (b)

(4) St (A x B) =st (a) x St (b)

(5) if St (b) is not equal to zero, then there is St (a) =st (a)/st (b)

(6) If a≤b, there is St (a) ≤st (b)

If B is a real number, then St (b) =b and, generally speaking, when B is a super real number, b=st (b) +ε, where ε is an infinitesimal.

Note: We present a very interesting question: if this "infinite proximity" concept can be accepted by middle school students, then they must be able to understand and grasp the "st" algorithm. Thus, the derivation (including differential), the definite integral is not a problem. The initial knowledge of calculus, it is not entirely possible to "devolve" to the middle school to teach. We put the words in, calculus (in essence) is not the "thing".