Super integer and Its Usage

As you know, the integer (integers) set Z contains positive, negative integers, and zero, which is a frequently used number system. In the infinitely small calculus, to expand the theoretical needs (only for this purpose), we must expand the integer number system Z. What should we do?

Professor J. Keisler defines the nature of Continuous Functions in section 3.8 of basic calculus (you can search for them) as follows:

Definition

A hyperinteger(Super integer)Is
A hyperreal number y such that Y = [x] for some hyperreal X.

Function Y =
[X] is usually called the "maximum integer function". Its mark "[]" is a symbol of a specific function, indicating an integer not greater than X. That is to say, "When
X
Varies over the hyperreal numbers ,[X]
Is the greatest hyperintegerY
Such thatY
≤
X. "And, according to the transfer principle," Every
Hyperreal numberX
Is between two hyperintegers [X]
And [X] + 1 ", that is, the following formula is established:

【X] ≤
X≤ [X] + 1.

It can be seen that the super integer is a specific super real number, which is a natural expansion z * of the traditional integer system Z *. The positive infinite super real number x generates a positive infinite super integer, which is recorded as K, H, and so on. We assume that [a, B] * (Note: This interval is not a real interval [a, B], and the upper right corner of the interval contains an asterisk (*). divide it infinitely (in the case of an infinitely small calculus, The equals score is enough) to obtain an infinitely multiple (H) and other long subintervals (the length isDelta):

[A, A + Delta],[A + delta, A + 2 Delta],……, [A +(K-1)Delta, A + k Delta],......,[A +(H-1)Delta,
B]

The endpoint (split point) of each subinterval is:

A, A + delta, A + 2 Delta ,......, A + k Delta,......,A + H Delta =
B,

The super integer k changes between 1 and H. It is a very useful structural model for the theory of expanded calculus, especially the infinite series and integral theory. In traditional calculus, this is unimaginable.

In fact, the infinitely small ε and Delta, super integer h and K are two "magic weapons" for the study of Infinitely small calculus. As long as you really understand and grasp the essence and usage of the two, learning the infinitely small calculus is almost done. The specific values of an infinitely small number and an infinite number (Super integer) are not important. They are just a theoretical symbol (inference tool) as long as they exist. A. robinson invented "non-standard analysis", which is really a "trick". He invented the infinitely small ε and Delta, defined the super integer h and K, but no matter how small their values are, how big is it. Wonderful!