超实数的演算

  进入二十一世纪，我国普通高校教授微积分不引入超对数，就是“装糊涂”。

  简而言之，无穷小放飞互联网已经1七年了，而且几乎每天都有超实数的文章放飞互联网。

难道我国普通高高校数学教师从来不上网？

  百度一下“无穷小”，进入“无穷小微积分”专业网站，下载相关文章，看看其中的内容目录即可知晓一切，… …

    本文附件文章，只有18页A4纸长度，讲解了无穷小微积分的基本概念与理论。看与不看，请随便。

袁萌  陈启清 12月23日

附件：

hyperreal calculus

Abstract

This project deals with doing calculus not by using epsilons and deltas, but by using a number system called the hyperreal numbers. The hyperreal numbers is an extension of the normal real numbers with both innitely small and innitely large numbers added. We will rst show how this system canbecreated,andthen shows omebasicpropertiesofthe hyperreal numbers. Then we will show how one can treat the topics of convergence, continuity, limits and dierentiation in this system and we will show that the two approaches give rise to the same denitions and results.

Contents

1 Construction of the hyperreal numbers 3 1.1 Intuitive construction . . . . . . . . . . . . . . . 3

1.2 Ultralters . . . . . . 3

1.3 Formal construction . .  4

1.4 Innitely small and large numbers . . . . .. 5

1.5 Enlarging sets . . . 5

1.6 Extending functions . . . . . .  . 6

2 The transfer principle  6

2.1 Stating the transfer principle . . . . . . . . . .  6

2.2 Using the transfer principle . . . . . .  7

3 Properties of the hyperreals 8 3.1 Terminology and notation . . . .  . 8

3.2 Arithmetic of hyperreals . . . . . 9

3.3 Halos . . . . . 9

3.4 Shadows . . . . .10

4 Convergence 11

4.1 Convergence in hyperreal calculus. . . .  .

4.2 Monotone convergence  . 12

5 Continuity 13

5.1 Continuity in hyperreal calculus . . . . . 13

5.2 Examples . . . . . . . . . . . . . . . . . . 14

5.3 Theorems about continuity . . . . . . . . .. 15

5.4 Uniform continuity . . . . . . .. . 16

6 Limits and derivatives 17 6.1 Limits in hyperreal calculus . . . . . . .  17

6.2 Dierentiation in hyperreal calculus . . . . . .18

6.3 Examples . . . . . .  18

6.4 Increments . . . . . . . . . . . . . . . . . . 19

6.5 Theorems about derivatives . .  . 19

1

1 Construction of the hyperreal numbers

1.1 Intuitive construction We want to construct the hyperreal numbers as sequences of real numbers hrni = hr1,r2,...i, and the idea is to let sequences where limn→∞rn = 0 represent innitely small numbers, or innitesimals, and let sequences where limn→∞rn =∞ represent innitely large numbers. However, if we simply let each hyperreal number be dened as a sequence of real numbers, and let addition and multiplication be dened as elementwise additionandmultiplicationofsequences, wehavetheproblemthatthisstructure is not a eld, since h1,0,1,0,...i
h0,1,0,1,...i=h0,0,0,0,...i. The way we solve this is by introducing an equivalence relation on the set of real-valued sequences. We want to identify two sequences if the set of indices for which the sequences agree is a large subset of N, for a certain technical meaning of large. Let us rst discuss some properties we should expect this concept of largeness to have. • N itself must be large, since a sequence must be equivalent with itself. • If a set contains a large set, it should be large itself. • The empty set ∅ should not be large. • We want our relation to be transitive, so if the sequences r and s agree on a large set, and s and t agree on a large set, we want r and t to agree on a large set.