00后大学生：学习超实数无穷小，赢在学习微积分的起跑线上

打开网站“无穷小微积分”，点击“Elementary Calculus”按钮，等待数秒钟，问题的答案就会从遥远的互联网“云端”飞到你的视屏上。我国的00后大学生就是懂网络、会英文的一代新人。

容易想象，读者不出几秒钟的时间，就会找到该书的第一章的1.4节“实数与超实数”，其中有下面的一段文字（请见本文附件）。这段文字正面回答了微积分为什么需要引入超实数无穷小，我国的中学生都能读懂这段文字。该书作者的学术水平就表现在这里！书如其人也。

    感言：如果今年全国数百万大学新生都采用这本微积分参考书，该有多好啊！菲氏微积分可以休矣！  

袁萌  6月16日

附：

1.4  REAL AND HYPER REAL NUMBERS

But for a very small increment of time Lit, the velocity will change very little, and the average velocity Liy/ Lit will be close to the velocity at time t0 . To get the velocity v0 at time t0 , we neglect the small term Lit in the formula Dave = 2to + Lit,

and we are left with the value

v0 = 2t0 .

When we plot y against t, the velocity is the same as the slope of the curve y = t2, and the average velocity is the same as the average slope. The trouble with the above intuitive argument, whether stated in terms of slope or velocity, is that it is not clear when something is to be "neglected." Nevertheless, the basic idea can be made into a useful and mathematically sound method of finding the slope of a curve or the velocity. What is needed is a sharp distinction between numbers which are small enough to be neglected and numbers which aren't. Actually, no real number except zero is small enough to be neglected. To get around this difficulty, we take the bold step of introducing a new kind of number, which is infinitely small and yet not equal to zero. A number e is said to be irifinitely small, or infinitesimal, if

-a < e

for every positive real number a. Then the only real number that is infinitesimal is zero. We shall use a new number system called the hypeiTeal numbers, which contains all the real numbers and also has infinitesimals that are not zero. Just as the real numbers can be constructed from the rational numbers, the hyperreal numbers can be constructed from the real numbers. This construction is sketched in the Epilogue at the end of the book. In this chapter, we shall simply list the properties of the hyperreal numbers needed for the calculus. First we shall give an intuitive picture of the hyperreal numbers and show how they can be used to find the slope of a curve. The set of all hyperreal numbers is denoted by R*. Every real number is a member of R*, but R* has other elements too. The infinitesimals in R* are of three kinds: positive, negative, and the real number 0. The symbols Lix, Liy, ... and the Greek letters e (epsilon) and Ci (delta) will be used for infinitesimals. If a and bare hyperreal numbers whose difference a - b is infinitesimal, we say that a is irifinitely close to b. For example, if Lix is infinitesimal then x0 + Lix is infinitely close to x0 . If e is positive infinitesimal, then - e will be a negative infinitesimal. 1/e will be an irifinite positive number, that is, it will be greater than any real number. On the other hand, - 1/e will be an infinite negative number, i.e., a number less than every real number. Hyperreal numbers which are not infinite numbers are called finite numbers. Figure 1.4.3 shows a drawing of the hyperrealline. The circles represent "infinitesimal microscopes" which are powerful enough to show an infinitely small portion of the hyperrealline. The set R of real numbers is scattered among the finite numbers. About each real number c is a portion of the hyperrealline composed of the numbers infinitely close to c (shown under an infinitesimal microscope for c = 0 and c = 100). The numbers infinitely close to 0 are the infinitesimals. In Figure 1.4.3 the finite and infinite parts of the hyperrealline were separated from each other by a dotted line. Another way to represent the infinite parts of the hyperrealline is with an "infinite telescope" as in Figure 1.4.4. The field of view of an infinite telescope has the same scale as the finite portion of the hyperreal line, while the field of view of an infinitesimal microscope contains an infinitely small portion of the hyperreal line blown up.