# 无理数的定义和实数理论的建立

1. 无理数最初来源于几何上的发现，那为什么不采用几何的方式来定义无理数呢？是什么原因使得康托（Georg Cantor）和戴德金（Richard Dedekind）的无理数或实数定义都不采用几何的方式？用几何的方式真的不行吗？

2. 算术法则或数学是真理吗？

3. 什么是数的连续体？实数系为什么是连续的？

4. 为什么无理数也有像有理数那样的乘法结合律和分配律等算术法则？如何证明？

5. 矩形的面积为什么是两邻边长度的乘积？

$$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{{bd}}$$$

$$$\frac{a}{b} \cdot \frac{c}{d} = \frac{{ac}}{{bd}}$$$

$$$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{{bd}} = \frac{cb + da}{{db}} = \frac{c}{d} + \frac{a}{b}$$$

## 两种定义实数的方式及其成因分析

Cantor对实数的定义 4 ：实数是一个有理数的柯西数列（Cauchy sequence of rational numbers）。其中，是“有理数的柯西数列”的条件是：该数列中的各项都是有理数，并且数列中除有限多项外的其它任意两项的差都小于一个预先任意指定的正有理数。任何一个满足这种条件的数列都是有理数的柯西数列，也是这种定义下的一个实数。

## 实数的加减乘除定义

（1）对于任意正实数x，总存在正整数n使得 $$$\frac{1}{n} < x$$$

$$$c_{0} \leq x \leq c_{0} + 1,$$$

$$$x \in I_{1} = \left\lbrack c_{0} + \frac{1}{10}c_{1},c_{0} + \frac{1}{10}c_{1} + \frac{1}{10} \right\rbrack$$$

$$$x \in I_{2} = \left\lbrack c_{0} + \frac{1}{10}c_{1} + \frac{1}{100}c_{2},c_{0} + \frac{1}{10}c_{1} + \frac{1}{100}c_{2} + \frac{1}{100} \right\rbrack$$$

$$$x \in I_{n} = \left\lbrack c_{0} + \frac{1}{10}c_{1} + \cdots + \frac{1}{10^{n}}c_{n},c_{0} + \frac{1}{10}c_{1} + \cdots + \frac{1}{10^{n}}c_{n} + \frac{1}{10^{n}} \right\rbrack$$$

$$$c_{1},c_{2},\ldots\ ,c_{n}$$$ 均是 $$$0,1,2,\ldots,9$$$ 中的其中一个，闭区间 $$$I_{n}$$$ 的长度是 $$$\frac{1}{10^{n}}$$$ 。如此无止境地构造下去便会得到无限多个区间

$$$I_{1},I_{2},\ldots,I_{n},\cdots$$$

$$$a$$$ $$$b$$$ 的乘除法还有待定义，你可能已经想到：可以考虑用数列 $$$\left\{ a_{n}b_{n} \right\}$$$ $$$\left\{ \frac{a_{n}}{b_{n}} \right\}$$$ 唯一逼近的值来分别定义 $$$a \cdot b$$$ $$$\frac{a}{b}$$$ 、来作为 $$$a \cdot b$$$ $$$\frac{a}{b}$$$ 的结果，但是除非我们能在不用到 $$$a \cdot b$$$ $$$\frac{a}{b}$$$ 的情况下证明 $$$\left\{ a_{n}b_{n} \right\}$$$ $$$\left\{ \frac{a_{n}}{b_{n}} \right\}$$$ 都有极限、都分别唯一逼近一个数，否则比如说在未定义 $$$a \cdot b$$$ 的情况下证明 $$$\left\{ a_{n}b_{n} \right\}$$$ 有极限的过程中如果用到 $$${ab}$$$ （例如出现 $$$\left| a_{n}b_{n} - ab \right| < \varepsilon$$$ ）就犯了循环论证（circular reasoning）的错误 50 。为了在没有定义 $$$a \cdot b$$$ $$$\frac{a}{b}$$$ 的情况下证明 $$$\left\{ a_{n}b_{n} \right\}$$$ $$$\left\{ \frac{a_{n}}{b_{n}} \right\}$$$ 都有极限，我们需要引入柯西收敛准则（Cauchy's Convergence Criterion），而这有必要从极限点（limit point）说起。

## 实数算术法则和不等关系的建立

1. $$$x \leq x$$$ .

2. $$$x \leq y$$$ $$$y \leq x,$$$ $$$x = y$$$ .

3. $$$x \leq y$$$ $$$y \leq z,$$$ $$$x \leq z$$$ .

4. 不是 $$$x \leq y$$$ 就是 $$$y \leq x$$$ .

5. $$$x \leq y,$$$ $$$x + z \leq y + z$$$ .

6. $$$x \leq y$$$ $$$0 \leq z,$$$ $$$x \cdot z \leq y \cdot z$$$ .

7. $$$x + (y + z) = (x + y) + z$$$ .

8. $$$x + 0 = 0 + x = x$$$ .

9. 对于 $$$x$$$ 总有 $$$y$$$ 使得 $$$x + y = y + x = 0$$$ .

10. $$$x + y = y + x$$$ .

11. $$$x \cdot (y \cdot z) = (x \cdot y) \cdot z$$$ .

12. $$$x \cdot 1 = 1 \cdot x = x$$$ .

13. $$$x \neq 0$$$ ，总有 $$$y$$$ 使得 $$$x \cdot y = y \cdot x = 1$$$ .

14. $$$x \cdot y = y \cdot x$$$ .

15. $$$x \cdot (y + z) = x \cdot y + x \cdot z$$$ .

## 综述

1. What Is Mathematics? Second Edition, Courant and Robbins, p54 ↩︎

2. What Is Mathematics? Second Edition, Courant and Robbins ↩︎

3. Julian Havil, The Irrationals, p241 ↩︎

4. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p984 ↩︎

5. Dedekind本人并不同意这种表述，详情请看Morris Kline, Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, p986 ↩︎

6. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, (Chap. 41, sec. 3) ↩︎

7. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p987 ↩︎

8. Richard Dedekind, Essays on the Theory of Numbers ↩︎

9. Real Numbers, Generalizations of the Reals, and Theories of Continua, Editors: Ehrlich, P. (Ed.)，page x ↩︎

10. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p979 ↩︎

11. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p979 ↩︎

12. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p982 ↩︎

13. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p1024 ↩︎

14. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p956 ↩︎

15. The History of the Calculus and Its Conceptual Development, Carl Benjamin Boyer, p284-286 ↩︎

16. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p1025 ↩︎

17. Introduction to Calculus and Analysis, Volume I, Reprint of the 1989 edition, Richard Courant, Fritz John, p88 ↩︎

18. Richard Dedekind, Essays on the Theory of Numbers ↩︎

19. 笔者不懂德语，这是在谷歌翻译的辅助下阅读后的结论 ↩︎

20. Contributions to the Founding of the Theory of Transfinite Numbers, Georg Cantor; Philip E B Jourdain, p26 ↩︎

21. Richard Dedekind, Essays on the Theory of Numbers ↩︎

22. Mathematics: The Loss of Certainty, Morris Kline, p84 ↩︎

23. Mathematics: The Loss of Certainty, Morris Kline, p88 ↩︎

24. Mathematics: The Loss of Certainty, Morris Kline, p88 ↩︎

25. Mathematics: The Loss of Certainty, Morris Kline, p95 ↩︎

26. Mathematics: The Loss of Certainty, Morris Kline, p95 ↩︎

27. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p980 ↩︎

28. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p1025-1026 ↩︎

29. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p879-880 ↩︎

30. Mathematics: The Loss of Certainty, Morris Kline, p92 ↩︎

31. What Is Mathematics? Second Edition, Courant and Robbins, p54 ↩︎

32. Mathematics: The Loss of Certainty, Morris Kline, p92 ↩︎

33. Mathematics: The Loss of Certainty, Morris Kline, p97 ↩︎

34. Mathematical Thought from Ancient to Modern Times, Vol. 3, 1990 edition, Morris Kline, p862 ↩︎

35. Mathematics: The Loss of Certainty, Morris Kline, p98 ↩︎

36. Mathematics: The Loss of Certainty, Morris Kline, p95 ↩︎

37. What Is Mathematics? Second Edition, Courant and Robbins, p54 ↩︎

38. Hans Niels Jahnke，A History of Analysis，p306 ↩︎

39. S. C. Malik， Principles of Real Analysis，P18; Editors: Ehrlich, p. (Ed.)，Real Numbers, Generalizations of the Reals, and Theories of Continua，page viii ↩︎

40. https://www.mathsisfun.com/geometry/construct-segment3.html ↩︎

41. What Is Mathematics? Second Edition, Courant and Robbins, p60 ↩︎

42. 这种定义方法在Richard C006Furant, Fritz John, Introduction to Calculus and Analysis Volume I, Reprint of the 1989 edition, p7有提及，但书中并未发展这种方法。 ↩︎

43. Richard Dedekind, Essays on the Theory of Numbers , p5 ↩︎

44. Richard Dedekind, Essays on the Theory of Numbers , p5 ↩︎

45. Richard Dedekind, Essays on the Theory of Numbers , p9 ↩︎

46. Introduction to Calculus and Analysis, Volume I, Reprint of the 1989 edition, Richard Courant, Fritz John, p2 ↩︎

47. Real Mathematical Analysis, Charles Chapman Pugh, 1st Edition, p20 ↩︎

48. Calculus with analytic geometry, 2nd Edition, George F. Simmons, p789 ↩︎

49. David French Belding, Kevin J. Mitchell, Foundations of Analysis, 2nd Edition, p21 ↩︎

50. 严格来讲从这里开始实际上有必要先介绍绝对值这个基本概念，但是通过几何的角度来理解绝对值也很简单，所以就不作介绍了。 ↩︎

51. Introduction to Calculus and Analysis, Volume I, Reprint of the 1989 edition, Richard Courant, Fritz John, p95-96 ↩︎

52. Introduction to Calculus and Analysis, Volume I, Reprint of the 1989 edition, Richard Courant, Fritz John, p97 ↩︎

53. Introduction to Calculus and Analysis, Volume I, Reprint of the 1989 edition, Richard Courant, Fritz John, p120 ↩︎

54. https://www.goteachmaths.co.uk/area-of-a-rectangle/ ↩︎

55. What Is Mathematics? Second Edition, Courant and Robbins, p399-400 ↩︎

56. Classic Set Theory: For Guided Independent Study , D.C. Goldrei, p15-16 ↩︎

57. What Is Mathematics? Second Edition, Courant and Robbins, p54 ↩︎

58. From Numbers To Analysis, Inder K Rana, p180 ↩︎