# 正规数

b是大于1的整数x实数。考虑以b为底的位值记数法x的数字序列。若s是以b为底的有限数字序列，我们以N(s,n)表示字串sx的开首n个数字出现次数。数x称为b为底正规若对任意长度k的字串s

$\lim_{n\to\infty} \frac{N(s,n)}{n} = \frac{1}{b^{k}}$

（即是说在x的数字中找到字串s概率，就像在完全随机生成的数字序列中的一样。）x称为正规数（有时称为绝对正规数） 如果以任何b为底x都是正规。

0.1234567891011121314151617...

0.235711131719232931374143...

## 参考

1. ^ Weisstein, Eric W. Normal Number. MathWorld. 2005-12-22 [2007-11-10].
2. ^ Preuss, Paul. Are The Digits of Pi Random? Lab Researcher May Hold The Key. Lawrence Berkeley National Laboratory. 2001-07-23 [2007-11-10].
• Bailey, D. H. and Crandall, R. E. "On the Random Character of Fundamental Constant Expansions." Experimental Mathematics 10, 175-190, 2001. online version
• Becher, V. and Figueira, S. "An example of a computable absolutely normal number", Theoretical Computer Science, 270, pp. 947-958, 2002.
• Borel, E. "Les probabilités dénombrables et leurs applications arithmétiques." Rend. Circ. Mat. Palermo 27, 247-271, 1909.
• Champernowne, D. G. "The Construction of Decimals Normal in the Scale of Ten." Journal of the London Mathematical Society 8, 254-260, 1933.
• Sierpiński, W. "Démonstration élémentaire d'un théorème de M. Borel sur les nombres absolutment normaux et détermination effective d'un tel nombre." Bull. Soc. Math. France 45, 125-144, 1917.