0.999…

在数学上， 考虑0.999...是否是等于1或是一个非常接近于1的数字是一个陷阱。下面是两者在實數集相等的证明。

证明

$0.999\ldots$	$={\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+\cdots$
	$=-9+{\frac {9}{1}}+{\frac {9}{10}}+{\frac {9}{100}}+{\frac {9}{1000}}+\cdots$
	$=-9+9\times \sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}$
	$=-9+9\times {\frac {1}{1-{\frac {1}{10}}}}$
	$=1.\,$

解释

关键的一步是理解无限等比数列的收敛性:

$\sum _{k=0}^{\infty }\left({\frac {1}{10}}\right)^{k}={\frac {1}{1-{\frac {1}{10}}}}.$

問題

這種証明的問題是，暗地裡使用了極限法。

极限证法

$0.999\ldots =\lim _{n\to \infty }\left(0.9+0.09+0.009+\ldots +0.9\times 0.1^{n}\right)=\lim _{n\to \infty }\sum _{k=0}^{n}{\frac {0.9}{10^{k}}}$

$=\lim _{n\to \infty }{0.9-0.9\times 0.1^{n} \over 1-0.1}$

$={0.9 \over 1-0.1}$

$=1$

• 在无穷等比数列$\left\{a_{n}\right\}$中$a_{1}$为首项，公比q满足$\left|q\right|<1$时
• 该无穷等比数列的和$S_{n}=\lim _{n\to \infty }\left({\frac {a_{1}-a_{1}q^{n}}{1-q}}\right)={\frac {a_{1}}{1-q}}$

另一种证法

这里还有一个没有使用很多数学知识的证明方法:

证明:

假设x=0.999...

∵

10x−x = 9.999... − 0.999...

即

9x = 9

∴

x = 1

这个证明使用了实数的一個性质──沒有非零無限小。

分数证法

证明1:

∵

1/9=0.111...

2/9=0.222...

3/9=0.333...

4/9=0.444...

5/9=0.555...

6/9=0.666...

7/9=0.777...

8/9=0.888...

∴

1=9/9=0.999...

证明2:

9/9= 1/9 + 8/9

= 0.111... + 0.888...

= 0.999...

证明3:

${\begin{aligned}0.333\dots &={\frac {1}{3}}\\3\times 0.333\dots &=3\times {\frac {1}{3}}={\frac {3\times 1}{3}}\\0.999\dots &=1\end{aligned}}$

問題

這類証明假定了 1/9=0.111... 、 1/3 = 0.333... 這類分數轉小數成立。

而

$0.111\ldots ={1 \over 10}+{1 \over 100}+{1 \over 1000}+\ldots ={1 \over 10}+{1 \over 10^{2}}+{1 \over 10^{3}}+\ldots$

$0.333\ldots ={3 \over 10}+{3 \over 100}+{3 \over 1000}+\ldots ={3 \over 10}+{3 \over 10^{2}}+{3 \over 10^{3}}+\ldots$

都要用到極限法或其他一些與實數相關的性質去証明。

最简单的证明

证明

∵

1/9=0.111...

∴

0.999...=1

Q.E.D.

反證法

证明

若 1≠0.999...

即是 1 與 0.999... 之間有數值存在

設 x 為 1 與 0.999... 之間的任意數值，使得 0.999... < x < 1

x 不存在，兩數之間沒有數值存在，故兩數相等

换言之，假定0.999...与1是不同的实数。那么，在(0.999..., 1)区间内必然存在无穷个实数。但实际上并不存在这样的实数；因此，原先的假设错误: 0.999... 与1并非不同的实数，它们相等。

参见

• 循环小数
• 等比数列
• 收敛数列
• 无限数列
• 极限
• 無窮小
• 實數
• 超實數

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外部连接

• Repeating Nines
• 为什么0.9999...=1?