\(\huge\color{red}{\textbf{十进制的实数群}}\)

APB先生

十进制的实数群

\[\left\langle \frac{1}{10^n}\right\rangle_{n=0}^{\ \infty}=\left\{ \cdots\cdots{,}\ \frac{-2}{10^n}{,}\ \frac{-1}{10^n}{,}\ 0{,}\ \frac{1}{10^n}{,}\ \frac{2}{10^n}{,}\ \cdots\cdots\right\}\]
      \(n=0\) 时：\[\left\langle 1\right\rangle=\left\{ \cdots{,}-2{,}-1{,}0{,}1{,}2{,}\cdots\right\}\]
      \(n=1\) 时：\[\left\langle \frac{1}{10}\right\rangle=\left\{ \cdots{,}\frac{-2}{10}{,}\frac{-1}{10}{,}0{,}\frac{1}{10}{,}\frac{2}{10}{,}\cdots\right\}\]
      \(1<n<\infty\) 时: ………………
      \(n=\infty\) 时：\[\left\langle \frac{1}{1\dot{0}}\right\rangle=\left\{ \cdots{,}\frac{-2}{1\dot{0}}{,}\frac{-1}{1\dot{0}}{,}0{,}\frac{1}{1\dot{0}}{,}\frac{2}{1\dot{0}}{,}\cdots\right\}\]

APB先生

\[\left\langle \frac{1}{10^n}\right\rangle_{n=0}^{\ \infty}=\left\{ \cdots\cdots{,}\ \frac{-2}{10^n}{,}\ \frac{-1}{10^n}{,}\ 0{,}\ \frac{1}{10^n}{,}\ \frac{2}{10^n}{,}\ \cdots\cdots\right\}\]

APB先生

\[\mathbb{R}=\bigcup_{n=0}^{\infty}\left\langle \frac{1}{10^n}\right\rangle\]