数学中国

用户名  找回密码
 注册
帖子
热搜: 活动 交友 discuz
查看: 4320|回复: 0

无限循环小数化成分数的另类方法

[复制链接]
发表于 2023-4-7 21:21 | 显示全部楼层 |阅读模式
根据分数性质、有理数的特性,无限循环小数化成分数(数学另类方法):本文不运用等比数列极限求和方法,避开许多人不能理解接受的数学逻辑工具所涉及到的无限——实无限?潜无限?:
(1)譬如:将无限循环小数0.9˙化成分数(整数),
解:
0.9˙=(0.9+0.09˙)——1式
    令0.9˙=(0.9+x)——2式
    2式两边同乘以10:
9.9˙ =(9+10x)——3式
(3式)-(2式):
9= (8.1+9x)
9x=9-8.1
9x=0.9
将x=9/90代入2是:
0.9˙=9/10+9/90
  =90/90
  =9/9
=1/1
=1
∴0.9˙=1
(2)将无限循环小数 0.˙123˙化成分数:
解:
0. ˙123˙=(0.123+0.000˙123˙)——1式
  令0. ˙123˙=(0.123+x)——2式
      (2)式*1000:
123. ˙123˙=(123+1000x)——3式
(3式)-(2式):
123=[123-(123/1000)-999x]
999x=123/1000
x=123/999000
将x=123/999000代入2式:
0. ˙123˙=(0.123+ 123/999000)
=123000/999000
=123/999
=41/333
∴0. ˙123˙=41/333
(3)把无限循环小数0.123˙化成分数:
解:
0.123˙=(0.12+0.003˙)——1式
令0.123˙=(0.12+ x)——2式
(2式)*10:
   1.23˙=(1.2+10x)——3式
(3式)-(2式):
1.11=1.08+9x
9x=0.03
x=3/900
x=1/300
将x=1/300代入2式:
0.123˙=(0.12+1/300 )
       =(12 /100+1/300)
       =(36/300+1/300)
=37/300
∴0.123˙=37/300
注:等式两边乘以10 ,100,1000,…,首位数1,后边是0,0的个数要和循环节数字位数相等,例如:无限循环小数0.˙123˙循环节数字是3位数,所以乘以1000,等等,目的是为了消除无限循环小数的无限循环节,便于计算。
您需要登录后才可以回帖 登录 | 注册

本版积分规则

LaTEX预览输入 教程 符号库 加行内标签 加行间标签 
对应的 LaTEX 效果:

Archiver|手机版|小黑屋|数学中国 ( 京ICP备05040119号 )

GMT+8, 2025-7-11 15:36 , Processed in 0.085183 second(s), 15 queries .

Powered by Discuz! X3.4

Copyright © 2001-2020, Tencent Cloud.

快速回复 返回顶部 返回列表
\frac{\square}{\square}\sqrt{\square}\square_{\baguet}^{\baguet}\overarc{\square}\ \dot{\baguet}\left(\square\right)\binom{\square}{\square}\begin{cases}\square\\\square\end{cases}\ \begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}\to\Rightarrow\mapsto\alpha\ \theta\ \pi\times\div\pm\because\angle\ \infty
\frac{\square}{\square}\sqrt{\square}\sqrt[\baguet]{\square}\square_{\baguet}\square^{\baguet}\square_{\baguet}^{\baguet}\sum_{\baguet}^{\baguet}\prod_{\baguet}^{\baguet}\coprod_{\baguet}^{\baguet}\int_{\baguet}^{\baguet}\lim_{\baguet}\lim_{\baguet}^{\baguet}\bigcup_{\baguet}^{\baguet}\bigcap_{\baguet}^{\baguet}\bigwedge_{\baguet}^{\baguet}\bigvee_{\baguet}^{\baguet}
\underline{\square}\overline{\square}\overrightarrow{\square}\overleftarrow{\square}\overleftrightarrow{\square}\underrightarrow{\square}\underleftarrow{\square}\underleftrightarrow{\square}\dot{\baguet}\hat{\baguet}\vec{\baguet}\tilde{\baguet}
\left(\square\right)\left[\square\right]\left\{\square\right\}\left|\square\right|\left\langle\square\right\rangle\left\lVert\square\right\rVert\left\lfloor\square\right\rfloor\left\lceil\square\right\rceil\binom{\square}{\square}\boxed{\square}
\begin{cases}\square\\\square\end{cases}\begin{matrix}\square&\square\\\square&\square\end{matrix}\begin{pmatrix}\square&\square\\\square&\square\end{pmatrix}\begin{bmatrix}\square&\square\\\square&\square\end{bmatrix}\begin{Bmatrix}\square&\square\\\square&\square\end{Bmatrix}\begin{vmatrix}\square&\square\\\square&\square\end{vmatrix}\begin{Vmatrix}\square&\square\\\square&\square\end{Vmatrix}\begin{array}{l|l}\square&\square\\\hline\square&\square\end{array}
\to\gets\leftrightarrow\nearrow\searrow\downarrow\uparrow\updownarrow\swarrow\nwarrow\Leftarrow\Rightarrow\Leftrightarrow\rightharpoonup\rightharpoondown\impliedby\implies\Longleftrightarrow\leftharpoonup\leftharpoondown\longleftarrow\longrightarrow\longleftrightarrow\Uparrow\Downarrow\Updownarrow\hookleftarrow\hookrightarrow\mapsto
\alpha\beta\gamma\Gamma\delta\Delta\epsilon\varepsilon\zeta\eta\theta\Theta\iota\kappa\varkappa\lambda\Lambda\mu\nu\xi\Xi\pi\Pi\varpi\rho\varrho\sigma\Sigma\tau\upsilon\Upsilon\phi\Phi\varphi\chi\psi\Psi\omega\Omega\digamma\vartheta\varsigma\mathbb{C}\mathbb{H}\mathbb{N}\mathbb{P}\mathbb{Q}\mathbb{R}\mathbb{Z}\Re\Im\aleph\partial\nabla
\times\cdot\ast\div\pm\mp\circ\backslash\oplus\ominus\otimes\odot\bullet\varnothing\neq\equiv\not\equiv\sim\approx\simeq\cong\geq\leq\ll\gg\succ\prec\in\ni\cup\cap\subset\supset\not\subset\not\supset\notin\not\ni\subseteq\supseteq\nsubseteq\nsupseteq\sqsubset\sqsupset\sqsubseteq\sqsupseteq\sqcap\sqcup\wedge\vee\neg\forall\exists\nexists\uplus\bigsqcup\bigodot\bigotimes\bigoplus\biguplus\bigcap\bigcup\bigvee\bigwedge
\because\therefore\angle\parallel\perp\top\nparallel\measuredangle\sphericalangle\diamond\diamondsuit\doteq\propto\infty\bowtie\square\smile\frown\bigtriangledown\triangle\triangleleft\triangleright\bigcirc \wr\amalg\models\preceq\mid\nmid\vdash\dashv\nless\ngtr\ldots\cdots\vdots\ddots\surd\ell\flat\sharp\natural\wp\clubsuit\heartsuit\spadesuit\oint\lfloor\rfloor\lceil\rceil\lbrace\rbrace\lbrack\rbrack\vert\hbar\aleph\dagger\ddagger

MathQuill输入:

Latex代码输入: