無限小数

I prove that infinite decimal of rational number is recurring decimal. It is decimal of rational number but is not finite decimal.

While the process of division may seem intuitive, I will provide a more concrete explanation as a formal proof.

Feeling

For example, when 89 is sequentially divided by 13, including decimal places, the remainders are as follows:

\begin{array}{rcl} 89 \div 13 & = & 6 remainder is 11 \\ 110 \div 13 & = & 8 remainder is 6 \\ 60 \div 13 & = & 4 remainder is 8 \\ 80 \div 13 & = & 6 remainder is 2 \\ 20 \div 13 & = & 1 remainder is 7 \\ 70 \div 13 & = & 6 remainder is 5 \\ 50 \div 13 & = & 3 remainder is 11 \\ 110 \div 13 & = & 8 remainder is 6 \end{array}

$\begin{eqnarray*} 89 \div 13 & = & 6 \; \textrm{remainder is} 11 \\ 110 \div 13 & = & 8 \; \textrm{remainder is} 6 \\ 60 \div 13 & = & 4 \; \textrm{remainder is} 8 \\ 80 \div 13 & = & 6 \; \textrm{remainder is} 2 \\ 20 \div 13 & = & 1 \; \textrm{remainder is} 7 \\ 70 \div 13 & = & 6 \; \textrm{remainder is} 5 \\ 50 \div 13 & = & 3 \; \textrm{remainder is} 11 \\ 110 \div 13 & = & 8 \; \textrm{remainder is} 6 \end{eqnarray*}$

On the 7th division, the same remainder as in the 1st division appeared again. Since there are only 12 possible remainers when divided by 13, ranging from 1 to 12, if the division doesn’t result in a perfect quotient after 13 divisions, it becomes evident that at some point, the same remainder would recur, leading to a repeating decimal.

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