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.
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For example, when 89 is sequentially divided by 13, including decimal places, the remainders are as follows:
89÷13=6remainder is11110÷13=8remainder is660÷13=4remainder is880÷13=6remainder is220÷13=1remainder is770÷13=6remainder is550÷13=3remainder is11110÷13=8remainder is6On 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.