Let $0, a_1a_2a_3\dots$ the decimal expansion of a number. Can you suggest such a way that when the order of the digits is changed the new number is rational ?
Solution
The digits $a_1a_2a_3\dots$ enherit their values from the set $\mathcal{A}=\{0, 1,2, \dots, 9\}$. It is obvious that these, in the expansion of the number, are repeated either infinite times or finite times. In the first places of the expansion we place the numbers, if those exist, that are repeated finite times. Now, if $b_1, b_2, \dots, b_n$ such that $b_1<b_2<\cdots<b_n$ are the infinite repeated digits then we place them as follows:
$$b_1b_2\dots b_n b_1b_2\dots b_n b_1b_2\dots b_n$$
The number that arises is periodical with a period of $b_1b_2\dotsb_n$ hence rational.
Solution
The digits $a_1a_2a_3\dots$ enherit their values from the set $\mathcal{A}=\{0, 1,2, \dots, 9\}$. It is obvious that these, in the expansion of the number, are repeated either infinite times or finite times. In the first places of the expansion we place the numbers, if those exist, that are repeated finite times. Now, if $b_1, b_2, \dots, b_n$ such that $b_1<b_2<\cdots<b_n$ are the infinite repeated digits then we place them as follows:
$$b_1b_2\dots b_n b_1b_2\dots b_n b_1b_2\dots b_n$$
The number that arises is periodical with a period of $b_1b_2\dotsb_n$ hence rational.
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