Decimal Representation in Series

Consider the series

 a + \frac{{{a_1}}}{{10}}  + \frac{{{a_2}}}{{{{10}^2}}} +  \cdots +  \frac{{{a_n}}}{{{{10}^n}}} + \cdots  ,\,\,\, - - - (*)

Where a \in \mathbb{Z} and {a_1},{a_2}, \ldots ,{a_n}, \ldots are integers lying between 0 and 9.

From monotonically non-decreasing bounded partial sums, if follows that (*) is convergent. The series (*) is often abbreviated as a.{a_1}{a_2} \ldots  {a_n} \ldots (called decimal representation) and is used for the real number which is the sum of the series (*). Thus every decimal representation represents a real number.

On the other hand, every real number has a unique decimal representation. Consider any x \in \mathbb{R}. Then \exists {\text{ }}a \in \mathbb{Z}, such that a + 1 > x \geqslant a.

Next we have integers {a_1},{a_2},  \ldots ,{a_n}, \ldots lying between 0 and 9 such that

{a_1} + 1 > \left( {x -  a} \right)10 \geqslant {a_1},
{a_2} + 1 > \left( {x -  a} \right){10^2} - {a_1}10 \geqslant {a_2},
…………………………………….
…………………………………….
{a_n} + 1 > \left( {x -  a} \right){10^n} - {a_1}{10^{n - 1}} -  \cdots - {a_{n - 1}}10 \geqslant  {a_n}

Hence, a.{a_1}{a_2} \cdots  \left( {{a_n} + 1} \right) > x \geqslant a \cdot {a_1}{a_2} \cdots  {a_n}{\text{ }}\forall n.

Since a.{a_1}{a_2} \cdots \left( {{a_n} + 1} \right) - a  \cdot {a_1}{a_2} \cdots {a_n} = \frac{1}{{{{10}^n}}} and\left\langle {a.{a_1}{a_2} \ldots {a_n}}  \right\rangle converges, therefore, x  = \lim a.{a_1}{a_2} \ldots {a_n}. i.e. x = a.{a_1}{a_2} \ldots {a_n} \ldots .

It is simple to see that the decimal representation of every rational number is either terminating or recurring. Every non-recurring infinite decimal represents an irrational number.

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