# 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.