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.