For questions about decimal expansion, both practical and theoretical.

A number can be represented in many different ways, but the most common is via its decimal expansion. Such a representation takes the form

$$a_na_{n-1}\dots a_1a_0.a_{-1}a_{-2}\dots$$

where $a_n \in \{1, 2, \dots, 9\}$ and $a_i \in \{0, 1, 2, \dots, 9\}$ for $i = n - 1, n - 2, \dots, 1, 0, -1, -2, \dots$. In the case that there is $N > 0$ such that $a_i = 0$ for all $i < -N$, these numbers are supressed in which case the decimal expansion usually appears as

$$a_na_{n-1}\dots a_1a_0.a_{-1}a_{-2}\dots a_{-N}.$$

Note that concatenation does not represent multiplication, it is just a part of the notation. The . between $a_0$ and $a_{-1}$ does not represent multiplication either; it is sometimes called the decimal point.

To put the notation on a rigorous footing, the expression $a_na_{n-1}\dots a_1a_0.a_{-1}a_{-2}\dots$ is shorthand for

$$\sum_{i = 0}^na_i10^i + \sum_{i=1}^{\infty}a_{-i}\frac{1}{10^i} = \sum_{i = -\infty}^na_i10^i$$

which can be shown to be convergent irrespective of the choice of $a_i$.

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