decimal expansion

A decimal representation of a non-negative real number r is its expression as a sequence of symbols consisting of decimal digits traditionally written with a single separator:

Here . is the decimal separator, k is a nonnegative integer, and




b

0


,
…
,

b

k


,

a

1


,

a

2


,
…


{\displaystyle b_{0},\ldots ,b_{k},a_{1},a_{2},\ldots }
are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly,




b

k


≠
0


{\displaystyle b_{k}\neq 0}
if



k
>
1.


{\displaystyle k>1.}
The sequence of the




a

i




{\displaystyle a_{i}}
—the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all




a

i




{\displaystyle a_{i}}
are 0, the separator is also omitted, resulting in a finite sequence of digits, which represents a natural number.
The decimal representation represents the infinite sum:

Every nonnegative real number has at least one such representation; it has two such representations (with




b

k


≠
0


{\displaystyle b_{k}\neq 0}
if



k
>
0


{\displaystyle k>0}
) if and only if one has a trailing infinite sequence of 0, and the other has a trailing infinite sequence of 9. For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of 9 are sometimes excluded.

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