Decimal representation

A decimal representation of a

non-negative In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...

real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...

is its expression as a

sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called ...

of symbols consisting of decimal digits traditionally written with a single separator:

r = b_k b_\ldots b_0.a_1a_2\ldots

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

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\neq 0

k > 1.

The sequence of the

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

are , the separator is also omitted, resulting in a finite sequence of digits, which represents a

natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...

. The decimal representation represents the infinite sum:

r=\sum_^k b_i 10^i + \sum_^\infty \frac.

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

b_k\neq 0

k>0

)

if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

one has a trailing infinite sequence of , and the other has a trailing infinite sequence of . For having