Decimal Representation

A decimal representation of a non-negative real number r is an expression of the form of a series, traditionally written as a sum

where a0 is a nonnegative integer, and a1, a2, … are integers satisfying 0 ≤ ai ≤ 9, called the digits of the decimal representation. The sequence of digits specified may be finite, in which case any further digits ai are assumed to be 0. Some authors forbid decimal representations with a trailing infinite sequence of 9 digits. This restriction still allows a decimal representation for each non-negative real number, but additionally makes such a representation unique. The number defined by a decimal representation is often written more briefly as

That is to say, a0 is the integer part of r, not necessarily between 0 and 9, and a1, a2, a3, … are the digits forming the fractional part of r.

Both notations above are, by definition, the following limit of a sequence:

.

Read more about Decimal Representation:  Finite Decimal Approximations, Non-uniqueness of Decimal Representation, Finite Decimal Representations, Recurring Decimal Representations

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