Decimal expansion

A decimal expansion is natural number $N$ followed by an infinite sequence $d$ of digits (or rather, a sequence whose values are constrained to the natural numbers less than 10) that is used in representing a non-negative real number. By extension, all real numbers can then be represented by signing decimal expansions with $+$ or $-$, although positive numbers are generally left unsigned.

A decimal expansion is denoted by:

$N.d_1d_2d_3d_4\ldots$

And represents the real number that is the limit to the following:

$N+\lim_{n\rightarrow\infty}\sum_{k=1}^n \frac {d_k}{10^k}$

Due to the infinite nature of decimal expansions, expressions of such are in all practicality limited to those decimal expansions that terminate (see below), or at the very least have a discernible pattern.

A decimal expansion $N.d_1d_2d_3d_4\dots$ is said to terminate if there is a positive integer $K$ such that $d_k=0$ for all $k>K$. In this case, the decimal representation can be represented finitely as $N.d_1d_2d_3d_4\ldots d_{K-1}d_K$.

A decimal expansion $N.d_1d_2d_3d_4\dots$ is eventually repeating if there are positive integers $K$ and $r$ such that $d_{k+r}=d_k$ for all $k \ge K$. In this case, the following representations will suffice:

• $N.d_1d_2d_3d_4\ldots d_{K-1}\overline{d_Kd_{K+1}d_{K+2}\ldots d_{K+r-1}}$
• $N.d_1d_2d_3d_4\ldots d_{K-1}\left(d_Kd_{K+1}d_{K+2}\ldots d_{K+r-1}\right)$

A case for using decimal expansions can be made by observing the following two facts:

• Any decimal expansion $N.d_1d_2d_3d_4\ldots$ represents a unique real number (that is, the limit always exists).
• Any nonnegative real number can be represented by some decimal expansion (and thus, any real number can be represented by some signed decimal expansion).

However, while it is true that "almost all" nonnegative real numbers possess unique decimal expansions, it is not absolutely true for all real numbers. For an example of this, consider that $0.\overline 9 = 0.999\ldots = 1$.