Loterre: Mathématiques: decimal representation

mathematical analysis > calculus > series > decimal representation

number > real number > Diophantine approximation > decimal representation

Terme préférentiel

decimal representation

Définition(s)

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:
${\\displaystyle r=b_{k}b_{k-1}\\ldots b_{0}.a_{1}a_{2}\\ldots }$
Here . is the decimal separator, $k$ is a nonnegative integer, and ${\\displaystyle b_{0},\\ldots ,b_{k},a_{1},a_{2},\\ldots }$ are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly, ${\\displaystyle b_{k}\ eq 0}$ if ${\\displaystyle k\\geq 1.}$ The sequence of the ${\\displaystyle a_{i}}$ —the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all ${\\displaystyle a_{i}}$ 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:
${\\displaystyle r=\\sum _{i=0}^{k}b_{i}10^{i}+\\sum _{i=1}^{\\infty }{\ rac {a_{i}}{10^{i}}}.}$
Every nonnegative real number has at least one such representation; it has two such representations (with ${\\displaystyle b_{k}\ eq 0}$ if ${\\displaystyle k>0}$ ) 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.

(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Decimal_representation)