Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of GraphSearch.
Concept
Decimal representation
Summary
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 are digits, which are symbols representing integers in the range 0, ..., 9.
Commonly, if The sequence of the —the digits after the dot—is generally infinite. If it is finite, the lacking digits are assumed to be 0. If all are , 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 if ) if and only if one has a trailing infinite sequence of , and the other has a trailing infinite sequence of . For having a one-to-one correspondence between nonnegative real numbers and decimal representations, decimal representations with a trailing infinite sequence of are sometimes excluded.
The natural number , is called the integer part of r, and is denoted by a0 in the remainder of this article. The sequence of the represents the number
which belongs to the interval and is called the fractional part of r (except when all are ).
Any real number can be approximated to any desired degree of accuracy by rational numbers with finite decimal representations.
Assume . Then for every integer there is a finite decimal such that:
Proof:
Let , where .
Then , and the result follows from dividing all sides by .
(The fact that has a finite decimal representation is easily established.)
0.999...
Some real numbers have two infinite decimal representations. For example, the number 1 may be equally represented by 1.000... as by 0.999... (where the infinite sequences of trailing 0's or 9's, respectively, are represented by "..."). Conventionally, the decimal representation without trailing 9's is preferred.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.