# decimal place

A decimal place of a number (a real number) $r$ is the position of a digit in its decimal expansion relative to the decimal point. Let us write $r$ as a decimal number:

$\xymatrix@C=0pt@H=10pt{A_{n}A_{n-1}\ldots A_{1}&\bullet&D_{1}D_{2}\ldots D_{m}% \ldots\inner@par&\save*\txt{thedecimalpoint}\restore\ar[u]&}$

where each of the $A_{i}$ and $D_{j}$ is a digit in the decimal system (one of $0,1,2,3,4,5,6,7,8,9$). In this decimal expansion,

• the $i$-th decimal place to the left of the decimal point is the position of $A_{i}$, and

• the $j$-th decimal place to the right of the decimal place is the position of $D_{j}$.

The digits $A_{i}$ and $D_{j}$ are called the decimal place values. $A_{i}$ is the decimal place value corresponding to the $i$-th decimal place to the left of the decimal point, while $D_{j}$ is the decimal place value corresponding to the $j$-th decimal place to the right of the decimal point.

For decimal places closer to the decimal point, specific names are used. Below are some of the most commonly used names:

name position from direction from example
the decimal point the decimal point (position of 7)
ones 1st left $7.2$
tens 2nd left $71(=71.0)$
hundreds 3rd left $735(=735.0)$
thousands 4th left $7126(=7126.0)$
tenths 1st right $0.7$
hundredths 2nd right $12.47$
thousandths 3rd right $9.837$
ten thousandths 4th right $6.0037$

${}\end{center}\inner@par\textbf{Remark}.Insteadofsayingthe$n$-thdecimalplacetotheleftorrightofthedecimalpoint,wesimplysaythe$n$-thdecimalplacetomeanthe$n$-thdecimalplacetothe\emph{right}ofthedecimalpoint,andsay$(-n)$-thdecimalplacetomeanthe$n$-thdecimalplacetothe\emph{left}ofthedecimalpoint.\inner@par Forexample,in$1632.9758$,thedigit$5$islocatedonthethirddecimalplace,while$6${{{islocatedonthenegativethirddecimalplace.\begin{flushright}\begin{tabular}[]{|% ll|}\hline Title&decimal place\\ Canonical name&DecimalPlace\\ Date of creation&2013-03-22 17:27:24\\ Last modified on&2013-03-22 17:27:24\\ Owner&CWoo (3771)\\ Last modified by&CWoo (3771)\\ Numerical id&12\\ Author&CWoo (3771)\\ Entry type&Definition\\ Classification&msc 00A05\\ Related topic&MetricSystem\\ Defines&decimal place value\\ Defines&hundreds\\ Defines&thousands\\ Defines&tenths\\ Defines&hundredths\\ Defines&thousandths\\ Defines&ten thousandths\\ \hline}\end{tabular}}}\end{flushright}\end{document}$