00A05-DecimalPlace.tex

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\pmtitle{decimal place}
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\pmdefines{ten thousandths}

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A \emph{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:
\begin{center}
$\xymatrix @C=0pt @H=10pt {A_nA_{n-1}\ldots A_1 & \bullet & D_1D_2\ldots D_m \ldots \\
& \save *\txt{the decimal point} \restore \ar[u] & }$
\end{center}
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,
\begin{itemize}
\item the $i$-th decimal place to the left of the decimal point is the position of $A_i$, and 
\item the $j$-th decimal place to the right of the decimal place is the position of $D_j$.  
\end{itemize}
The digits $A_i$ and $D_j$ are called the \emph{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:

\begin{center}
\begin{tabular}{|c||r|r|r|}
\hline name & position from & direction from & example \\
 & the decimal point & the decimal point & (position of 7) \\
\hline \hline ones & 1st & left & $7.2$ \\
\hline tens & 2nd & left & $71(=71.0)$ \\
\hline hundreds & 3rd & left & $735(=735.0)$ \\
\hline thousands & 4th & left & $7126(=7126.0)$ \\
\hline tenths & 1st & right & $0.7$ \\
\hline hundredths & 2nd & right & $12.47$ \\
\hline thousandths & 3rd & right & $9.837$ \\
\hline ten thousandths & 4th & right & $6.0037$ \\
\hline
\end{tabular}
\end{center}

\textbf{Remark}.  Instead of saying the $n$-th decimal place to the left or right of the decimal point, we simply say the $n$-th decimal place to mean the $n$-th decimal place to the \emph{right} of the decimal point, and say $(-n)$-th decimal place to mean the $n$-th decimal place to the \emph{left} of the decimal point.  

For example, in $1632.9758$, the digit $5$ is located on the third decimal place, while $6$ is located on the negative third decimal place.
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