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:

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:

$\text{𝐑𝐞𝐦𝐚𝐫𝐤}.Insteadofsayingthe$n$-thdecimalplacetotheleftorrightofthedecimalpoint,wesimplysaythe$n$-thdecimalplacetomeanthe$n$-thdecimalplacetothe\text{<em class="ltx\_emph ltx\_font\_italic">right</em>}ofthedecimalpoint,andsay$(-n)$-thdecimalplacetomeanthe$n$-thdecimalplacetothe\text{<em class="ltx\_emph ltx\_font\_italic">left</em>}ofthedecimalpoint.Forexample,in$1632.9758$,thedigit$5$islocatedonthethirddecimalplace,while$6