Rounding 2007-08-08 16:57:59 2008-06-09 10:39:57 Definition rounding up rounding down symmetric arithmetic rounding rounding error truncation rounded to banker's rounding \usepackage{amssymb,amscd} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{mathrsfs} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions \usepackage{amsthm} % making logically defined graphics \usepackage{xypic} \usepackage{pst-plot} \usepackage{psfrag} % define commands here \newtheorem{prop}{Proposition} \newtheorem{thm}{Theorem} \newtheorem{ex}{Example} \newcommand{\real}{\mathbb{R}} \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} \emph{Rounding} is a general technique for approximating a real number by a decimal fraction. There are several ways of rounding a real number, five of which are the most common: \emph{rounding up}, \emph{rounding down}, \emph{truncation}, \emph{ordinary rounding} (or \emph{rounding} for short), and \emph{banker's rounding}. \subsubsection*{Rounding to an Integer} The simplest kind of rounding is that of rounding a real number to an integer. Let $r$ be a real number. Then \begin{description} \item[rounding up:] rounding up of $r$ is taking the smallest integer that is greater than or equal to $r$. This integer is denoted by the ceiling function $$\lceil r \rceil:=\min\lbrace n\in \mathbb{Z}\mid n\ge r \rbrace.$$ Examples: $\lceil 2.1 \rceil = 3$, and $\lceil 62.672 \rceil=63$. \item[rounding down:] rounding down of $r$ is taking the largest integer that is less than or equal to $r$. This integer is denoted by the floor function \begin{displaymath} \lfloor r \rfloor:=\max\lbrace n\in \mathbb{Z}\mid n\le r\rbrace = \left\{ \begin{array}{ll} \lceil r \rceil & \textrm{if $r$ is an integer}\\ \lceil r\rceil - 1 & \textrm{otherwise.} \end{array} \right. \end{displaymath} Examples: $\lfloor 1.24 \rfloor = 1$, and $\lfloor -2.63 \rfloor = -3$. \item[truncation:] rounding by truncation is done by ignoring all decimals to the right of the decimal point, which is equivalent to taking only the integer part of $r$. The truncation of $r$ is denoted by $[r]$. In terms of rounding up and rounding down: we have \begin{displaymath} [r] = \left\{ \begin{array}{ll} \lfloor r \rfloor & \textrm{if $r\ge 0$}\\ \lceil r \rceil & \textrm{if $r< 0.$} \end{array} \right. \end{displaymath} If we write $r$ as a decimal number using decimal expansion, then $[r]$ is the integral portion of $r$. Examples: $[2.354]=2$, and $[-81.67]=-81$. \item[ordinary rounding:] this is the most commonly used of the rounding methods described so far. (Ordinary) rounding of $r$ is finding the closest integer to $r$, and if $r$ is exactly half way between two integers, use the larger of the two as the result. Let $R(r)$ represents the ordinary rounding of $r$. It is easy to see that $$R(r)=\lfloor r+0.5 \rfloor.$$ Examples: $R(-3.37)=-3$, while $R(7.5)=8$. There is an easy algorithm of rounding $r$ to the nearest integer. \begin{enumerate} \item write $r$ as a decimal number using decimal expansion \item if the tenths decimal place value is less than $5$, then $R(r)=[r]$ \item if the tenths decimal place value is at least $5$, then $R(r)=[r]+1$. \end{enumerate} \item[banker's rounding:] a variant of the ordinary rounding is the banker's rounding: if $r$ is exactly half way between two integers, and the integer portion of $r$ is even, round down $r$. Otherwise, use ordinary rounding on $r$. If $B(r)$ denotes the banker's rounding of $r$, then it can be defined as \begin{displaymath} B(r) = \left\{ \begin{array}{ll} \lfloor r \rfloor & \textrm{if $[r]$ is even, and $2r\in \mathbb{Z}$}\\ R(r) & \textrm{otherwise.} \end{array} \right. \end{displaymath} For example, $B(3.5)=4$, while $B(2.5)=2$. \item[stochastic rounding:] this rounding method requires the aid of a random number generator. Rounding of $r$ may be done using any of the above methods when $r$ is not exactly half way between two consecutive integers. Otherwise, $r$ is randomly rounded up or down based on the outcome of randomly selecting a number between 0 and 1 using a random number generator. The choice of rounding up (and thus down) depends on how numbers are in $[0,1]$ are allocated for rounding up (or down). \item[alternate rounding:] this rounding method, like the last one, uses other available methods except when the number in question $r$ is exactly half way between two consecutive integers. However, this method is used in a situation where a sequence of numbers needs to be rounded: \begin{enumerate} \item the first number in the sequence is rounded using any of the above methods; \item when the $n$-th number is rounded, the $(n+1)$-th number is rounded as follows: if the number is exactly half way between two consecutive integers, then it is rounded down if the $n$-th number is rounded up, and vice versa. Otherwise, use the rounding method used to round the first number in the sequence. \end{enumerate} \end{description} \subsubsection*{Rounding to a Decimal Fraction} More generally, the three methods described can be applied to rounding of $r$ to a decimal fraction. The general procedure is as follows: \begin{enumerate} \item First, specify how accurately we want to round $r$. This can be accomplished by specifying to what decimal place we want to approximate $r$. Let this place be $n$ (note that $n>0$ if it is to the right of the decimal point and $n<0$ otherwise). \item Write $r$ as a decimal number using decimal expansion. \item Multiply $r$ by $10^n$. By doing this, we are basically moving the decimal point so it is positioned between the $n$-th decimal place and the $(n+1)$-th decimal place. \item Use any of the four methods above to round $10^nr$. \item Divide the rounded number by $10^n$ to get the result. \end{enumerate} In practice, steps $3$ through $5$ can be combined into one step, simply by performing the rounding operation at the specified decimal place as if it were the ones place. For example, rounding $\pi=3.14159...$ to the nearest thousandths place is $3.142$, the thousandths place value $1$ is increased to $2$ because the ten thousandths place is $5$. \textbf{Remark}. In general, rounding to the $n$-th decimal place can be thought of as a function $f$ from $\mathbb{R}$ to $D$, the set of all decimal fractions, such that \begin{itemize} \item $|f(r)-r|\le 10^n$, and \item $f(r)=r$ if $10^nr\in \mathbb{Z}$. \end{itemize} If $g:\mathbb{R}\to \mathbb{Z}$ denotes any of the four rounding methods described in the previous section, and $g_n$ corresponds to rounding to the $n$-th decimal place using method $g$ in step $4$ above, then the entire rounding process can be summarized by a single formula: $$g_n(r)=\frac{g(10^n r)}{10^n}.$$