Definition:Rounding
Contents
Definition
Rounding is the process of approximation of a value of a variable to a multiple of a given power of whatever number base one is working in (usually decimal).
Let $n \in \Z$ be an integer.
Let $x \in \R$ be a real number.
Let $y \in \R$ such that:
- $y = 10^n \floor {\dfrac x {10^n} + \dfrac 1 2}$
or:
- $y = 10^n \ceiling {\dfrac x {10^n} - \dfrac 1 2}$
where $\floor {\, \cdot \,}$ denotes the floor function and $\ceiling {\, \cdot \,}$ denotes the ceiling function.
Then $y$ is defined as $x$ rounded to the nearest $n$th power of $10$.
Both of these definitions amount to the same thing, except for when $\dfrac x {10^n}$ is exactly halfway between $\floor {\dfrac x {10^n} }$ and $\ceiling {\dfrac x {10^n} }$.
How these instances is treated is known as the treatment of the half.
Rounding to Nearest Integer
When $n = 0$, the operation is referred to as rounding to the nearest integer:
Let $y \in \R$ such that:
- $y = \floor {x + \dfrac 1 2}$
Then $y$ is defined as $x$ rounded to the nearest integer.
Treatment of Half
Consider the situation when $\dfrac x {10^n} + \dfrac 1 2$ is an integer.
That is, $\dfrac x {10^n}$ is exactly midway between the two integers $\dfrac x {10^n} - \dfrac 1 2$ and $\dfrac x {10^n} + \dfrac 1 2$.
Recall that the general philosophy of the process of rounding is to find the closest approximation to $x$ to a given power of $10$.
Thus there are two equally valid such approximations:
- $\dfrac x {10^n} - \dfrac 1 2$ and $\dfrac x {10^n} + \dfrac 1 2$
between which $\dfrac x {10^n}$ is exactly midway.
There are a number of conventions which determine which is to be used.
Also known as
When $n < 0$, the terminology used is usually:
- $x$ rounded to the nearest $m$th decimal place
where $m = -n$.
Examples
$72 \cdotp 8146$ to $2$ Decimal Places
$72 \cdotp 8146$ rounded to $2$ decimal places is $72 \cdotp 81$.
This is because $72 \cdotp 8146$ is closer to $72 \cdotp 81$ than $72 \cdotp 82$.
Sources
- 1972: Murray R. Spiegel and R.W. Boxer: Theory and Problems of Statistics (SI ed.) ... (previous) ... (next): Chapter $1$: Rounding of Data
