Definition:Rounding

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:

Also known as
When $n < 0$, the terminology used is usually:


 * $x$ rounded to the nearest $m$th decimal place

where $m = -n$.