# 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:

- $y = 10^n \left\lfloor{\dfrac x {10^n} + \dfrac 1 2}\right\rfloor$

Then $y$ is defined as **$x$ rounded to the nearest $n$th power of $10$**.

### Rounding to Nearest Integer

When $n = 0$, the operation is referred to as **rounding to the nearest integer**:

Let $y \in \R$ such that:

- $y = \left\lfloor{x + \dfrac 1 2}\right\rfloor$

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.

The convention on $\mathsf{Pr} \infty \mathsf{fWiki}$ is that the greater of the two is used:

- $y = 10^n \left\lfloor{\dfrac x {10^n} + \dfrac 1 2}\right\rfloor$

but other systems exist.

## Also known as

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

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

where $m = -n$.

## Sources

- 1961: Murray R. Spiegel:
*Theory and Problems of Statistics*... (previous) ... (next): Chapter $1$: Rounding of Data