Definition:Rounding/Treatment of Half

Definition
Let $n \in \Z$ be an integer.

Let $x \in \R$ be a real number.

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 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.