Definition:Rounding/Error

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

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

Let $X \in \Q$ be equal to $x$ rounded to the nearest $n$th power of $10$.

The rounding error caused by the rounding of $x$ to $X$ is defined as:
 * $e_R = \size {x - X}$

Also defined as
Some sources define the rounding error to be $X - x$.

Other sources define the rounding error to be $x - X$.

Using such conventions, the rounding error can be either positive or negative.

Using the convention $\size {x - X}$, the rounding error can only be positive.