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 a rounding error to be $X - x$.
Other sources define a rounding error to be $x - X$.
Using such conventions, a rounding error can be either positive or negative.
Using the convention $\size {x - X}$, a rounding error can only be positive.
Also known as
A rounding error is also known as a roundoff error.
Examples
Example: $73.854 \, \mathrm {mm}$
Let $x$ be a length expressed as $73.854 \, \mathrm {mm}$ to $5$ significant figures.
The maximum rounding error in $x$ is $0.0005 \, \mathrm {mm}$.
Example: $0.09800 \ \mathrm {m^3}$
Rounding Error/Examples/0.09800 m^3
Example: $3.687 \times 10^8 \, \mathrm {km}$
Rounding Error/Examples/3.687 x 10^8 km
Also see
- Results about rounding errors can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors