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
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): rounding
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): error: 1. (in numerical computation) Rounding (or roundoff) errors
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): rounding