# Definition:Error

## Definition

Let $x$ be an approximation to a (true) value $X$.

The error $\varepsilon$ is a measure of how much difference there is between $x$ and $X$.

### Relative Error

The relative error in $x$ is defined as:

$\dfrac {\size {X - x} } X$

where $\size {X - x}$ denotes the absolute value of $X - x$.

### Absolute Error

The absolute error $\varepsilon$ is the difference between $x$ and a $X$, and can be defined in one of three ways:

 $\text {(1)}: \quad$ $\ds \varepsilon$ $:=$ $\ds X - x$ $\text {(2)}: \quad$ $\ds \varepsilon$ $:=$ $\ds x - X$ $\text {(3)}: \quad$ $\ds \varepsilon$ $:=$ $\ds \size {X - x}$

where $\size {X - x}$ denotes the absolute value of $X - x$.

Different sources use different conventions.

## Also known as

An error is also, in some branches of mathematics, known as a residual.

## Also see

Not to be confused with a mistake.