Numerical computation usually replaces an exact mathematical quantity $x$ by an approximation $\tilde x$.

The absolute error is

$$|\tilde x-x|,$$

while the relative error measures the error compared with the scale of the exact value:

$$\frac{|\tilde x-x|}{|x|}$$

when $x\ne0$.

Two major error sources are different in origin. Roundoff error comes from finite-precision arithmetic. Truncation or discretization error comes from replacing an exact mathematical operation by an approximation, such as a finite step or finite polynomial.

An error estimate is meaningful only relative to the quantity being computed and its scale. A small absolute error can be large in relative terms near zero, while a large absolute error can be negligible for a very large quantity.

Numerical analysis studies not only how to obtain approximations, but how their errors arise, propagate and decrease as the method is refined.