Ill-Conditioned Systems and the Condition Number

A system $Ax=b$ is ill-conditioned when a tiny change in the data ($b$ or $A$) produces a huge change in the solution $x$. The condition number $\kappa(A)=\lVert A\rVert\,\lVert A^{-1}\rVert\ge 1$ measures this sensitivity: it is the worst-case amplification factor from relative input error to relative output error.