1.1 Accuracy

Measurements

Two basic measurements:

1. Absolute Error: $|\tilde{p}-p|$ where $\tilde{p}$ is the approximation of $p$
2. Relative Error: $\frac{|\tilde{p}-p|}{|p|}$,weighing the absolute error against the magnitude of the number being approximated.

Significant Digits

We say that $\tilde{p}$ is accurate to $n$ significant digits if the leading $n$ significant digits of $\tilde{p}$ match those of $p$.

Sources

1. Algorithmic: due to the method itself $|\tilde{p}-p|$
2. Floating-point (round-off): since computers save a finite amount of the decimal expansion. Small errors compound. If two number share significant digits, if you subtract them, the next value will have 2 less significant digits. $|{\tilde{p}}^{\ast }-\tilde{p}|$