Significant Digits of Approximations

Definition: Let $x_A \in \mathbb{R}$ be an approximate value of $x_T \in \mathbb{R}$. Then $x_A$ is said to have $m$ Significant Digits if the $m$ leading digits of $x_A$ are accurate relative to $x_T$.

For example, suppose that $x_T = 234.2202239$. If we did an experiment and found that $x_A = 234.2102239$ then we would say that $x_A$ has $4$ significant digits because the first four digits of $x_A$ matches that of the first four digits of $x_T$. Note that the fifth digit of $x_A$ is $1$ while the fifth digit of $x_T$ is $2$.

More generally, we say that $x_A$ has $m$ significant digits relative to $x_T$ if:

(1)

\begin{align} \quad \biggr \rvert \frac{x_T - x_A}{x_T} \biggr \rvert ≤ 5 \times 10^{-m-1} \end{align}

For our example above, we said $x_A$ had $4$ significant digits. We can verify this as:

(2)

\begin{align} \quad \biggr \rvert \frac{x_T - x_A}{x_T} \biggr \rvert = \biggr \rvert \frac{234.2202239 - 234.2102239}{234.2202239} \biggr \rvert \approx 4.2694... \cdot 10^{-5} ≤ 5 \cdot 10^{-5} \end{align}

Note that if we suspected that $x_A$ had $5$ significant digits (the incorrect answer), then the inequality given above does not hold as:

(3)

\begin{align} \quad \biggr \rvert \frac{x_T - x_A}{x_T} \biggr \rvert = \biggr \rvert \frac{234.2202239 - 234.2102239}{234.2202239} \biggr \rvert \approx 4.2694... \cdot 10^{-5} ≥ 5 \cdot 10^{-6} \end{align}