The number $p$ is said to approximate $p*$ to $t$ significant digits if $t$ is the largest nonnegative integer for which $\frac{|p-p*|}{|p|}\le 5×10^{-t}$. Returning to the machine representation $fl(y)$ for the number $y$ has the relative error $|\frac{y-fl(y)}{y}|$.
If $k$ decimal digits and chopping are used for the machine representation of $y=0.d_1d_2...d_kd_{k+1}...×10^n$,
then
$|\frac{y-fl(y)}{y}|=|\frac{0.d_{k+1}d_{k+2}...}{0.d_1d_2...}|×10^{-k}$.
Since $d\neq0$, the minimal value of denominator is $0.1$. The numerator is bounded above by $1$. As a consequence $|\frac{y-fl(y)}{y…