|p-p*|/|p|<=5*10^(-t)
said by numerical analysis textbook burden
then
how we define significant digits? do we have to choose any integer t?
example of this text book
p=0.54617, p*=0.5462 how we get significant digits by definition?
this text book say four significant digits but I don't know why
You take the largest $t$ that satisfies the requirement. Just plug into the definition $$\frac {|p^*-p|}p=\frac {|0.54617-0.5462|}{0.54617}=\frac{0.00003}{0.54617}\approx 0.000055$$ This is greater than $0.5\cdot 10^{-4}$, so it should only be $3$ significant digits. Most people will just count the number of accurately rounded digits and say this has four, but that is not the definition you have quoted.