Answer
There are 4 significant digits.
Work Step by Step
To find the number of significant digits in the measurement $0.001005~m$, we follow the rules on page 162 of the book.
We use rule #1: all non-zero digits are significant. Thus "1,5" are significant.
We also use rule #2: all zeros between significant digits are significant; that is, we have here 2 zeros which are significant.
Rule #6 tells us to ignore zeros to the left of the $1$.
Thus, there are 4 significant digits.
