In my textbooks, significant figures are defined as:
“Significant figures by definition are the reliable digits in a number that are known with certainty.”
“A significant figure is the one which is known to be reasonably reliable.”
Reliable means giving the same result on successive trials or reliable information can be trusted to be accurate.
However, at the end of each of the two definitions it is also written that ‘the last digit of a number is generally considered uncertain in the absence of qualifying information.'
For example if the mass of an object is 12.248 gm, the last digit which is ‘8’ is uncertain by plus or minus 0.001 gm. This uncertainty is unreliability in the information. So, the digit ‘8’ is uncertain, and thus according to the definition 1 it is not a significant figure. However, the rules for significant figures say that ‘non-zero digits are all significant’. Due to the last digit ‘8’, if the object was weighed with careful handling minimizing the chances of error, the figures preceding ‘8’ that are (1, 2, 2, 4) are certain and reliable. If the mass of another object is 2 gm, then it is uncertain by plus or minus 1 gm i.e. its mass could be 3 gm or could be 1 gm. ‘2’ is therefore not certain!
What is the reason that significant figures are defined to be the reliable digits in a number? In what sense they are said to be reliable?