Recently, I am learning about numerical methods and I found this question in the textbook to find absolute, relative, and percentage errors if x is rounded-off to three decimal digits. Given $$x = 0.005998$$ In the solution they have rounded off x to 0.006 but this is rounding off to three decimal places.
Is decimal digit and decimal places equivalent in case of fractional numbers? Please correct where I am making the mistake. I am thinking to round off the number to three decimal digits as 0.00. Please tell am I correct to round off as 0.00 (this is round off to three decimal digits not three decimal places) or not?
Textbook solution:
Number rounded-off to three decimal digits =.006
Error = .005998 – .006 = – .000002
Absolute error $E_a$ = | error | = .000002
Relative error $E_r$ = .0033344
Percentage error $E_p$ = $E_r × 100 = .33344%$
Also, in the textbook, they haven't followed the rules for add/sub/multiplication/division of significant figures. Isn't it necessary to follow the significant rules in final answer to be calculated?
For a given floating-point number (a simplified definition of this is a number with a decimal point included), the terms decimal places and decimal digits are used interchangeably.
This is because a decimal digit literally refers to ta digit following the decimal point. So, for the number you wanted to round off, $x = 0.005998$, the correct answer would be: $$ x(round-off \ to \ 3 \ decimal \ places) = 0.006 $$
The key point is identifying that decimal places and decimal digits are the same. When asked to round off to a certain decimal place, you must have that many digits following the decimal point.
As for your query on the rules of mathematical operations based on the significant digits, it is important to understand that the final digit in a number is unknown. Meaning that the right-most digit always is uncertain and has the possibility of being different. It appears that the purpose of this exercise is to introduce you to how rounding off can create errors. In this case, you would consider the following: $$ \begin{align} x & = 0.005998 \\ x(rounded) & = 0.006000 \\ Error (absolute) & = 0.000002 \end{align} $$
You need to fill in the decimal digits to the right of the rounded-off representation with zeros in order to perform some mathematical operation. You are absolutely correct in pointing out that the rules of significance are not followed and that the answers should have also been rounded off to the 3rd decimal place. However, it is assumed in most cases and not explicitly mentioned, which is what appears to have been assumed in this question.