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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?

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  • $\begingroup$ For some basic information about writing mathematics at this site see, e.g., here, here, here and here. $\endgroup$ Jun 20, 2022 at 19:02
  • $\begingroup$ @DavidK is it necessary to follow rules of significant figures in numerical computations as I have seen many question in the textbook that clearly ignores the rules for add/sub/multiplication/division of significant figures? or we can take some liberties while calculating the numerical answer ignoring significant figures? $\endgroup$ Jun 20, 2022 at 19:59
  • $\begingroup$ If the original number is not exact but is itself known only to four significant digits, the relative error should be $0.003$ (one significant digit, the same as for the absolute error) if we are keeping track of significant digits. It is mere coincidence that this is three decimal digits; the percent error is $0.3\%.$ Since I do not have the same textbook I do not know what its overall level of quality is. $\endgroup$
    – David K
    Jun 20, 2022 at 20:14

2 Answers 2

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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.

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  • $\begingroup$ So, let's take number as 8.00598, after rounding off it to three decimal digits, will it be 8.006? $\endgroup$ Jun 20, 2022 at 19:15
  • $\begingroup$ I disagree strongly that the errors should also have been rounded to three places after the decimal. In the case of the absolute error, that would result in an answer of zero for every problem, making the term "absolute error" meaningless. While the other error measurements would not also go to zero, we are asked for measurements based on the original knowledge of the number to more than three decimal places, and we should use that knowledge. $\endgroup$
    – David K
    Jun 20, 2022 at 19:31
  • $\begingroup$ @DavidK But that's the whole point of the error or meaning of uncertainty! The right-most digit is always uncertain with a varying probability of being incorrect. So in this case, while the intent is to have the user compute error to the 6th decimal place, OP is correct in assuming that it should be rounded off to 3 places. So, the absolute error of 0.000 indicates that the final digit is uncertain and may not necessarily be zero. I have pointed out here how the question seems to have assumed what you're trying to say, but the OP is actually correct in saying the rules haven't been followed. $\endgroup$
    – Ajay Menon
    Jun 20, 2022 at 19:43
  • $\begingroup$ @HarshitSingh That's correct, if you have to round off 8.00598 to 3 decimal places, it becomes 8.006. If you found this answer helpful, kindly upvote it so that I know it's been of use. Thanks! $\endgroup$
    – Ajay Menon
    Jun 20, 2022 at 19:44
  • $\begingroup$ We will just have to disagree then. I have never seen your interpretation from any reliable source. $\endgroup$
    – David K
    Jun 20, 2022 at 19:51
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X=0.005998 upto three decimal point=0.006

  • Absolute error=0.005998-0.006
  • Relatable error =(0.005998-0.006)/0.005998
  • Percentage error= {(0.005998-0.006)/0.005998}.100th% Kindly calculate all those calculation by ownself
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