# Thread: Approximate numbers and rounding

1. ## Approximate numbers and rounding

I hope this is the proper spot for this! I did a little bit of searching and couldn't quite find what I was looking for, so here goes (it's more of a procedure question than a specific problem):

The chapter I'm on in my text covers calculators and approximate numbers. The rules given for operations with approximate numbers are:

1) When approximate numbers are added/subtracted the result is expressed with the precision of the least precise number (precision being the number of decimal places, if I'm reading the text correctly).

2) When approximate numbers are multiplied/divided the result is expressed with the accuracy of the least accurate number (accuracy being the total number of significant digits).

3) When the root of an approximate number is found the result is expressed with the accuracy of the number.

I'm confused about the procedure when these operations are combined, however. The text only says "where there is a combination of operations, the final operation determines how the final result is to be rounded off" with the example: $\displaystyle 38.3 - 12.9(-3.58) = 84.482 = 84.5$ (which unfortunately isn't very comprehensive). In this example I'm lead to believe that the final answer is rounded to tenths because 38.3 only has a single decimal place of precision (the final operation being subtraction), but this assumption doesn't hold true for later questions.

$\displaystyle -(-26.5)^2 - (-9.85)^3$ for example has a book answer of 253, which seems to indicate that the rounding that would take place on the multiplication before the subtraction has an effect on the final rounding. I don't think I can round at each step, though, as that would introduce error into the problem, much like chain rounding? Plus given that this is all in the context of using a calculator, it doesn't make much sense to stop at the end of each operation to round. Do I just kind of earmark the least number of significant digits if there's muliplication, then apply that to the final result? And vice-versa with the precision if the final operation is multiplication/division? I guess what I really need is a more in-depth explanation of the rounding procedures used with the mixed operations in a problem like this

2. Originally Posted by redshirt
$\displaystyle -(-26.5)^2 - (-9.85)^3$ for example has a book answer of 253, which seems to indicate that the rounding that would take place on the multiplication before the subtraction has an effect on the final rounding.......
Rounding ALWAYS takes place after the full operation is performed.
-(-26.5)^2 - (-9.85)^3 = 253.421625

to nearest whole number: 253
to 1 decimal place: 253.4
to 2 decimal places: 253.42
to 3 decimal places: 253.422
to 4 decimal places: 253.4216
to 5 decimal places: 253.42163
to 6 decimal places: 253.421625

3. Originally Posted by Wilmer
Rounding ALWAYS takes place after the full operation is performed.
-(-26.5)^2 - (-9.85)^3 = 253.421625

to nearest whole number: 253
to 1 decimal place: 253.4
to 2 decimal places: 253.42
to 3 decimal places: 253.422
to 4 decimal places: 253.4216
to 5 decimal places: 253.42163
to 6 decimal places: 253.421625
Do I still need to keep track of what rounding would have been done to the individual operations that make up the whole, though? I'm still looking for the process to use to figure out what rounding to apply to the final answer for these type of problems in general. In that example's case, for instance, the final result's been rounded to 3 significant digits. The only way I can see getting that is by following the multiplication rule - but the final operation's subtraction

4. 'Scuse the bump, I just noticed that this was buried in the move from the basic forum

5. Originally Posted by redshirt
Do I still need to keep track of what rounding would have been done to the individual operations that make up the whole, ....
No, non, nyet, ...

6. Originally Posted by Wilmer
No, non, nyet, ...
I do understand that I'm not supposed to actually perform these intermediary operations as that would introduce rounding errors, but what I don't understand is how to figure out what rounding to do on a complex problem based on the rules listed in the first post. Back to the example again, the final answer is supposed to be rounded to 253 - 3 significant digits, no decimal points. As I see it, if I followed the "last operation determines the rounding" line from the book to the letter, I'd have an answer of 253.4, one decimal point from the final operation being subtraction and the least precise number being -26.5. I just need a better explanation of how to carry those rounding rules through problems that have multiple sets of operations.

Is it that this kind of thing isn't standard at all? Searching around I've found a few pages and documents that list the same rules for precision and accuracy, but nothing that's gone into any detail at all about how they apply to problems with combined operations. "The final operation determines how the final result is to be rounded off" is the only explanation I've found. Frustrating, as there are instances of rounding throughout the entire text that seem to depend on these rules.

(Sorry about the slow reply, by the way!)