# Significant figures

## Main Question or Discussion Point

I am trying to convert a mole quantity into a mass. $m_{CO_2}$ will represent mass, $M_{CO_2}$ will represent molar mass and $n_{CO_2}$ will represent mole quantity.
I have $n_{CO_2} = 3.3$ and $M_{CO_2} = (12.01 + 2(16.00))$
So, $m_{CO_2} = 3.3(12.01 + 2(16.00))$
If I compute the value of $M_{CO_2}$ first,
$m_{CO_2} = 3.3(44.01)$ I don't have to round yet, since I am still only using significant figures.
I then complete my multiplication and due to 3.3 only having 2 significant figures, I get $m_{CO_2} = 1.5 * 10^2$.

If I distribute in the following way:
$m_{CO_2} = 3.3(12.01) + 3.3(2)(16.00)$
I have to make sure each term only have 2 significant figures
$m_{CO_2} = 40 + 110 = 1.5 * 10^2$
My textbook suggest that I reduce rounding errors by grouping similar operations. Is one way better than the other?

DrClaude
Mentor
Personally, I don't think you should round anything until the final result is obtained. In that case, the order of operations is irrelevant.

Borek
Mentor
My textbook suggest that I reduce rounding errors by grouping similar operations. Is one way better than the other?
In ancient times, when multiplication was done on paper, using less digits and tricks that allowed to maintain accuracy with using less digits were valuable as they could be use to speed up calculations. As of today they don't matter.

That being said, in numerical methods sometimes it is important to know what you are doing to not loose accuracy, but that's a completely different thing.

So let me get this straight, I can do all my operations and then only involve my significant figures at the end? Regardless if there's both addition and multiplication?

Borek
Mentor
I can do all my operations and then only involve my significant figures at the end?
Yes.

Actually it is the only correct way.

DrDu
Actually, if you really want to know the precision of your calculation, you should take into account that the number of significant figures is typically due to measurement errors, e.g. the amount of moles has only been measured with a certain precision and the molar mass is only known with some uncertainty e.g. due to variations in isotopic composition.
Typically, the uncertainty is of the order of the last figure given e.g. n=3.3 (+/- 0.1).
Then you could use error propagation to determine the uncertainty of your final result.
The number of significant figures is a way to approximate this method.
There is lots to be found on the internet, e.g.:
http://www.rit.edu/~w-uphysi/uncertainties/Uncertaintiespart2.html [Broken]

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Thanks guys !