# Significant figures

DarthRoni
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?

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.

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.

DarthRoni
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?

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.