[bert]

I understand the sig fig rules that pertain to addition and multiplication problems, however, in multi-step problems, I have not been able to find a systematic approach to getting the right amount. Everybody who I have asked just said "it's tricky so be careful".

If the problem is: (5.01 + 4.1) / (1.00 x 1.00) Is the answer 9.11 or 9.1? The addition part indicates 2 sig figs because the tenths place is the last accurate decimal, while the multiplication part indicates 3 sig figs. Does it depend on whether the result inside the parentheses is added/subtracted or multiplied/divided from the other result?

What about a problem like: (2.00 x 30) + 5.01 + 2.0 Is the answer 60. or 67.0? The multiplication indicated the former, the addition indicates the latter.

Problems where the numerator has numbers being added and the denominator has numbers being multiplied like: (5.01 + 2.0 - 3) / (60.01 x 3.0) are even more confusing, especially when scientific notion is mixed in. I want to know an approach that goes step-by-step and works 100% of the time.

[Borek]

The addition part indicates 2 sig figs because the tenths place is the last accurate decimal, while the multiplication part indicates 3 sig figs.

Whichever is lower.

What about a problem like: (2.00 x 30) + 5.01 + 2.0 Is the answer 60. or 67.0? The multiplication indicated the former, the addition indicates the latter.

Again, whichever is lower.

Don't worry too much. It is good to understand where the idea comes from to not abuse numbers by writing out all these nonsensical digits, but significant figures are not a good way of expressing the accuracy, and the only place people pretend to care abut them are initial chemistry courses.

[bert]

Thank you. That's a bit reassuring. I'm worried that i'll have points deducted from an exam because I wrote 1 more digit than necessary. I've seen problems where the total sig figs are "supposed" to go up because addition is the last step, problems where writing a number in scientific notation reduces the sig figs for some reason, and I've seen one explanantion online which said results found within a parenthesis should be rounded by sig figs (instead of rounding the final answer), and the rounded result should be used in the next step and so on.

On a slightly unrelated note, is it more correct to write 2.23 x 10^3 or 22.3 x 10^2?

[Borek]

Thank you. That's a bit reassuring. I'm worried that i'll have points deducted from an exam because I wrote 1 more digit than necessary.

Unfortunately that can happen, if your teacher treats them too seriously. In my experience most people don't care too much, besides, in most problems it is enough to check all data and use the lowest number of sigfigs in the answer. Definitely don't abuse digits, as it catches attention.

I've seen problems where the total sig figs are "supposed" to go up because addition is the last step, problems where writing a number in scientific notation reduces the sig figs for some reason, and I've seen one explanantion online which said results found within a parenthesis should be rounded by sig figs (instead of rounding the final answer), and the rounded result should be used in the next step and so on.

The main rule is: do calculations with all digits you have, round ONLY the final result.

On a slightly unrelated note, is it more correct to write 2.23 x 10^3 or 22.3 x 10^2?

In scientific notation significand (the number before the exponential part) should be from the <1,10) range (or 1≤x<10).

Sometimes you will see numbers written in so called engineering notation, where the exponent is multiply of three (so 12.3×10³, which in scientific notation is correctly written as 1.23×10⁴). That's just another convention.

[Mitch]

I care about significant digits...

[Borek]

I care about significant digits...

Would you use them for any serious research, when it comes to the measurement accuracy and reporting statistics? Would you use them to calculate 6 sigma results?

It definitely makes sense to not report too many digits (as I wrote: understand the idea and don't abuse digits), but have you ever followed strict addition rules when doing calculations? Or have you just looked at the numbers involved and chose more or less 2-3-4 digits that looked like they make sense?

Sig figs are faulty by design. They are typically understood as "the last digit is accurate to ±0.5". Then 1 M solution is anything between 0.5 and 1.5 M (50% discrepancy) while 9M solution is anything between 8.5 and 9.5M (about 5% discrepancy). 50% vs 5% discrepancy for the same number of digits reported. That doesn't make sense and should be avoided. Interestingly, as far as I am aware chemistry is the only science where they are used. Physicists don't care too much, whenever it matters they rather report variance and write as many digits as make sense in the context (σ/value), or write just about 3-4 digits.

[Mitch]

Honestly, I have always used them religiously since I learned about them in Highschool in 1996. Since I'm consistent about it, it lets me not have to use my gut and trust the results I get. I very rarely not use them, and feel really guilty when I choose not to. I do ignore them if I have to display an average or standard deviation, as you indicated, it doesn't make sense in all circumstances.

[bert]

Today I had a physics lecture. The professor kept going on about the importance of significant figures. They show how accurate the measurment is, writing more than necessary misleads people as to how accurate it is, etc, etc, etc. He said he saw a student get an F on his paper once because he put the wrong number of significant figures on it and the professor was furious. After class I asked him how important significant figures are when there are no measurments. He responded everything comes from some measurment, so I asked him about problems with no units what so ever. He didn't have an answer, but reiterated how crucial they are in general.

I asked this question on Chemistry Stack Exachange, and one person answered:
The step-by-step approach is to consider each step separately (observing the correct order of priority), keeping track of significant figures of the intermediate results. Then, calculate the number without any rounding of the intermediate results, and record it with the appropriate number
2.00 x 30) + 5.01 + 2.0
The product has two significant figures, least significant figure is the ones.
Adding 5.01 will have least significant figure as the ones.
Adding 2.0 will have least significant figure as the ones.
The arithmetic result (unrounded) is 67.01, so you write down 6.7⋅10^1. Depending who your teacher is, 67 or 67. is fine as well.

But 30 only has 1 significant figure. It doesn't make sense. I feel like there's something really obvious nobody is telling me.
Edit:He fixed it and wrote the answer should be 7 x 10^1. That makes a lot more sense. My rounding was off. I think I have a slightly better handle on this now.

[Mitch]

Hi Bert, what exactly are you confused on, maybe I can help.

[bert]

Hi Bert, what exactly are you confused on, maybe I can help.

Thank you for the offer, but i'm not confused about anything specific. I have already detailed why I was having trouble and the kind of help I was looking for: when given a problem with no units where I am expected to give an answer with the correct amount of significant figures, what do I do? That's as specific as it gets.

[Mitch]

It's hard to think of an instance in reality where there isn't a unit. If you are using arbitrary units, and you had a value of 100. I would assume each digit is significant if they put a decimal point immediately after the 100 like this: 100.

If they don't put a decimal point, then you have to clarify the significance of trailing zeros by asking questions.