Significant Figures and Units in Physics Calculations

[PFuser1232]

Physics has always been my main interest; with more than a mere "favourite subject" label. Therefore I have made it one of my goals to not only be able to solve problems, but also express rigor and detail in my calculations. With regard to solving problems, I have 2 main questions:
1) Significant Figures
I am familiar with the common practice in all of science regarding significant figures - the lowest number of significant figures in a question determines the number of significant figures in the answer. But what if a question has several parts, in which one should use one answer in the next "part". In that case, should one use the "exact" value from a calculator (√2, for instance)? Or should one stick to the significant figures rule and use the "rounded up" value? Which is the correct method?
2) Units
I know and appreciate the importance of units in physics, and I understand that they should be explicitly stated in a final answer. But what about the working? I mean, while computing a derivative or an integral, should one include units for the coefficients? Do physicists actually show units while "doing the math"? Or do they consider this practice redundant and of no good use?

 

[olivermsun]

1) If you have an exact value like √2, then you use the exact value (or enough accuracy that it doesn't matter) so as not to affect the significant digits in your answer.

2) Well, derivatives and integrals change the units, so it's very important to keep track of the units through these operations. How you choose to keep track on your scrap paper is your business. If it's pretty obvious stuff, for example, if you are getting the acceleration a bunch of times from mass and velocity measurements in time, all in the same units, then sure, it's probably redundant to keep writing the units every time. If you're deriving an expression for the first time, you probably better make sure all your units work out correctly.

 

[jtbell]

In a multi-part problem, I always use the unrounded answer from each part as input for my calculations for the next part, by keeping it in my calculator (either in the display or in a memory register) so that I can easily use it when I start calculations for the next part.

When I report the answer for each part, I round it off as appropriate.

 

[PFuser1232]

1) If you have an exact value like √2, then you use the exact value (or enough accuracy that it doesn't matter) so as not to affect the significant digits in your answer.

2) Well, derivatives and integrals change the units, so it's very important to keep track of the units through these operations. How you choose to keep track on your scrap paper is your business. If it's pretty obvious stuff, for example, if you are getting the acceleration a bunch of times from mass and velocity measurements in time, all in the same units, then sure, it's probably redundant to keep writing the units every time. If you're deriving an expression for the first time, you probably better make sure all your units work out correctly.

So regarding significant figures: suppose I get a mass of √2 kg in part 1, but I express it as 1.4 kg while writing down the answer. In part 2, when I use that value for mass, I use √2, but not 1.4? [Despite the fact that my previous answer indicates that I know the mass to only two significant figures.]

 

[olivermsun]

I don't think you could ever get a mass of √2 kg if you were keeping track of significant digits properly. It could be an answer like 1.0 * √2 kg (multiplying a real measurement of 1.0 kg by a factor of √2), in which case you would report 1.4 kg.

jtbell's suggestion of holding intermediate values (without rounding) in your calculator is also what I usually do.

 

[PFuser1232]

I don't think you could ever get a mass of √2 kg if you were keeping track of significant digits properly. It could be an answer like 1.0 * √2 kg (multiplying a real measurement of 1.0 kg by a factor of √2), in which case you would report 1.4 kg.

jtbell's suggestion of holding intermediate values (without rounding) in your calculator is also what I usually do.

So I report 1.4 kg, but when using a value for this mass in the next part of the question, I use √2.