Should I round on each step when solving a physics problem?

[G-Man]

today in physics class our teacher told us that when we were doing a problem we had to round to the sig digs on each step. This semed like it would make the answer less accurate. Is the the right method to use or not.
And if it is correct why is it done this way?

 

[z-component]

In my first year of high school physics we used "sig figs" as we call them-- significant figures. This year in AP Physics 2/C, we don't. Significant figures are used when obtaining measurements using experimental equipment like weighing devices. The idea is to only give the recorded values to the level of accuracy that the device gives, making lab experiments as accurate as possible.

The level of accuracy depends on the instruments used. When doing calculations, sig figs are used to give a guideline of how many numbers in a solution should be used. If you're given a problem with 2.0 N of force or 17.83 m/s^2, there's a bit of a difference (according to some teachers) in using only 2 or 3 significant figures in your answer.

http://en.wikipedia.org/wiki/Significant_figures

 

[G-Man]

we were given the question "a ball is thrown upwards with a starting velocity of 25m/s and the acceleration due to gravity is -9.81m/s^2. At what time(s) will the ball be 3.0m above the starting position" and my teacher said that I got it wrong because when I was using the quadratic formula I did not round to the correct number of significant digits in the discrimant itself. While I was under the impression that I did not need to round until I got the answer.
Should I be rounding on each step of the question in this case?

 

[berkeman]

I don't know the rules for sig figs, but it seems like it would be different based on the operation. Like, the rounding rule for multiplication would be different from the one for addition. Maybe that's why you have to be careful in the middle of your calculations, instead of waiting until the end (I'm not sure about that obviously).

Like, say you have two numbers 1.1 and 1.1. The measurement can't be any more accurate than 0.1 whatevers. You can get 1.1 from round up 1.05 or rounding down 1.14.

1.1^2 = 1.21, 1.05^2 = 1.1025 and 1.14^2 = 1.2996 (+7/-9% from ideal)

1.1+1.1 = 2.2, 1.05+1.05 = 2.1 and 1.14+1.14 = 2.28 (+4/-6% from ideal)

So by not rounding the multiplication answers off when you get them, you can potentially carry through a bigger error than you would by not rounding off the additions. Just a thought. Guess I should have learned more about sig figs...

 

[Hurkyl]

If you can do a proper analysis of the error, it is better to wait and do all the rounding at the end.


However, in lieu of that, you should round at each step. For example, you don't want to accidentally claim:

7128 * (3.498 / 3.456 - 1.012) = 1.089

because you waited until the end to round!

It actually results in zero significant figures of information! Assuming each of those numbers on the LHS is correct to within 0.5, the result of this calculation could be anywhere within:

(-4.550, 7.773)

or even (just slightly) outside!

 

[z-component]

Yep berkeman that's exactly the idea behind significant figures-- carrying through errors. I was taught to pay attention to sig figs only for the final answer, referring back to the number of sig figs in the given information. Like I said, I don't need to worry about them anymore though. Your teacher obviously has different expectations but I suggest not worrying whether or not the way your teacher has you do your sig figs is correct or not. You probably won't use them if you take higher level physics courses and if you do, possibly only for lab experiments. They're a good thing to learn for the purpose of science but they're not very useful in the long run.

 

[Mattara]

No, no, no. You should never round to anything in the calculation. You should only do it in the answer. If you have 2 sig. digits in the question, it is less accurate to use more than 2 sig. digits in the answer.

 

[BobG]

we were given the question "a ball is thrown upwards with a starting velocity of 25m/s and the acceleration due to gravity is -9.81m/s^2. At what time(s) will the ball be 3.0m above the starting position" and my teacher said that I got it wrong because when I was using the quadratic formula I did not round to the correct number of significant digits in the discrimant itself. While I was under the impression that I did not need to round until I got the answer.
Should I be rounding on each step of the question in this case?

There are some cases where you want to round at each step. Hurkyl's answer is probably the best, in general. I don't think the problem above is one of the cases where you need to round each step of the way, though.

Common sense used to be the best guide as to when to round each step and when you're better off waiting until the end to round off. Unfortunately, common sense can only be learned by experience and will be almost impossible to learn if you've used a calculator your entire life. Hence, having to learn the rules for sig figs, rules for when to round off intermediate steps, etc.

In the old days, if a person tried to solve Hurkyl's example with a slide rule (where you always rounded off each step because of the limitations of a slide rule) it would become painfully obvious that you were multiplying 7128 by crap. You'd wind up having to solve the problem long hand using pencil and paper knowing all the time that your final answer was little more than a wag. While Hurkyl's example may be a little artificial, there definitely are problems where subtraction somewhere in the middle eliminates most of the significant digits, meaning your final answer is much less accurate than you'd be letting on if you arbitrarily rounded off at the end.

On the other hand, you have to be pretty unlucky for every round-off error to be the same direction, so rounding off every step of the way usually winds up very, very close to the answer you'd wind up with if you hold off. Either way, your answer should fall within the range of uncertainty. So, concerns about holding on to the calculator answer are overblown and show a lack of understanding about just how accurate those numbers will be in the real world.