One of the (many) things that makes middle aged academics like me grumpy, is students giving answers like speed = 4.738281 m/s, by just copying the number from the screen of their calculator/mobile-phone-app, without any thought at all. But I am actually paid to do something more constructive that be grumpy about this sort of poor practice, so this year I have done a little set of notes on accuracy for my final year class. They take the students through a couple of examples.
The notes are in answer to the questions students very reasonably ask me when they see me wincing in pain at answers like the one above. If you are given a formula like E = mc² , together with a mass of say 1.1 kg, and you know the speed of light c = 299 792 458 m/s (we know c very accurately indeed), how do you know how accurate will your answer be?
The short answer is that it is as accurate as the least accurate part of the bits you are using to get that answer. Here this is the mass of 1.1 kg which is only accurate to two figures, so your answer is only accurate to two figures, E = 9.9 × 1016 J. This is a simple example as the formula is exact, if the formula is approximate, then that can be limiting factor for how accurate is your answer.
So, working out how accurate your answer is does require a bit more thought that just plugging in numbers to a calculator, but it is not too bad. Another question I get that is a little harder to answer is: How accurate does an answer need to be?
This requires a bit more judgement, and comes with practice, but it is a very useful skill to work on. Working harder than you need to, to calculate an answer that it is more accurate that it needs to be, is a waste of your valuable time. Often students don’t realise that in many cases a rough answer is all you need?
For example, if the question is: Will converting 1.1 kg into energy cause a problem? Then answering this question can be done with a very rough estimate. Above we gave an answer to two significant figures, and that is more than enough. An estimate of 1017 J would easily be enough — this is an awful lot of Joules that would create a very big bang. Even if we had been lazy and just said well c is about 108 m/s, then our estimate of 1016 J would have been fine, it is out by a factor of ten but it is still an awful lot of Joules. So still same conclusion.