2—Infinite Series 502.9 Checking Results
When you solve any problem, or at least think that you’ve solved it, you’re not done. You still have to check
to see whether your result makes any sense. If you’re dealing with a problem whose solution is in the back of
the book then do you think that the author is infallible? If there is no back of the book and you’re working on
something that you would like to publish, do you think thatyou’reinfallible? Either way you can’t simply assume
that you’ve made no mistakes and you have to look at your answer skeptically.
There’s a second reason, at least as important, to examine your results: that’s where you can learn some
physics and gain some intuition. Solving a complex problem and getting a complicated answer may involve a lot
of mathematics but you don’t usually gain any physical insight from doing it. When you analyze your results you
can gain an understanding of how the mathematical symbols are related to physical reality. Often an approximate
answer to a complicated problem can give you more insight than an exact one, especially if the approximate
answer is easier to analyze.
The first tool that you have to use at every opportunity is dimensional analysis. If you are computing a
length and your result is a velocity then you’re wrong. If you have something in your result that involves adding
a time to an acceleration or an angle to a distance, then you’ve made a mistake; go back and find it. You can do
this sort of analysis everywhere, and it is the one technique that I know that provides an automatic error finding
mechanism. If an equation is dimensionally inconsistent, backtrack a few lines and see whether the units are
wrong there too. If they are correct then you know that your error occurred between those two lines; then further
narrow the region where the mistake happened by looking for the place at which the dimensions changed from
consistent to inconsistent and that’s where the mistake happened.
The second tool in your analysis is to examine all the parameters that occur in the result and to see what
happens when you vary them. Especially see what happens when you push them to an extreme value. This is
best explained by some examples. Start with some simple mechanics to see the procedure.