How to not mistakes with long calculations

[zmike]

Does anyone have any advice for working with long calculations such as integrals of equations with 10 or more terms. I've been doing of the questions in my book and 99% of the time I'll make calculation errors when I do it by hand with a manual calculator even when I try to slow down!

Do you try to calculate each term then combine them together OR plug them all in at once OR...?

thanks

 

[NobodySpecial]

Make an estimate of the answer - just an order of magnitude (or 3 orders of magnitude if you are doing astronomy/particle physics)

Combine several large constants 'GM' for example or 'hc' into a more manageable number

Write the results of intermediate steps.

Check the units balance - and that you have the right units

 

[phyzguy]

Use a computer! Software like Mathematica will do long calculations like these without error. Even if you have parts of the calculations that the computer can't do, you can still enter these parts by hand and use the computer to keep track of the terms. It might seem like it takes longer the first time to get it programmed in, but it is actually much faster than re-doing it 10 times by hand while you correct the errors.

 

[Deathcrush]

when its too long, it's okay to use a computer, however, don't become lazy, because probably you will need to know how to do it without a computer, for example, I've got a friend that is studying physics, and I am studying engineering, he would often talk about how great his math classes are, with lots of proffs and using a very formal approach, but he gets scared if he has to solve 5 equations with 5 unknowns :rofl:

 

[spyrustheviru]

yeah but deathcrush admit it, we engineers also don't know maths after we graduate :Pdo your calculations slowly, use a scrap paper, don't try to do things with your head, check every calculation the moment you finish it, write big numbers and leters, don't smudge and write messy, if you make an error and use a pen cross over the whole thing with an X and do it again.
And always have an estimate in your head about what you expect-if the outcome looks suspicious try it again to see if you where right to worry. I

 

[Deathcrush]

LOL I know, but I will make an effort not to forget my math, I love math too much to just forget everything

 

[Integral]

Repetition. Any one result is an accident. After you have completed the calulation go back and do it again. Once you have done it several times and got the same result it is very likely correct.

 

[Redbelly98]

Hey, I just came across this thread while poking around, sorry about the delayed response.

Redoing the problem is one way to catch mistakes, but that can be time consuming so if you can find a quicker way to check that is better. You wouldn't want, on an exam, to double the time it takes to do each problem.

How to check an answer quickly depends on what type of problem it is. The OP mentioned doing integrals, so a good check would be to differentiate the result and see if you get the original integrand back.

If you're asked to solve an equation for x, then you can substitute the value you get back into that equation and see if it's true.

If you're asked to simplify an expression, then substitute simple values like 0 or 1 into the initial and final expressions, and see if the values agree.

If you're asked to factor a polynomial, see if the roots give zero in the original polynomial.

 

[JohnSimpson]

I'm a pretty strong advocate of using computer software to do difficult calculations such as long integrals in their entirety. Once you've reduced a problem you're trying to solve to a clear but tedious and difficult calculation, you have to really ask yourself if it's worth your time. Do you care about the journey or the destination? Choosing the journey doesn't always make you noble IMO.

 

[arildno]

Intelligent compartmentilization of sub-expressions can also come in handy.

Suppose your original expression contains many sub-expressions, with different values of k, $$k*(\lambda-\mu)$$.
Here, introducing $$A=\lambda-\mu$$ reduces your writing burden (of the relevant sub-expressions) with two-thirds, thereby reducing the risk of calculating wrongly.

In order to maximize this beneficial effect, it might be smart to "complexify" your original expression, by adding zeroes and multiplying by ones in creative ways so that a number of such re-definable families of sub-expressions become apparent.

Sometimes, therefore, the best method to attack a complicated expression is to make it, as a first step, more complicated (having more terms/factors), rather than desperately, and naively, trying to reduce this complexity head-on.

 

[disregardthat]

Don't rely on computers too much. Do every single step by hand, and in time you will become much more efficient and thank yourself for that extra bit of work. Programs such as WolframAlpha are useful but not very healthy if used often, in my opinion. My advice is to carefully analyze and double-check each step as you go. When you get used to this way of working out long calculations you will become efficient at it and rarely do mistakes, and you won't waste much time desperately looking for errors which can prove fatal to your entire computation.

I second Arildno's last advice; introducing new variables to easen up the algebra/arithmetic can be very useful, and I do it often.