Handling elementary but messy computations in proofs

Question

Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious and error prone, yet the path to a solution is not always elegant (or, at the least, an elegant path is not always apparent to this amateur mathematician).

How would you advise a beginning mathmeatician to handle these cases? How do seasoned mathematicians handle this? Often I proceed forwards with pen and paper, but more often than not this results in elementary errors (sometimes simply because the amount of writing gets huge, my penmanship gets sloppy, and I misread my writing). Should I simply be more patient and explicit, and learn to do these computations by hand, accurately if laboriously? Should I learn to use software such as Sage to do them? Or should I take the computational ugliness as a sign that a more elegant proof should be approached?

Update

A good example of the algebraic manipulations I'm talking about are those referenced (but not spelled out) in this answer https://math.stackexchange.com/a/246288/

Answer 1

It all depends on the context and the goal.

If your goal is just to find answers and verify a result, then by all means use a CAS to do the algebra for you. (For example, some computation-heavy papers just say things like "by SAGE, the answer for this is step is...". But if you want to keep your algebra skills sharp, then maybe just use a computer to check your error-prone steps.

And if you want to learn more mathematics, maybe search out/ask people for relevant generalizations.

For example, if I weren't teaching someone induction, I would never write out the induction for the formula for $J_n$ alluded to in the post you linked, because I know a general theorem about homogenous linear recurrence relations. With that theorem in hand, it suffices to check $J_1$, $J_2$, and to solve $r^2=r+2$.

Answer 2

You asked

Often when working on a proof, I get to a computation which appears to be elementary (e.g. requiring only standard algebra and perhaps calculus) but messy. Solving this via pen and paper is tedious and error prone, yet the path to a solution is not always elegant (or, at the least, an elegant path is not always apparent to this amateur mathematician).

How would you advise a beginning mathematician to handle these cases?

My advice is that if you solving algebra/calculus problems, then a CAS can be a very valuable tool for a few reasons.

One reason you mentioned is "tedious" calculations. It can be much faster and less error-prone for even a simple calculator to do some of the calculations. The more capable the CAS, the better it is able to handle the task required.

Another reason is to check your arithmetic and algebraic equations no matter how they are derived. It is very easy to make sign errors, copy errors, and other kind of errors. We all make these mistakes, but a CAS can check for many of these errors. For example, if an equation is supposed to be true for all $\,x\,$, then you can use a CAS to test this for many numerical values of $\,x\,$

Yet another reason is that many CAS can perform operations that are not possible to do by manual methods. For example, factoring multi-variable polynomials is almost impossible with manual methods, yet is a standard capability of many CAS. This can lead to elegant proofs if you can find significant factors of a rational function. For example, suppose that $\,a,b,c\,$ are three angles such that $\,a+b+c=2\pi.\,$ Then they satisfy the equation $$\cos(a)^2+\cos(b)^2+\cos(c)^2-2\cos(a)\cos(b)\cos(c)-1=0.$$ This can be proved by setting $\,a=(\log A)/i,b=(\log B)/i,c=(\log C)/i.\,$ Substituting into the left side of the equation and factoring reveals a factor of $\,1-ABC\,$ which translates into the result that $\,a+b+c\,$ is a multiple of $\,2\pi.$ Note that in this example $\,a,b,c\,$ are the angle measures of the three sides of a spherical triangle which lies in a plane.