## Sufficient proofs

Hi Sir,

This is a general question I have always wanted answered. If in an exam, you were asked to prove a particular theorem, how rigorous does it have to be in order to get full marks for that particular part of the question? The reason I am asking this is because often in exams I try to make proofs as thorough as I can but end up not having much time for the other questions.

Thanks

A full answer to this question would be quite complex and perhaps not overly helpful at this point. The reason is that it’s not possible to give entirely algorithmic guidelines. As I’ve explained a few times in class and during office hours, it’s unavoidable that the overall structure of the questions and solutions and, sometimes, even the writing of the entire paper can affect the marking of a single part, especially when it comes to a question of detail. To mention just one example, you can imagine a a number of different solutions where along the way, someone writes,

`…the minimal polynomial is $(x-3)(x-2)$…’

without any justification.

Compare situations where

(1) The entire problem was to compute the minimal polynomial;

and

(2) The problem was to compute a Jordan basis and the minimal polynomial was a relatively easy step.

Clearly, the nature of the deduction can’t be the same. To get a sense of some other factors that may affect the marking, suppose the characteristic polynomial had already been computed to be $(x-2)(x-3)$, and furthermore, the person had already shown considerably sophisticated understanding of the relevant definitions and properties. Then the marker might be inclined to believe that the writer was aware of the `obvious fact’ that the characteristic and minimal polynomials are equal when the roots of the characteristic polynomial are distinct. (Especially if the solution read `… the minimal polynomial is *also* $(x-2)(x-3)$.’) However, if the entire solutions looks very much like an awkwardly memorized reproduction of something from the notes and then some of the justifications are omitted, the story becomes quite different. One can go on about this point endlessly, but I won’t.

For the moment, my suggestions are as follows:

(1) Never memorize proofs without a genuine understanding.
(2) If in doubt, justify.
(3) If you have a truly confident grasp of the material, then don’t worry too much if some simple details are omitted in the course of a long proof. Your strong understanding will be visible anyways.

Of course, if your understanding of the material is fragile, this whole discussion becomes irrelevant in a different way.

An important point that’s not emphasized often enough is that the solution to a math problem is also a piece of original writing. Rigor of detail is obviously necessary, but it’s also the whole thing that needs to come together in a coherent and convincing way. Incidentally, it’s not impossible that an unusual amount of not especially relevant detail can also be cause for deduction. So when asked what amount of justification is optimal, the only true answer I can give is `a reasonable amount.’

One thing I can say with a great degree of confidence is that in the course of three or four years at the university, it is eventually your correct level of skill that becomes apparent and is recognized.