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I am learning how to structure my proofs in such a way that others can read them with ease. It was pointed out to me several times on this site that my proofs are not very clear. Anyway, here goes:

The number r is rational iff -r is rational.

Suppose that r is rational.
    r can be written as a fraction a/b.
    (-1)*r = (-1)*(a/b) = (-a)/b.
    Thus (-1)*r can be written as a fraction.

Suppose that -r is rational.
Then -(-r) = r is (by the theorem above) also rational.

Any stylistic advice much appreciated. (It would be rather embarrassing if the proof were incorrect mathematically).

Adam
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  • It is supposed to be indented, but for some reason the indentation doesn't show. – Adam Dec 07 '13 at 00:09
  • That's about as good as I can get it at the moment - hope that is what you wanted. – Old John Dec 07 '13 at 00:13
  • It's perfect. One question: have you somehow recovered my indentation, or indented the thing yourself? – Adam Dec 07 '13 at 00:15
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    @Adam clicking edit allows you to see (and change) the "code" that goes into making the comment. What Old John did (I assume) was click edit, highlight your proof, and click the {} button, which causes indenting to be preserved. – Ben Grossmann Dec 07 '13 at 00:24
  • @Omnomnomnom Yep, that is what I did. I also added a couple of extra line-breaks to stop the nasty horizontal scroll bar from appearing. – Old John Dec 07 '13 at 00:25
  • @Adam I have a series of write ups on Proof writing, and would welcome any feedback on them. – Calvin Lin Dec 07 '13 at 01:26
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    You can shorten the second part: if $-r$ is rational, then, using the first part, $r=-(-r)$ is also rational.

    It is personal taste, but I find rather elegant to use the symmetry of the problem.

     – Taladris Dec 07 '13 at 01:29
  • @Adam I like some of the things you have done. A small suggestion might be to learn LaTeX and use it whenever possible. Note that we share the common burden of writing with more clarity. To that end, I leave you with a paper that really inspired me a long time ago by Paul Halmos. – J. W. Perry Dec 07 '13 at 05:00

2 Answers2

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My main stylistic advice would be to try to make your proof "read" more like prose. Maybe something like this:

Suppose that r is rational, so that r can be written as a/b. 
Then we have (-1)*r = (-1)*(a/b) = (-a)/b, and we see that 
(-1)*r is also rational.

Suppose conversely that r is irrational, but also assume that 
(-1)*r is rational. Then it would follow that (-1)*[(-1)*r] = r 
would also be rational, by our previous argument. This would contradict
our assumption that r is irrational, proving that our
assumption that (-1)*r is rational must have been wrong.
Old John
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  • I find this way more...personal. I am undecided about which way is better. – Adam Dec 07 '13 at 00:46
  • @Adam as you do these, I think you find a balance between the two. For the reader, too much of either can be off-putting; too much "prose" might make your proof an unreadable wall of text, but in a line-by-line proof the jump from one line to the other can be a bit jarring and hard to keep up with. I think that a well written proof is "approachable", which means different things to different people. – Ben Grossmann Dec 07 '13 at 04:34
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Some (hopefully helpful) criticism.

What you seem to be proving is the statement "r is rational if and only if $-r$ is rational". For your second paragraph, instead of making your goal to show "$r$ is irrational" implies "$-r$ is irrational", make the goal of the paragraph to show that "$-r$ is rational" implies "r is rational". That way, your second paragraph becomes a direct proof, rather than a proof by contradiction.

Ben Grossmann
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  • Well, I was aware of the possibility to do this, but for some reason I prefer this way. – Adam Dec 07 '13 at 00:40
  • I suppose that's a matter of taste. I always try to avoid a proof by contradiction where a direct proof will do just as well. The way I see it, adding the proof by contradiction just makes the proof a little bit longer. – Ben Grossmann Dec 07 '13 at 00:42
  • Well, I suppose that it is a personal idiosyncracy of mine. Given two proofs, one of which I think most people would think of first and another proof that I think has a bit of "me" in it - I prefer the latter. – Adam Dec 07 '13 at 01:08
  • Sometimes with the result that I prefer a proof that is 3 times the length of the standard proof... – Adam Dec 07 '13 at 01:10
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    If it is a proof that is to be read only by you, it is good to maximize your pleasure. – André Nicolas Dec 07 '13 at 01:26
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    @Adam I would definitely agree that the shortest proofs are not necessarily the best; often writing a proof in one-third the length makes a proof much harder to read. I think it's important to remember that your "job" as proof-writer is to convince your reader that the statement you want to prove is true. What I generally aim for is a proof that minimizes the reader's mental effort. I find that, more often than not, proofs by contradiction can be tasking because the reader has to "keep in mind", sometimes for a while, the statement to be contradicted. Also, it's harder to read more words. – Ben Grossmann Dec 07 '13 at 04:39