2
$\begingroup$

I need to write down a complicated proof for a paper, for which I need to employ equations that I established earlier for almost every new relation I show. I would consider it best for the reader, if I denoted the number of the employed relations to establish a new relation over each relation sign.

In a simplified example, I would like to write something like: $$ x+y-z \stackrel{3.1}{=} x+w, $$ where equation 3.1 gives me $y-z=w$.

However, I cannot remember seeing this or a similar notation in the mathematical literature, which may be for a good reason. My question therefore is: Is there any such good reason against writing down proofs like this or have I just been reading the wrong papers?

Also, should such a reason exist: What would be similar, acceptable ways to write this down? I would strongly dislike writing a sentence about the employed relations for every single relation I show, as this would make the proof very hard to follow in my opinion. (Leaving finding the employed relations as an exercise to the reader is not an option, given the vast amount of relations.)

$\endgroup$
  • 2
    $\begingroup$ I do see this, albeit rarely. If the equations used are collected somewhere earlier in your paper, and provided that it's not too hard to figure out from inspection which is being used in each equality, I might instead preface the equation block you want to annotate with a phrase like, "Using the established identities 3.1-3.9 gives..." If these caveats don't apply, I might do what you suggest, but perhaps put parentheses around the equation number over each equality sign, which helps further distinguish that number from the expressions around it. $\endgroup$ – Travis Nov 18 '14 at 12:03
  • $\begingroup$ That said, if it's sufficiently nonobvious how the identities are being used in each step in the equation block, it's probably a good idea to decompose the blocks (not necessarily one equality per block) and indicate at least the trickier of those steps in words too. $\endgroup$ – Travis Nov 18 '14 at 12:04
  • $\begingroup$ @Travis: If you can reference some papers using this notation, you might as well turn this comment into an answer. $\endgroup$ – Wrzlprmft Nov 20 '14 at 15:31
  • $\begingroup$ If I knew offhand of any particular paper that does, I would be happy to be. That I do not perhaps says something about the rarity of the notation. I'll post back here if I find such a reference in the near future. $\endgroup$ – Travis Nov 20 '14 at 16:37
1
$\begingroup$

The way I've seen it is to put numbers before or after the equations in parentheses with colons, like so:

$$3 + x = 1$$

$$\implies x = - 2 \quad :(1)$$

Let $t = -x$. Then:

$$(2):\quad t = 2 \quad \text{by (1)}$$

It follows that:

$$(3): \quad x + t = 0 \quad \text{by (1) and (2)}$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.