# User:KarlFrei

## Message for new users

You should be aware that this site is in some respects unfortunately actively hostile towards new contributors. There are a number of rules on this site that everybody is expected to adhere to. Unfortunately, not all of these rules are written down. There are help pages, but they are incomplete. It would be great if somebody would expand the help pages. However, unfortunately, those few people who have the rights to edit the help pages seem to be unwilling to do.

Personally, I am a trusted user, a status that I got because of some good contributions to this site. Trusted users used to be able to edit help pages. Unfortunately, this ability was removed as soon as I started editing the help pages, although my edits were in good faith, accurate, and constructive, as far as I can see: [1] [2] [3] (this last edit has already been undone: I do not know why). I have come to the conclusion that sadly, trusted users cannot be trusted to edit the help pages. This only leaves people that are not interested in editing help pages. At least, no editing of any significance was done in 2019: [4] [5] [6].

It is very unfortunate that the administrators of this site have set things up so that only they themselves can edit these important pages, but then decline to keep them in good shape ("Life is too short"). The alternative that is used on this website is to contact users directly (as you may have noticed already) and leave cleanup messages on the pages they contribute, leading to repetitive discussions. It is unclear to me why the administrators apparently prefer to have the same discussions over and over again, rather than allowing people that are actually interested in editing help pages to do so.

It is my view that trusted users should be able to edit help pages. For what it is worth, I am willing to give up my own trusted user status if that helps - it is unclear how this benefits me in any event.

Here are the most important unwritten house rules.

- Deleting of any material is strongly discouraged. (OK, this rule actually is listed in the help (because I added it myself), but I feel that this is a very important rule for new users to know, and not sufficiently emphasized.)
- (Added to help page by me, then removed by an administrator without reason:) There is to be
**no change**in any material on this website UNLESS- it is demonstrably wrong (and even then, it might be kept for historical interest)
- it is not in house style
- it is an expansion of existing material, by adding links to related concepts or definitions, adding missing steps in proofs, adding explanations for steps in proofs, etc. etc. ...

- (As a logical consequence of rules 1 and 2 above:) Editing on this website means
**expanding**and expanding**only**. Even though we have a general disclaimer stating that your text may be "altered" and even "edited mercilessly", that is not really accuratate. If you write anything on this website, and it is mathematically correct and in the proper style, you may rest assured that it will still be there in a hundred years. Never mind if your proof is circuitous or hard to follow or understandable only by you: if we cannot show it is actually wrong, it will stay. On the other hand, if you happen to see any step in a proof on this website that can be simplified, then unfortunately the only way to get the simpler step on ProofWiki is to write a completely separate proof with this change, no matter how small and easy the change or how big the proof.*Do not under any circumstances attempt to "improve" any proof by editing it.*(Mathematicians beware!) If you come here hoping to possibly make some proofs easier, you can do that by adding a new version, but the version which you think is bad MUST stay on the website next to it.*This should now be clearer on the editing help pages as well, though the motivation remains murky.* - Note that the page House style actually contains very strict and incredibly important
*rules*. The main feedback that you will get on your early contributions on this site is that the content you added is not in house style (at least in my experience; your mileage may vary, of course). - The house rules are fixed and not subject to discussion.

To summarize: even though the main page of ProofWiki has *collaboration* as a primary goal, this should not be taken to mean that you can just start and *change* text as you would on, say, Wikipedia. Remember the mantra: **editing means expanding** or putting articles into house style.

- I find it sad that you see it like this. If you are so terribly frustrated, please go somewhere else. People like Leigh.Samphier have demonstrated to be cunning in adopting all the "unwritten rules" and actively contribute by making the mathematics on this website more structured and more accessible.

- As long as you insist on going your own way and doing as you deem fit, there will be no place on this site for you. Unlike Wikipedia, where the aim is limited to providing on a lot of topics of interest a fairly detailed account, ProofWiki aims to provide a
*coherent*account. I should not need to convince you that the fluctuation in style, setup, and structure of pages on WP would have on ProofWiki the effect of throwing all textbooks randomly together without any consistency.

- Additionally the editing of proofs to make them "simpler" or "more elegant" is not ideal because it tends to make the material less accessible. Until at the extreme end any theorem is so simple that it is "left to the reader as a trivial application of the definitions". Maintaining accessibility at all relevant levels is
*hard*, like maintaining cohesion, and therefore needs to be approached with due respect. Personal style or preference are irrelevant in this respect, we should not strive to make a single editor the measure of all things.

- As such comparisons to Wikipedia are ill-fit, and any change should be backed by a source work or demonstrated to be an overall increase in nonlocal cohesion (in case of refactoring). And yes, because this kind of endeavour is quite complex, we have a culture of discussing before changing.

- If all that is not what you would like to see from a mathematics website, I suggest that you move on and write your own blog or whatever. — Lord_Farin (talk) 09:02, 3 January 2020 (EST)

## Motivations

I have started here to collect motivations for several house rules as I find them here and there on this website. This is meant as a public service. Anyone who has the time and inclination, please feel free to add to this page, or correct it in case of any mistakes.

1. Refactoring is only to be done by experienced contributors.

- One of the reasons for this concerns the links in the Sources section at the bottom of the page. Contributors have in the past demonstrated that they have difficulty in understanding or motivation when it comes to making sure that the flow is consistent. In particular, note that the "next" link from [one result] from [one source] no longer goes to the next page in the sequence for [that source], because it has been moved from [the original page] to [a newly created subpage for a proof] and so the flow has been broken.

- Because it takes considerable effort and care to make sure that the flow is not compromised, we respectfully request that you please do not do any of this "refactoring", and instead (unless a page already has several sub-proofs on separate pages) just add further proofs on the same page, leaving it up to the administration team to split them up into subpages. In that way, administrators (and in particular, those who have direct access to the specific works which cite material held on these pages) may ensure that the flow is no longer compromised.

- Note that it is not only that the links need to be updated to point to the particular (e.g.) "/Proof 1" page. It also needs to be determined whether the work in question actually documents that proof, or whether they just quote the result, in which case the citation link needs to stay on the original page.

2. The proofs on this website should match the proofs in the sources they were taken from, modulo house style.

- Motivation: I have no idea. It seems to have something to do with the high cost of maths textbooks.

3. Links to subpages should use redirects.

- Two reasons. Firstly, it's nicer to display (particularly applicable to theorems) and to type. Secondly, the fact that subpages are needed makes it highly more likely that the page structure will be refactored
*again*, in which case it just takes an update of the target of the redirect rather than updating all links everywhere. (To those familiar with OO-programming, the view of the redirect as an interface or API might be clarifying.)

## Proof 2

We have

\(\ds a x^2 + b x + c\) | \(=\) | \(\ds 0\) | dividing by $a$ | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds x^2 + \frac{b}{a} x + \frac{c}{a}\) | \(=\) | \(\ds 0\) |

Note that

- $(x-r)(x-s)=0 \Leftrightarrow x=r \vee x=s$

Our goal is to find $r,s$ (if possible) such that

- $x^2 + \frac{b}{a} x + \frac{c}{a} = (x-r)(x-s)$

for all $x$.

Then the roots are exactly $r$ and $s$.

We have

- $(x-r)(x-s)=x^2-(r+s)x+rs$

Hence the desired equality is equivalent to

- $\frac{b}{a} = -(r+s)$ and $\frac{c}{a} = rs$

The average of $r$ and $s$ is therefore

- $\frac{r+s}{2} = -\frac{b}{2a}$

This implies that there exists a value $z$ such that

- $r = -\frac{b}{2a} + z$ and $s = -\frac{b}{2a} - z$

We calculate the product

\(\ds rs\) | \(=\) | \(\ds \left(-\frac{b}{2a} + z\right)\left(-\frac{b}{2a} - z\right)\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds \) | \(=\) | \(\ds \frac{b^2}{4a^2} - z^2\) |

As the product is known to be $\frac{c}{a}$, we find

\(\ds z^2\) | \(=\) | \(\ds \frac{b^2}{4a^2} - \frac{c}{a} = \frac{b^2-4a}{4a^2}\) | ||||||||||||

\(\ds \leadsto \ \ \) | \(\ds z\) | \(=\) | \(\ds \sqrt{b^2-4a}/{2a}\) |