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Assuming any commonplace, but rigorous, definitions and properties of $\pi$, what could be a simple, rigorous and non-numerical argument for the following inequality?

$$\pi\neq\sqrt{2}+\sqrt{3}$$

By simple I mean within the framework of a typical elementary analysis course at university, say, and thereby excluding deeper properties like transcendence etc.

By non-numerical, I mean arguments that avoid calculating approximations.

Damian Reding
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    Related: https://math.stackexchange.com/q/701822/42969. – Martin R May 21 '20 at 18:21
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    The simplest argument might be to calculate $\pi,\sqrt{2},\sqrt{3}$ to five points of precision and seeing that $\pi$ is at least $0.004$ less than $\sqrt{2}+\sqrt{3}$ – JMoravitz May 21 '20 at 18:21
  • @JMoravitz: I meant to exclude this one by use of the word 'elegant', but I agree that this is debatable ;) – Damian Reding May 21 '20 at 18:22
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    Does this answer your question? How prove $ \sqrt{2}+\sqrt{3}>\pi$? – JMoravitz May 21 '20 at 18:23
  • @nicomezi Please click on JMoravitz's link. – amWhy May 21 '20 at 18:27
  • The answers provided in the linked question are not what I would call simple or elegant. I disagree with the duplicate. @amWhy – nicomezi May 21 '20 at 18:27
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    Then, @nicomezi, the OP needs to define what they call simple or elegant, and needs to include far more context, else the current question could also be closed as not suitable for math.se, due to your lack of context, effort, etc. Or correctly closed as "opinion-based" because it fails to define, objectively, what counts as "elegant" and/or "simple." – amWhy May 21 '20 at 18:29
  • I do not mind closing question when it is for the right reason. The author precised what a simple proof means for him in the first sentence between the parenthesis. Answers in the linked question used either : another inequality as a starting point (hence not complete proof), series (which is not elementary), approximations (which he spoke about in the comment above). The term elegant is indeed more subjective. @amWhy – nicomezi May 21 '20 at 18:33
  • I don't understand why you use precise as a verb, nor what you mean by "sine", which you clearly don't mean the $\sin$ function in your use. "Simple" is also subjective, because it depends on one's level of study, @nicomezi, and the OP doesn't clarify whether they are a highschool student in trigonometry, or an undergrad non-math major, or an undergrad math major, or a grad student. So, your assumptions about what the OP means can't be verified without the OP's added context. :-) – amWhy May 21 '20 at 18:38
  • @amWhy I appreciate what you are saying, but allowing only rigorously definable wording in the questions kind of restricts the value that can be gained by asking and sharing questions. There is much intuition in maths that cannot captured precisely. I doubt that simple/elegant can be defined satisfactorily and yet, between the lines, it is clear what is meant in the sense that for any potential answer it will almost certainly be clear whether these are satisfied. As for the numerical proofs in the link, they are not necessarily what I'd call elegant, but I agree that this is debatable. :) – Damian Reding May 21 '20 at 18:39
  • Then: Please edit your post, @Damian, to pin down the level of study at which you ask this question (please see my most recent comment prior to this) and provide more details of what you want in an answer. And also include your own efforts or research in attempting to answer the question. Now I am repeating myself, though. :-) But do not do that in a comment to me. Do that as an edit to your post. – amWhy May 21 '20 at 18:40
  • @nicomezi I'll defer to the OP to edit the post to clarify. If the edit improves the post and explains why the linked question is not sufficient, I'll be the first to vote to reopen it. But you're just speculating. – amWhy May 21 '20 at 18:44
  • @amWhy What would you like me to edit? I do not really see the need for additional clarifications, but since the question is closed anyway now, it doesn't matter. For clarification though, a while ago I posted the following question and I believe that a similar case of inprecision could have been made, but then an answer hit the nail on the head. https://math.stackexchange.com/questions/3222702/quartic-as-a-product-of-quadratics – Damian Reding May 21 '20 at 18:48
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    You should include more info about the required level or things you would define as elegant (such as not computing approximations). @DamianReding – nicomezi May 21 '20 at 18:49
  • A closed questions can be reopened, after your efforts to edit it to include more context and clarification, Damian. I agree with @nicomenzi's most recent comment! (+1) – amWhy May 21 '20 at 18:51
  • Are series considered as elementary analysis for you ? You should clarify this point, if yes there is indeed already an answer in the linked question. @DamianReding – nicomezi May 21 '20 at 18:58
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    They are, but they are also numerical approximations to me. I begin to realize that I may be asking for too much though and that they may be the best that's available, so not insisting on reopening the question. – Damian Reding May 21 '20 at 19:01

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Find a shape which contains a circle of diameter $1$ and has perimeter $\sqrt{2}+\sqrt{3}$. By some isoperimetric inequalities, this shows that $\pi < \sqrt{2} + \sqrt{3}$.

Bob Krueger
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