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This question already has an answer here:

The square $$\sqrt 2+ \sqrt{3} \approx 3.14$$ Is this a coincidence or is their some mathemtical significance?

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marked as duplicate by Asaf Karagila, Git Gud, JimmyK4542, Hakim, beep-boop Sep 12 '14 at 18:01

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ Just a coincidence. $\pi$ is transcendental, while $\sqrt{2}+\sqrt{3}$ is algebraic. $\endgroup$ – mweiss Sep 12 '14 at 17:56
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    $\begingroup$ $\pi$ is $not$ equal to $3.14$! $\endgroup$ – LearningMath Sep 12 '14 at 17:58
  • $\begingroup$ @mweiss: Great remark :-) $\endgroup$ – idm Sep 12 '14 at 18:01
  • $\begingroup$ This is the not a duplicate! The question linked to asked for a geometric explanation, whereas this one asks for any kind of explanation. There might be one from analysis that is more persuasive than the geometric arguments given in answer to the other question. $\endgroup$ – Dave Sep 12 '14 at 18:12
  • $\begingroup$ @Dave The OP wasn't specific, he just mentioned "some mathematical significance" and the question linked to presents answers fulfilling the needs of the OP. $\endgroup$ – Hakim Sep 12 '14 at 18:18
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I think that this and this are fairly relevant here. It is the case that $\sqrt 2 + \sqrt 3 \approx 3.14 \approx \pi$, but it is not the case that $\sqrt 2 + \sqrt 3 = \pi$.

You are falling victim to the "strong law of small numbers".

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