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Why do we consider $\pi$ as a irrational number? Why is that? We all know that $\pi$ is the solution of circumference / diameter of a circle and there could be infinite amount of circles which can have circumference and diameter in the form of positive integers? Then why is that?

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closed as off-topic by Eevee Trainer, José Carlos Santos, Javi, Joel Reyes Noche, Yanko Apr 24 at 12:57

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    $\begingroup$ Because someone proved it. $\endgroup$ – Mauro ALLEGRANZA Jan 24 at 10:31
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    $\begingroup$ We consider $\pi$ as an irrational number because it is an irrational number. $\endgroup$ – Joel Reyes Noche Jan 24 at 10:34
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    $\begingroup$ Can you recall what an irrational number is? What is your definition of a circle? What do you call consider in mathematics? Can you give an example of a circle having integers for both circumference and diameter? $\endgroup$ – mathcounterexamples.net Jan 24 at 11:34
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You say that

"there could be infinite amount of circles which can have circumference and diameter in the form of positive integers".

On the face of it, there could be. But it turns out that there is no circle which has both circumference and diameter in the form of a positive integer!

A first step in proving this, is to prove that the ratio of the circumference to diameter is the same for all circles. This ratio is called $\pi$. It has long been proven that this ratio is irrational, though there is no proof (yet) that is as elementary as the definition of $\pi$ itself. See the Wikipedia page for a few examples of proofs.

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