How was the irrationality of pi proven?

[Nerditude]

Here's a question. Pi is said to be the ratio of a circle's circumference to its diameter. If this is the case, what does it say about the circumference of a circle that pi is still irrational.

I get that pi is also used in the calculation of a circumference in the first place. Since this is true, it has to be the case that pi has been proven to be irrational independently of its definition.

So, the question is, how did we prove pi is irrational?

 

[Anti-Crackpot]

So, the question is, how did we prove pi is irrational?

Nothing I can tell you that a quick Google search won't tell you. But one thing you might want to be aware of is that pi is also transcendental, meaning that it is the solution to no polynomial equation, or more precisely, "that is not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients." See: http://en.wikipedia.org/wiki/Transcendental_number

This, in contrast, for instance, to the Golden ratio, which is irrational, but not transcendental.

Incidentally, here is a link to another thread on this forum where the same issue was discussed at some length:

Looking for "Easy" proof of Pi Irrational
https://www.physicsforums.com/showthread.php?t=8193

As for the below question...

Pi is said to be the ratio of a circle's circumference to its diameter. If this is the case, what does it say about the circumference of a circle that pi is still irrational.

It's kind of mind-bending, but if you were to snip a perfect circle and stretch it out into a straight line along an x-axis demarcated into measurement increments as small as you please, then, while that line it would be "this long" and no longer, or "this short" and no shorter, well, good luck being able to locate, or rather, specify, where exactly the endpoint of that line is relative to your 0 point.

 

[Number Nine]

Halls of Ivy mentioned the following theorem in the above linked thread...

Of course, that requires that one know the theorem:

Let c be a positive real number. If there exist a function, f, positive on the open interval from 0 to c, continuous on the closed interval from 0 to c, and such that all its anti-derivatives can be taken to be integer valued at 0 and c (by appropriate choice of the "constant of integration") then c is irrational.

The proof of that is elementary but long.

Does anyone know where I might find a proof? That seems like a wonderful theorem.

 

[SteveL27]

Halls of Ivy mentioned the following theorem in the above linked thread...



Does anyone know where I might find a proof? That seems like a wonderful theorem.

I just Wiki'd this.

http://en.wikipedia.org/wiki/Proof_that_π_is_irrational