Have you ever seen the proof that is irrational? If not, I highly recommend heading over to The Math Less Traveled. Blogger Brent Yorgey just posted the last of his six part series in which he gives Ivan Niven’s easy-to-follow 1947 proof of that famous fact. The proof uses only basic calculus.
While trying to understand Niven’s proof that pi is irrational, I decided to back up and refresh my memory on some more basic trigonometry and I stumbled on this statement at: http://www.themathpage.com/aTrig/circle.htm
“It should be intuitively clear that π cannot be a rational number, because it indicates the ratio of a curved line to a straight. And to name such a ratio exactly is impossible.”
This is not at all intuitively obvious to me. In fact, it seems to me that the statement above is obviously incorrect. If I take a straight line and bend it, the ratio of its bent length to its straight length will still be 1.
David Richeson: Division by Zero 
I’m retired and my math chops are pretty rusty.
While trying to understand Niven’s proof that pi is irrational, I decided to back up and refresh my memory on some more basic trigonometry and I stumbled on this statement at: http://www.themathpage.com/aTrig/circle.htm
“It should be intuitively clear that π cannot be a rational number, because it indicates the ratio of a curved line to a straight. And to name such a ratio exactly is impossible.”
This is not at all intuitively obvious to me. In fact, it seems to me that the statement above is obviously incorrect. If I take a straight line and bend it, the ratio of its bent length to its straight length will still be 1.
Am I missing something?
Dear Sir:
By this time, you have probably had several replies. When a line (segment) is bent, its shape changes. Tape-measures are useful.