Posted by: Dave Richeson | February 11, 2010

Proof that pi is irrational

Have you ever seen the proof that $\pi$ 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.

Responses

1. 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.