One of my twitter friends was complaining about something that's apparently making the rounds of Facebook for π-day. It annoyed me sufficiently to be worth ranting about a little bit.
Why isn't π rational if π=circumference/diameter, and both measurements are plainly finite?
There's a couple of different ways of interpreting this question.
The stupidest way of interpreting it is that the author didn't have any clue of what an irrational number is. An irrational number is a number which cannot be written as a ratio of two integers. Another way of saying essentially the same thing is that there's no way to create a finite representation of an irrational number. I've seen people get this wrong before, where they confuse not having a finite representation with not being finite.
π doesn't have a finite representation. But it's very clearly finite - it's less that 3 1/4, which is obviously not infinite. Anyone who can look at π, and be confused about whether or not it's finite is... well... there's no nice way to say this. If you think that π isn't finite, you're an idiot.
The other way of interpreting this statement is less stupid: it's a question of measurement. If you have a circular object in real life, then you can measure the circumference and the diameter, and do the division on the measurements. The measurements have finite precision. So how can the ratio of two measurements with finite precision be irrational?
The answer is, they can't. But perfect circles don't exist in the real world. Many mathematical concepts don't exist in the real world. In the real world, there's no such thing as a mathematical point, no such thing as a perfect line, no such thing as perfectly parallel lines.
π isn't a measured quantity. It's a theoretical quantity, which can be computed analytically from the theoretical properties derived from the abstract properties of an ideal, perfect circle.
No "circle" in the real world has a perfect ratio of π between its circumference and its diameter. But the theoretical circle does.
The facebook comments on this get much worse than the original question. One in particular really depressed me.
Just because the measurements are finite doesn't mean they're rational.
Pi is possibly rational, we just haven't figured out where it ends.
We know an awful lot about π. And we know, with absolute, 100% perfect certainty that π never ends.
We can define π precisely as a series, and that series makes it abundantly clear that it never ends.
That series goes on forever. π can't ever end, because that series never ends.
Just for fun, here's a little snippet of Python code that you can play with. You can see how, up to the limits of your computer's floating point representation, that a series computation of π keeps on going, changing with each additional iteration.
def pi(numiter): val = 3.0 sign = 1 for i in range(numiter): term = ((i+1)*2) * ((i+1)*2 + 1) * ((i+1) *2 + 2) val = val + sign*4.0/term sign = sign * -1 return val