180 Days of Ideas for Discussion in Math Class.
(as of 9July2014, we're in overtime!)
It's the ratio of two values, but they can't both be integers, which is the requirement for rational numbers. The problem is that they can't both be integers because one is the irrational multiple of the other.It's a very "Chicken or the egg" type of argument.
This is a great question to have students debate. If the whole class settled on Justin's response I'd have two ideas to play devil's advocate. First I'd say try it. Measure the circumference and diameter of a circle. (I see many "Pi Day" activities that suggest students do just that). No matter what they measure I'd be able to write the ratio they discover as a ratio of two integers. So pi must be rational. If some student or students was then able to convince the class that my argument was faulty because their measurements were only approximations, I'd counter that all measurements are approximations so how do we know that pi CAN'T be written as the ratio of two integers. I'm definitely going to try this next year when we start discussing pi.
Galois, when you do this you might also at some (later) point ask the same question about the length and width of their desk tops!
Check this linkhttp://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational