If you're having math problems, I feel bad for you, son...

[Joseph8th]

24. Nicely said... and...

... one might add, though, that transcendental irrationals are only ill defined when expressed as decimal expansions. Pi = C/d is completely precise expression of the abstract number Pi, but because it's irrational, no numerical expression is ever precise.

There's only a few irrational numbers out of an infinitude that have their own names. e & pi & phi are about it. The rest are things like the roots of primes and the like. Each is precise, though, in operation, since they are defined via operation.

That is, we should more properly write pi := C/d where the ":=" indicates definition of a map pi: +R --> R, and C and d are empirically measured quantities, so that's a separate issue of precision. Then, for instance, sqrt(pi) or 2*pi - 3 are both also precise irrational numbers, since we can get pi = (sqrt(pi))^2 precisely by operations.

Of course, we can always approximate pi by inscribing a circle of radius r inside a square, then throwing darts at random, and computing the ratio of darts in the circle only to the number in the square total, which turns out to be pi/r. Every way you cut it, though, the continued fraction or series expansion expression of an irrational number will be the most both precise and useful definitions, since they do not rely on empirical measurement, but were derived via other maths.