Although I can’t really advance the level of discourse here, which is admittedly impressive, I thought I’d throw in a couple of definitions for anyone who’s reading this and scratching their heads.
“Irrational” refers to a nonterminating decimal, and nothing more. 2/3, for example, translates to 0.66666666666… forever. That’s “irrational.”
“Transcendental” refers to an irrational number that cannot be expressed using a polynomial equation with rational coefficients – or, in English, a simple (terminating) equation with regular numbers. (In other words, the well-known 1/2 + 1/3 + 1/4 + 1/5 +… 1/n doesn’t count.) 0.66666… is therefore not a transcendental number, because it can be expressed as 2/3.
The two best-known transcendental numbers are e and pi, although some dude once proved that there are an infinite number of them. Oh, by the way, seeing that pi is transcendental – if you learned in school that 22/7 equals pi, be aware it’s an approximation that’s certainly close enough for most real-world purposes, but it ain’t pi; it diverges after just a few decimals.
Anyway, that’s my meager contribution to a fascinating thread. That’s what I love about the SDMB – whether the topic is math theory or toejam, the discussion will be enlightened and entertaining. It’s a beacon of rationality amid the sea of pseudointellectual detritus that comprises the rest of the 'net…
“I’m not an actor, but I play one on TV.”