[karmakaze]

I don't think "first forty digits of pi" should be admissible. As it depends on an external definition that's not computed by the representation. I could come up with a mathematical definition for any prefix of digits, give it a name and say the 'first x digits of C'.

[skybrian]

That's just because it's a loose description in English. It would actually be a short computer program that calculates pi, and instead of the first 40 digits you could ask for, say, a megabyte of data, or take the limit as the amount of data grows.

Transmitting a program to compute pi would be shorter than the data needed by any compression algorithm that isn't somehow based on knowing the trick.

The same trick could be used for any mathematically interesting number. The point is that incompressible random sequences exist that are not like that. You can't do better than transmitting the sequence itself.

[karmakaze]

Yes the point was merely 'referring to an external' isn't a good example of a minimal size description.

[alisonkisk]

Complexity is defined by size over an entire class of objects, not a single one. Pi is reused many many times at constant cost, wile your C becomes C2, C3, C4,...