[rcxdude]

Assuming the length of the rod is 1m, and you have a resolution of one plank length, there are 10e35 possible ratios you can express (because that's the number of possible locations of the notch). that's about 2^113, which is a number which fits in 15 bytes of information. As discussed below, if you also allow for the bar to be the size of the observable universe, this doesn't increase by much. A notch or ratio is linear, information and combinatorics grows a lot lot faster than that.

[jtrip]

How would you say a rod to notch ratio of 22/7 compares? How much information would you say that has?

[rcxdude]

Meaningless on its own. What matters is how many distinct ratios are possible.

[stevenhuang]

> Meaningless on its own.

That's the crux. What we're saying is that, of these set of ratios, there exists one whose infinite decimal representation contains our intended data. Yes we'd be limited to Planck length resolution, but the idea is that we would determine such a ratio and construct the notch and bar in such a way that it yields the decimal sequence desired; the # of particular ratios is not relevant.

I wouldn't be surprised if it can be proven this can't be done, but then that would be proper question to ask.