> Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's

> Case in point, take a unit square, it will have a diagonal who's digits have a long "infinite" string of digits but sqrt(2) isn't "infinite information", it's very finite (ie., one number!) It's tricky to talk about "information" when using arbitrary human conveniences, like decimal arabic numerals, etc. In particular: - The side length of a unit square also has infinite digits: 1.00000... - If adding trailing zeros seems superfluous, how about writing the side length as 0.9999...? - The diagonal has length 1 on a ruler which measures sqrt(2)ths - The diagonal length can be written very compactly as 'x | x^2 = 2', and there's no reason a numeral system couldn't represent that "natively" A lot of definitions of "information" (e.g. Shannon information) depend on identifying an element out of a set. Such definitions break down when we're dealing with constants, since e.g. sqrt(2) couldn't be anything else. It's like trying to apply computational complexity to a fixed input, e.g. it only takes constant time to sort a particular list, or to find the shortest path for a particular travelling salesman, etc. To avoid degenerating into zero, there needs to be some set/space of possibilities which all need handling. The author's definitions seem to do this: e.g. if we're considering the set of all possible real numbers, then they do indeed "contain infinite information"; or, a phrasing I would prefer, they would require an infinite amount of information to distinguish them exactly.