You'd probably need to watch Numberphile's episode on the Infinite Orchard problem to really get the gist, but here's the crux of it.

We're going to need to plot an infinite graph, with x and y axes. The intersection of the two axes will be (0,0), and we'll plot a point at every pair of integers. Now, if we draw a line from (0,0) to (1,1), and then extend it, we'll see that what we're doing here is drawing a line through every pair of integers that has a ratio of 1:1. If we draw the lin from (0,0) to (1,2), next point we hit will be (2,4), then (3,6), and this is a different constant ratio line, this time 1:2.

It doesn't take much to conclude that what we are doing when we draw a straight line through a point is drawing a rational number line.

An irrational number line will never hit a point, it will find a way through the infinity of points without directly hitting one. If the line hits a tree, you just found a way to express that ratio in terms of integers. The pi line will go close to (22,7), closer yet to (355,113), and closer yet to even more accurate approximations.

The phi line is the most interesting one though, since it manages to maintain the furthest possible distance from a point. Phi, aka the golden ratio, is the most irrational number, because it is the least well approximated. You'll need huge denominators to approximate phi to the same accuracy as 355/113 does for pi.

Anyway, it's this that I think the Cantor diagonal theory fails to realise... that irrational numbers have different levels of irrationality, and the more irrational a number is, the rarer it is.

But here's the kicker... the guy on Numberphile pulls a number pretty much out of his arse, with no explanation, which is a shame, but he basically says the probability of taking a random line of sight and hitting a point is 6/pi^2, which is a really interesting number for reasons you might appreciate. It's the reciprical of the inverse square infinite fraction. But I digress... if that's the probabiltiy of looking at a tree in an infinite orchard, then it's the same as the probability of picking a number at random, and it's rational. So the idea that "almost all" numbers are irrational seems to me very, very wrong. Actually most numbers are rational.

Tell me where I'm being dumb!