What's your favourite paradox?

[aviator]

Right, here's a more visual representation of what he has done:


The last line shows 3 different ways of expressing the exact same number: 0.999..., 9/9 and 1.

Another way of expressing 0.999... or 0.9 recurring is by the following infinite (and converging sum), which is equal to 1:


The above notation effectively means: 0.9+0.09+0.009+... going on forever.

I'll show you that the above sum is true as it is a geometric series, so the sum to infinity is:

where a is the first term (in this case 0.9) and r is the common ratio (what we multiply to get the next term) which is 0.1 in this case.
When we apply the formula:

Hopefully this clears it up a bit more.