except that it makes logical sense, unlike whatever you people use.
and yes i know the subtraction in that pic is totally wack. it's late and i don't feel like fixing it
the pic just illustrates a point
 nm better edit before **** ppls cry
Imagine a square with side length 1. The perimeter is 4.
Now remove the top right portion of the square, removing 25% of its total area. The perimeter is still 4.
For each of the two points you have, remove a similarly proportioned square. The perimeter is still 4.
You can continue on for as long as you like, and the perimeter of the polygon never changes. It is still 4.
However, as soon as you cut it infinitely many times, you are left with simply an isosceles triangle with sides equal to 1, which as we know has a hypotenuse of length sqrt(2), which means the perimeter is 2 + sqrt(2), which is most certainly not 4.
I have attached below a reason why I am an engineer and not an artist.
The black lines are the original square.
The red box is what you remove for the first iteration.
The green boxes are what you remove for the second iteration.
The blue boxes are the third, and so forth...
Until you reach your state at infinity, which is the purple line.
Had I drawn this perfectly to scale, none of the box's edges would have crossed the purple line (in fact their bottom left corners would all be exactly touching it).
The .9... question is fun because the question is very simple for people to understand. But it actually requires quite a hefty amount of math to really believe. You need to understand...
1. That numbers with an infinite number of decimal places are just as real and valid as any natural number.
2. That you can perform all the same arithmetic operations on these numbers, including such strange-looking operations as .9... * 10 = 9.9... and 9.9... - .9... = 9. These are true, but they are admittedly odd, and you are right to question their legitimacy.
3. That patterns (more formally "invariants") do not hold at infinity sometimes, and that these are on a case-by-case basis.
4. That infinite series are not a process. There are no "steps" to infinite summations, the entire statement is exactly equivalent to its end result at all times. There is not a time when .9... is .9, and a time when it is .99, and then .999, and .9999, and so on. The "process" is how we think about it sometimes, but that doesn't mean the number itself is one.
...and probably more that I'm not thinking of. It's a complicated problem even though its proofs are relatively short. It's why it sparks so much debate--because everyone can pitch in.
And that, to both trolls, and to people who love math, is really pretty fun.
Edit: Fixed my silliness.