What's your favourite paradox?

[aviator]

IT IS JUST A THEORY...

Well, first of, multiplying something by ten always requires the number to end with a zero, (this does not matter for now as you see in the example, just wanted to point that out)
If 0.999... = x
then 10x = 9.999...0 <- This zero is meaningless as it stands behind a comma
so 10x - x would result in (9.999...0 - 0.999... = 9)
9x = 9 so x = 1

But, doesnt multiplying by ten also mean that moving its content forward by one decimal? Meaning that,
10x = 9.999...0 <- On this xth position (the theoritically last position of this not ending number) is a zero
01x = 0.999...9 <- On the same position (the theoritically last position of this not ending number) must be a nine
so 10x - x would result in (9.999...0 - 0,999...9 = 8,999...1)
9x = 8,999...1 so x = 0,999...9

These are my thoughts about this topic, this is pure theory.

If you're interested, I've attached 2 links about this. These aren't exactly "rigorous" proofs, because if you were to do that you'd be getting into degree level maths. One of them touches on your point about multiplying by 10. Effectively, multiplying 0.999... which has an infinite number of nines by 10 to get 9.999... will still have an infinite number of nines. The first link touches on this. The second one is a little discussion involving infinite sums as 0.999... can also be represented by the infinite sum from 1 to +∞ of  9/(10)^n, the sum's limit is 1.

You can also follow the same method as before: 1/9=0.111... (x9) => 9/9=0.999... => 1=0.999..., albeit these do have assumptions, but a rigorous proof of this would be almost impossible for people without completing a university course in maths to understand.

https://www.physicsforums.com/insights/w...are-equal/
https://www.physicsforums.com/threads/is...-1.542160/