Why 0.999... doesn't equal to 1...

Everybody tries to convince that if you divide 1 by 3, you get 0.333..., which is repeating number.
If you add that number to itself, you get 0.666..., which is another repeating number.
This is when it get's interesting... One would expect that if you add these two numbers together, one would get 1 again, because (1/3)*3 = 3/3 = 1... But the real answer is 0.999..., yet another repeating number.

Let me explain where laws of mathematics go wrong...

If you think smallest and largest possible positive number, you have to first define what infinity is.

Infinity is defined as number that doesn't follow normal laws of mathematics. If you add together or multiply infinity with any number, you still get infinity, but if you subtract from or divide infinity, the result is undefined...

When I define infinity, I use formula: infinity = 10^n-1, because if you forget that -1, you can always multiply by any number larger than 1 to get larger number. The first digit of infinity has to be 9, which is the largest of the digits in base 10. n here is actually the undefined part, as there is no known rational number that satisfies the formula.

When I define smallest possible positive number, I use formula 1/(infinity+1), or 1/(10^n)... One has to add 1 to infinity because otherwise the result would be repeating number, which would make it possible to have smaller number, as one can always truncate repeating number to get smaller number, repeating fraction never ends with digit 0.

To sum things up, number 1 subtracted by repeating number 0.999... is the smallest possible positive number, or result of f(n)=1/(10^n) that satisfies 1/(infinity+1).