[split] 0.999... equals 1

[edk]

That is why infinity is a problem. In scientific notation one can show these infinitely long strings (including the next number up or down), but they are all still considered to be equal to 0.

[1 ≤ |x| < 10] xE-infinity=0

Infinity - x, where x is any number still equals infinity. Any attempt to use infinity in an equation results in invalid answers.

In the equation you posted, f(0.999...) is also undefined.

Using literal infinity sometimes results in invalid answers, in the same way using zero sometimes does. In general it is fine as a coefficient but you can't reduce an equation with it because all you prove is that infinity == infinity. By your logic, the square and square root operations are invalid: (-1)^2=1 => sqrt((-1)^2)=1 => -1=1

I think you may have missed my point that the infinite zeroes thing precludes the difference between 0.99... and 1 from being a number (by your definition of 0.99...). This means we can add an irrational number and something that isn't a number at all and come out with a rational number. Ignoring this though, would you agree that if we scale this up by, say, a billion, (10^9)*(0.0...1) + (10^9)*(0.99...) = 10^9? If so, either the infinitesimals are different sizes (a contradiction in terms) or 0.99... is not consistent with the rules of multiplication.

Also: What is 0.99...^2?

But we're not dealing with infinity here, we're dealing with infinite repetition, in this case of a single digit. You clearly understand that repeating decimal representations where the repeating unit has a finite number of digits evaluate to a rational number, so why there is a special case for when all the digits are equal is beyond me. I demonstrated a method for finding the rational number that's equal to a given repeating decimal.


It's clear that 0.900900900900... =900/999; 900.900900... - 0.900900 = 999(0.900...) = 900 => 0.900... = 900/999. It is easy to see how this can be generalised to a sequence of any number of zeroes; 0.900000900000... = 900000/999999, for example. When applied to 0.99..., a special case of the above where the length of the string of zeroes is 0, we see that 0.9... = 9/9. No infinity is required anywhere.