I've ended up in a debate on a proof for .999˜ being equal to 1 (in the real number system.) I think the arguments have boiled down to whether or not we can preform variable substitution on one side of the equation, while keeping that variable rather than substituting it on the other side of the equation.
What we are looking at:
Given: x = 0.999˜ (Using the ˜ symbol to represent repeating) .. A 10x - x = 9.999˜ - x 9x = 9.999˜ - x 9(.999˜) = 9.999˜ - (.999˜) 8.999˜ = 9 .. OR .. B 10x - x = 9.999˜ - x 9x = 9.999˜ - (0.999˜) // Preform a substitution on ONLY the RHS 9x = 9 x = 1 ..
What's so confusing is that both should be right...since 8.999˜ IS equal to 9. But, since this is a proof you should expect the same representation of the number at the most simplified step (I believe?)
So for simplicity sake my questions are...:
- Is the algebra valid in both examples A and B?
- Does example A act as proof regardless of giving different representations of the same number at the most simplified step?