dsgrue3 said:
Zkuq said:
dsgrue3 said:
I don't want to discuss this any further. It isn't helping with the concept of the OP that 0.999... = 1, which I fully agree with. I just disagree with the initial "proof."
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The initial proof has no errors in it, it's standard equation manipulation and if we interprete your 'counter-example' your way, I'm fairly scared about the consequences to solving equations in general. Your 'counter-example' (or rather, your interpretation of it) pretty much says that whenever we modify equations, there's a chance we screw up just by modifying the equation even if all steps are correct. To me, it seems Jaydi is right here.
And besides, this whole thing is on pretty shady ground in the sense that you should be able to point out the error in the original proof considering it's so simple. For example, in 1=2 "proofs" you can find a step where the equation was multiplicated or divided by zero, but in this case, I doubt you can find a single step where an error is made. This explanation doesn't exactly prove anything but it should make you think if you've done something wrong when you can't actually find the error in such a simple process. And then there's the fact that two seemingly different things are often equal, even if the seem unrelated to each other.
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This has been bothering me since I read the OP.
Seems like there should be an obvious flaw and something about going from 10x = 9.999... to 9x = 9 rubs me the wrong way. But when we simply do as he says and subtract x, it yields 9x = 9.999... - x, then substituting the x yields the 9. So everything seems okay, despite substituting on only one side of the equation, which seems incredibly odd.
I think the issue is with that step though, because:
9(0.999...) = 9.999... - x does not seem valid. If we substitute here we have 9.999... = 9.
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I fail to see how you get 9.999... = 9 if you substitute there. Substitution brings the equation to this form:
9(0.999...) = 9.999... - 0.999... = 9,
from which we get (divide by 9) 0.999... = 1. Right?
