This argument more or less assumes its conclusion; after all, if it weren’t the case that 1 − 0.999… were zero, then it would be some positive number x, so you could pick epsilon = x.
Again, that’s assuming the conclusion; what if 1 − 0.999… weren’t zero, and you picked that as epsilon? You’re skipping steps. It’s worth writing down exactly what you think is happening more carefully.
(To be clear, I’m not claiming that you’ve asserted any false statements, but I think there’s an important sense in which you aren’t taking seriously the hypothetical world in which 1 − 0.999… isn’t zero, and what that world might look like. There’s something to learn from doing this, I think.)
Alice: 1 = 0.999... Bob: No, they’re different. Alice: Okay, if they’re different then why do you get zero if you subtract one from the other? Bob: You don’t, you get 0.000...0001. Alice: How many zeros are there? Bob: An infinite number of them. Then after the last zero, there’s a one.
Alice is right (as far as real numbers go) but at this point in the discussion
she has not yet proved her case; she needs to argue to Bob that he shouldn’t
use the concept “the last thing in an infinite sequence” (or that if he does
use it he needs to define it more rigorously).
you aren’t taking seriously the hypothetical world in which 1 − 0.999… isn’t zero
In this (math) world it is zero only because for every nonzero positive epsilon, you can pick a FINITE number of 9s, such that 1-0.999999...99999 (a FINITE number of 9s) is already SMALLER than that epsilon.
For EVERY real number greater than zero, you have a FINITE number of 9s, such that this difference is smaller.
Therefore the difference cannot by a number greater then 0.
For every epsilon greater than zero, the difference 1-0.99999999… is even smaller. Smaller than any positive number.
Then, if it’s not negative, then it’s zero. This difference is zero.
This is the most correct way to put it, I believe.
Yes. Still this is the concept of limits and it is a significant step for most people. I think the most common first reaction is “Huh?”.
But people will make the effort if you explain this is a solution to the mysteriousness of “infinitesimals”.
This argument more or less assumes its conclusion; after all, if it weren’t the case that 1 − 0.999… were zero, then it would be some positive number x, so you could pick epsilon = x.
And in certain constructions, epsilon is a distinct number—so it’s actually fallacious without going back to the definitions!
No, it does not!
Whatever epsilon you might choose, you can easily take enough 9s (nines) after the 0. - to have the difference smaller than this epsilon of yours.
Again, that’s assuming the conclusion; what if 1 − 0.999… weren’t zero, and you picked that as epsilon? You’re skipping steps. It’s worth writing down exactly what you think is happening more carefully.
(To be clear, I’m not claiming that you’ve asserted any false statements, but I think there’s an important sense in which you aren’t taking seriously the hypothetical world in which 1 − 0.999… isn’t zero, and what that world might look like. There’s something to learn from doing this, I think.)
If I may, let me agree with you in dialogue form:
Alice: 1 = 0.999...
Bob: No, they’re different.
Alice: Okay, if they’re different then why do you get zero if you subtract one from the other?
Bob: You don’t, you get 0.000...0001.
Alice: How many zeros are there?
Bob: An infinite number of them. Then after the last zero, there’s a one.
Alice is right (as far as real numbers go) but at this point in the discussion she has not yet proved her case; she needs to argue to Bob that he shouldn’t use the concept “the last thing in an infinite sequence” (or that if he does use it he needs to define it more rigorously).
There is no “after the last” zero.
In this (math) world it is zero only because for every nonzero positive epsilon, you can pick a FINITE number of 9s, such that 1-0.999999...99999 (a FINITE number of 9s) is already SMALLER than that epsilon.
For EVERY real number greater than zero, you have a FINITE number of 9s, such that this difference is smaller.
Therefore the difference cannot by a number greater then 0.