What if our numbering system was not based on 10?

[matt grime]

You've just introduced new and completely extraneous information into representatives of elements of an analytically constructed algebraic object. You don't feel just a little compelled to explain why it "works" in terms of the algebra and analysis of the reals? I don't think it does work, for what it's worth, and cannot see why you'd introduce the idea. I also don't agree with saying "infinity is defined with limits" since "infinity" is an ambiguous term that we ought to avoid using.

 

[StatusX]

I should have said infinitely repeating decimals. And as for algebraic rules: drop everything after the first repeating group and then proceed as normal.

So yes, there is extraneous information, and I certainly don't intend for anyone to use something like this for any practical or theoretical purpose. The only reason I mentioned 0.499...5 is because to a lot of people new to the subject, this seems intuitively to be half of 0.999..., and in a sense, they aren't completely wrong. If you think of 0.999... as the limit of the sequence {0.9, 0.99, 0.999,...}, then the sequence in my first post is obtained by halving each term. In a way, this sequence can be thought of as 0.499...5, and the limit of the sequence is 0.5. It is just another way of showing that the system is consistent; that whether you choose to use 0.999...=1 and half that to get 0.5 or do it the way I did, you get the same answer.

 

[matt grime]

"In a way, this sequence can be thought <WRONGLY> of as 0.499...5," my addition.

Why teach people bad habits and wrong maths?

 

[StatusX]

Ok this is my last try to get you to see what i was doing. Are you really suggesting there isn't an intuitive way in which the sequence {0.45, 0.495, 0.4995, ...} could be represented as 0.4999...5? Forget anything else, just think "representing the sequence." Of course there is. But when you see how the decimal notation is defined with limits, you see this is no different than 0.4999... (this is no more useful, by the way), which is no different than 0.5. Once they see that, they'll see why they should never use 0.4999...5 again, instead of just having to take a mathematicians word for it. But since no one seems to have benifited from it, forget I said anything.

 

[matt grime]

I see perfectly what you're doing, and it is wrong. Why teach something that is just plain wrong? It isn't intuitive to me since it creates new unexplained symbols in positions that clearly cause confusion. So don't teach them something that is wrong. If you wish to think of it in terms of limits of the finite examples then do it properly:

0.\underbrace{9 \ldots 9}_n
divided by two gives a number that differs from 0.5 by k*10^{-n} for some constant k, hence in the limit the difference shrinks to zero.

And then if your attitude is that they see why they shouldn't use it, why teach them to think in terms of it? Teach them the proper definitions and deductive reasoning and half the cranks who can't grasp these simple things wouldn't have a leg to stand on since no reasonably educated person would fail to see why they're wrong.

 

[StatusX]

You're right, I should have been more careful about it, and showed why their intuition of 0.499..5 is wrong without ever using that symbol myself. Instead, I should have just referred to the sequence and talked about it's limit. In any case, it seems all the confused people are gone, so this is all academic.

 

[ahrkron]

This seems a good point to close this incarnation of the .99... = 1 issue.
As usual, most probably it won't take long before it's reborn... oh, well.