.9... =1 (Revisited)

I’ve read the past threads on .9… equaling one and followed them pretty much, and thought I understood the deal (board consesus, they are are the same).

Unfortunately, I recently finished “The Ghost From The Grand Banks” by Arthur C. Clarke, where during the story he alludes several times that -1.9… is not the same as -2.

In fact, in the “Sources and Acknowledgments”, he explicity says:

Now I don’t drool when I talk, but I ain’t Arthur C. Clarke, either. And if he’s saying they ain’t the same, we’ll I’m confused again.

Is .999… = 1 under all conditions, for all time for ever and ever no matter how nitpicky you wanna be regardless of the equation you are plugging it into?

(I say that because the big part of the the story was the Mandelbrot set, and perhaps that makes a difference? - I’m not a mathematician so forgive my ignorance)

Let X = .99999…


10X = 9.99999…

10X - X = 9.99999… - .99999…

9X = 9

X = 1


X = 1 = .99999…

QED

Yep. At least, given the common definitions of “=” and “…” used in mathematics. I suppose it’s possible to define them such that it’s not true, but I don’t know that there’s any reason that anyone would want to do this.

And also, -1.999… is equal to -2. I don’t know what Clarke was talking about.

I haven’t read the story, but it sounds to me like the distinction Clarke is making concerns the difference between the interval (-2,x) and the interval [-2,x); the former includes numbers just larger than -2, but not -2 itself, the latter is the same interval, only it actually includes -2.

It sounds to me like in Chapter 18, Clarke was saying that the set (-2,0) is part of the M-set, and that the M-set does not include -2. In “Sources and Acknowledgements” it appears he’s correcting that, and stating that -2 is in the M-set, after all (which it is, in fact).

It looks like he’s a little confused on what the left endpoint of the interval of (-2,x) is. I think he’s saying that the left endpoint of (-2,x) is -1.99…, while the left endpoint of [-2,x) is -2. This is true, actually, but is no different from stating the fact that the left endpoint of both intervals is simply -2 (since -2=-1.999…)–it’s just that -2 is omitted from one of the intervals (but it’s still the left boundary of that interval). It’s misleading of him to refer to the left endpoint of (-2,x) as -1.999…, while referring to the left endpoint of [-2,x) as -2; the left endpoints are, after all, the same, single value, the distinction is that one interval includes -2, one doesn’t, it has nothing to do with repeating 9’s whatsoever.

So the bottom line is that the interval (-2,x) is different from the interval [-2,x), but the left most boundary in either case is -2, which is equal to -1.999…, after all. I hope all that made sense, I think it was probably more verbose than necessary, looking back at it.

Libertarian’s proof is right. Alternatively, once can define 0.9999… as the limit of the sequence:

.9, .99, .999, .9999,…
That is, term A[sub]n[/sub] = 1 - 10[sup]-n[/sup]

Then it’s straightforward to show that for all epsilon there exists an N such that n > N implies
1 - A[sub]n[/sub] < epsilon.

(Just choose N such that 10[sup]-N[/sup] < epsilon.)

The proof Libertarian offers is the easiest one for a poop-slinging simian such as myself to follow.

It was frustrating to read through the story, because I kept asking myself why the hell Clarke kept on remarking on the difference between -1.999… and -2.

In fact, when I came across the first mention, I assumed he was going to explain they were the same and got all excited because I was smart for once!

You know, I distinctly remember getting taught Libertarian’s proof of this issue when I was in Junior High.

Unless you go beyond the standard real numbers (perhaps to something like Abraham Robinson’s non-standard analysis), there is no difference between -1.999… and -2. While it’s possible that Clarke meant to allude to something like Robinson’s infinitesimals, I don’t think that’s likely. I think that Clarke’s just wrong here. Unfortunately, The Ghost from the Grand Banks is one of the few Clarke books I don’t own.

I agree with Wendell, Arthur C. Clark is wrong. By any mathematical definition of -1.9999… it is equal to negative two. -1.999… is not in the set (-2, x).

You could also prove this with limits.

lim -2+({10^-x})
(x->infinity)

is equal to 2.

-Soup

From a site as dedicated to the fight against ignorance as this one . . .

A mess of proofs.

Including the ones already presented, of course.