0.9_ = 1?

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look up non-standard analysis and hyper-real numbers.

ruveyn

Sounds like a least upper bound problem to me...but what do I know? Nothing yet...I will soon, though.

if 0.9_ = 1, then there would exist a set that is open and bounded? This means that a limit point would forever exist? (Given that we have a metric space) Hmm... can that be possible? I know that a closed set is bounded (look at Rudin's famous book for a proof)

Please don't respond to this, as I'm just another idiot.

I thought this was a new thread, but it is the same thread from before. Is it really all that paradigm-shifting?

The way I see it is the following: If two numbers stray off by only a marginal gap, then the newer number is NOT equivalent of the original number. The way to solve this dilemna, I suppose, is that instead of using decimals to represent thirds, one should write out thirds as fractions.

1/3 = 0.33_

2/3 = 0.66_

3/3 =/= 0.99_

Obviously, we must represent the latter-most as a fraction to prevent what I might call a 'numeral paradox' from happening.

-TiureJabba

There is no "numeral paradox", though. Implicit in what you say is an assumption that every real number has an unique decimal representation, which isn't true. Implicit in 0.9_=1 is proof that there exist real numbers with more than one decimal representation.

.99999999 * 10 = 9.999999
9 * .9999999 = .99999999 * 10 - .9999999 * 1 = 9.999999 - .9999999 = 9
9 * .9999999 = 9
1 * .9999999 = 1
.9999999 = 1

Nope.

ruveyn

I've used a sort of modular arithmetic to calculate digits more than a googol from the decimal point and it matters in my application a lot, whether there be nines or zeros there.

Think of the difference in binary:
can 0.11111111111111111111111111111111111111111111111111111111111
be the same as
1.000000000000000000000000000000000000000000000000000000000000

NO! They are the exact opposite. If it was a gray picture file, one would be a
black picture with a white dot and the other would be a white picture with
a black dot, and the same could be true if you made the binary into a
decimal by multiplying it by 9.

But as sound files, they would all sound the same and not be copies.

Agreed. Of course, you must be a perfectionist not to onsider it the same because it ultimately becomes so close to 1 it's totally irrelevant for any purposes to suppose it's anything other than 1, but theoretically, yes, 0,9_ is 0,9_, not 1.

How can a countable sequence have an infinite number of zeros preceding a terminating 1. A countable infinite sequence of ordinal omega does not have a terminating element. The expression 0.0_1 is meaningless.

ruveyn

OK.

It makes no sense to try to make 1 out of infinity 9's.

"0.infinity zeros and then a 1" makes no sense.

It only makes sense if you know how many 0s or 9s there are,
and you can have as many as you want.

It does make sense for fractions like 1/3=0.33333333_
because if you do a long division it will make 3's forever unless
you stop and say the remainder is 3. If the remainder was 0 then
it wouldn't be 1/3 it would be 33333333/10000000 = 0.33333333 exactly.

Wrong. The limit of the infinite sequence

.9, .99, .999, .... .999999...9 (k 9s), .... is 1. This sequence is the sequence of partial sums of the infinite series:

sum (k = 1 to infinity) [9/10^k] = 9*(1/10 + 1/100 + 1/1000 + ...) =
9 * 1/9 = 1.

I leave it to you to prove sum (k = 1 to infinity) x^k = x/(1-x) where x < 1.

ruveyn

Last edited by ruveyn on 21 Aug 2009, 7:49 pm, edited 1 time in total.

duplicate deleted

Last edited by ruveyn on 19 Aug 2009, 8:42 am, edited 1 time in total.

um... infinity... means... forever until you stop?... and you can have as many as you want but you have to know how many their are...

...and that makes sense?

.9_ does not equal 1

You can debate it all you want to. And people surely will long past we are all dead and buried. But it doesn't. It's close enough for any practicle and even most impracticle purposes. But, it simply isn't the same. It is the closest number there is to 1, that is also less than 1. But it is not one.

And fyi. The following has several errors.

.9_ x 10 = 9.9_

9.9_ - .9_ = 9

9 / 9 = 1

1 = .9_

I don't expect anyone to understand it, I expect you to assume that I am wrong. I dedicated nearly a month to the complete understanding of this, and similar paradoxial math problems before I finally understood, in entirety. I've never been able to convey this understanding however....maybe I need to make up some new math functions >.<

Anyway, goodluck to anyone who wishes to wrap thier head around it, it's a toughy.

People have understood convergence of infiite series since the early 19th century.

ruveyn