0.999999999.... = 1? Watch

JMonkey
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#201
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#201
Lol thanks for the neg Franklin and for signing it. Sincerely. I think you'll note I had cut my losses long before you negged me, but ah well, such is life. That's a mighty twitchy trigger finger you have there, perhaps you should try not being so judgemental in future. I have at no point been rude or not admitted my fault. But never mind, I actually prefer negs to positives. Let's face it all the positives I get are as unsigned and unmessaged as the negs, but at least you can work out who did the negs most of the time.

I think I know which direction I intend to go in with rep, it's the cool way.
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CHEM1STRY
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#202
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#202
Ah you mathematicians. No-one knows!
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Unbounded
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#203
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#203
Perhaps an argument which may have already been put forward, but here:
0.99999... is a geometric series:

 0.99999\cdots = 0.9 + 0.09 + 0.009 \cdots  = \displaystyle\sum_{n=1}^{\infty} 0.9 \times \left(\frac{1}{10}\right)^n = \frac{0.9}{1-\frac{1}{10}} = \frac{0.9}{0.9}  = 1

Side-note: Someone should write a wikipedia page on common myths or misunderstandings in maths, like dividing by zero, 0.999=1, square roots of negatives, and mad ideas of proofs that 1=2 and that lot, which involve the first topic mentioned usually..
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sonofdot
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#204
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(Original post by GHOSH-5)
Perhaps an argument which may have already been put forward, but here:
0.99999... is a geometric series:

 0.99999\cdots = 0.9 + 0.09 + 0.009 \cdots  = \displaystyle\sum_{n=1}^{\infty} 0.9 \times \left(\frac{1}{10}\right)^n = \frac{0.9}{1-\frac{1}{10}} = \frac{0.9}{0.9}  = 1

Side-note: Someone should write a wikipedia page on common myths or misunderstandings in maths, like dividing by zero, 0.999=1, square roots of negatives, and mad ideas of proofs that 1=2 and that lot, which involve the first topic mentioned usually..
http://en.wikipedia.org/wiki/0.99999999
http://en.wikipedia.org/wiki/Invalid_proof
http://en.wikipedia.org/wiki/Division_by_zero
:p:
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Oh I Really Don't Care
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#205
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#205
... lulz
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Unbounded
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#206
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#206
oh fair enough.
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Zhen Lin
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#207
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(Original post by JMonkey)
I see then perhaps the guy who made the preposition was being a bit disingenuous. It was the history of maths from ancient to modern times, was on BBC4. Quick google reveals it was called The Story of Maths. Obviously I can't argue the toss, so I'll bow to greater knowledge. Thanks for the information.
That would be Marcus du Sautoy, and my history of mathematics lecturer likes to say everything he says is wrong.
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Kolya
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#208
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(Original post by Zhen Lin)
That would be Marcus du Sautoy, and my history of mathematics lecturer likes to say everything he says is wrong.
What if du Sautoy was to say: something your history of mathematics lecturer says is right? :wink2:
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Zhen Lin
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#209
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#209
Uhu, but what if the law of excluded middle does not hold?
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JMonkey
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#210
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Can whoever negged me grow a pair in future. I'm getting tired of people taking pot shots at me, but not even having the courage to tell me why? Thanks. If your mummy says you don't have to sign it fine, but leave a message, it's not like I can even neg you back. I don't know bunch of worthless idiots use the rep system. waste of my time.

(Original post by jmonkey)
No I gave them a definition beneath, it's easier if you use some obscure symbol, that is allowed you know provided you clearly denote your terms beneath they can mean whatever you want.

And it is true that the infinite sum of fractions and the sum of integer numbers are equal. Plus it's also true that the decimals are a greater infinity. You probably need to study up on it or something? That was the whole basis of the introduction to the infinite sets, the proofs of fractions and so on.
Even better why not post your disapproval to me, and the rest of the world to show how much of a grown up you are. Big girls blowse, your mother was a warthog and your father smelled of elderberries, I fart in your general direction...
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DFranklin
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#211
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#211
In an effort to give this thread a chance of staying on topic, I've removed the last 10 posts or so. Please let's have no more throwing of insults and discussion of who neg-repped who. Thanks.

Edit: Make that the last 15 posts. Carry on with this, and people will start getting warnings.
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acm345
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#212
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#212
Just bored so just gonna say the obvious, 1/9=0.11111111111.........., 2x1/9=0.222222222......., therefore 9 x 1/9 =0.9999999999........ and 9 x 1/9 is 9/9 is 1
Therefore 0.99999999999.......... =1
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Zii
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#213
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#213
I can't understand how people still can't see that 0.999... = 1. If you are one of these people, why not think about it this way instead?

Take the number 1. Take 0.9 away from it. You get 0.1. Take 0.09 away from this. You get 0.01. Take 0.009 away from this. You get 0.001. Take 0.0009 away from this. You get 0.0001. Keep going. You can see that what we're basically doing is:

1 - (0.9 + 0.09 + 0.009 + 0.0009 + ...).

Now, the use of the ... does not imply that there is "always" one more number to take away. The use of the ... does not imply that you are "very close but not quite there". The ... means we ARE there, we ARE in the limit, and as we can clearly see, we get 0.000...000...000... ... ... ... . Zero.

You've seen all the rigorous proofs in this thread; I know this is not one. But no-one can argue that 0.000... does not, in fact, equal zero. There is not an "infinite" amount of zeroes, and then a one. There are just zeroes. So we get:

1 - (0.9 + 0.09 + 0.009 + ...) = 1 - 0.999... = 0.000... = 0

which means that

1 = 0.999....

They are one and the same.
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Creole
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#214
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#214
You could also argue the classic, "well if they aren't the same number, find me a number that's larger than one and smaller than the other".
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majikthise
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#215
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#215
Or even better, the classic *close thread*
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Planto
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#216
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#216
The reason 0.999... = 1 is because 0.999... does not really exist. It's the result of a flaw in the decimal system. When you look at it this way, it's barely worth considering.

Also, this thread is old.
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The Bachelor
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#217
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(Original post by Planto)
The reason 0.999... = 1 is because 0.999... does not really exist. It's the result of a flaw in the decimal system. When you look at it this way, it's barely worth considering.

Also, this thread is old.
Of course it exists, if you define a decimal number as a sum. I.e ...a_2a_1a_0.a_{-1}a_{-2}... is defined as \displaystyle \sum_{n \in \mathbb{Z}} 10^n a_n.
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JMonkey
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#218
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I hate these threads.

Infinity does not really exist physically as far as we know it is a concept based on the set of everything of which nothing is divisible or additional to or can be subtracted from or U.

Taking an apple away from the Universe is impossible but if you could and there was a multiverse, then you have the Universe as a set of possibly infinite Universes called Big U or somit.

Everything, get it all there is or ever can be. That's what we mean by infinite. Not God, not your deranged imaginings after 10 pts of special brew and a stomach pump, just \infty unbounded and beyond anything.

Therefore by axiom anything that approaches it infinitely, cannot be smaller than it's limit, therefore its extent of .999... never ends, therefore by definition it = it's limit 1. It's really not that hard, however some people think that they can play fast and loose with axioms because they are special. Yes you are special: needs.

If you don't grasp why that axiom is allowed in maths, then take up shrubbery and leave the maths to the people who can grasp the more advanced numerical concepts.

.999... exists, the concept of infinity exists. Get over it.
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JMonkey
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#219
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http://en.wikipedia.org/wiki/Cardina..._the_continuum

This might be of interest to some of the more pointy headed pointy folk.

Uncountability

Georg Cantor introduced the concept of cardinality to compare the sizes of infinite sets. He famously showed that the set of real numbers is uncountably infinite; i.e. {\mathfrak c} is strictly greater than the cardinality of the natural numbers, \aleph_0:

 \aleph_0 < \mathfrak c

In other words, there are strictly more real numbers than there are integers. Cantor proved this statement in several different ways. See Cantor's first uncountability proof and Cantor's diagonal argument...
...The famous continuum hypothesis asserts that {\mathfrak c} is also the second aleph number \aleph_1. In other words, the continuum hypothesis states that there is no set A whose cardinality lies strictly between \aleph_0 and {\mathfrak c}

\not\exists A : \aleph_0 < |A| < \mathfrak c

However, this statement is now known to be independent of the axioms of Zermelo–Fraenkel set theory with the axiom of choice (ZFC). That is, both the hypothesis and its negation are consistent with these axioms. In fact, for every nonzero natural number n, the equality {\mathfrak c} = \aleph_n is independent of ZFC. (The case n = 1 is the continuum hypothesis.) The same is true for most other alephs, although in some cases equality can be ruled out by König's theorem on the grounds of cofinality, e.g., \mathfrak{c}\neq\aleph_\omega. In particular, \mathfrak{c} could be either \aleph_1 or \aleph_{\omega_1}, where ω1 is the first uncountable ordinal, so it could be either a successor cardinal or a limit cardinal, and either a regular cardinal or a singular cardinal
What happens when you let maths geniuses at maths, and then maths geniuses at maths geniuses.

Oddly enough I think I agree with Cantor, only because that way it makes infinity more than just a boring axiom without sets. But it's really only an academic exercise in figer twiddling so meh who cares.

See also:

http://en.wikipedia.org/wiki/Continuum_hypothesis
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The Bachelor
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#220
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What do you mean you "agree" with Cantor?
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