.9999 ... = 1
there seems to be general consensus about this.
However i don't know if that proof in OP is valid.
Name:
Anonymous2007-02-18 17:39
So does this mean .7 repeating = .infinite 7s and then one 8?
Name:
Anonymous2007-02-18 18:04
Rather than saying "giving infinity a value," it's perhaps a bit
clearer to say, "giving the concept of a limit of an infinite sequence
of numbers a value."
.9 is not 1; neither is .999, nor .9999999999. In fact if you stop the
expansion of 9s at any finite point, the fraction you have (like .9999
= 9999/10000) is never equal to 1. But each time you add a 9, the
error is less. In fact, with each 9, the error is ten times smaller.
You can show (using calculus or other methods) that with a large
enough number of 9s in the expansion, you can get arbitrarily close to
1, and here's the key:
THERE IS NO OTHER NUMBER THAT THE SEQUENCE GETS ARBITRARILY CLOSE TO.
Thus, if you are going to assign a value to .9999... (going on
forever), the only sensible value is 1.
There is nothing special about .999... The idea that 1/3 = .3333...
is the same. None of .3, .33, .333333, etc. is exactly equal to 1/3,
but with each 3 added, the fraction is closer than the previous
approximation. In addition, 1/3 is the ONLY number that the series
gets arbitrarily close to.
And it doesn't limit itself to single repeated decimals. When we say:
1/7 = .142857142857142857...
none of the finite parts of the decimal is equal to 1/7; it's just
that the more you add, the closer you get to 1/7, and in addition, 1/7
is the UNIQUE number that they all get closer to.
Finally, you can show for all such examples that doing the arithmetic
on the series produces "reasonable" results:
Since:
1/3 = .333333...
2/3 = .666666...
1/3 + 2/3 = .999999... = 1.
By the way, there is nothing special about 1 as being a non-unique
decimal expansion. Here are a couple of others:
This is _the_ classic math troll. Just let the children play.
Name:
Anonymous2007-02-20 22:12 ID:SPnDISQ2
>>19
Please explain how the concept of an infinite geometric series is misusing infinity.
And you can take your shitty hotel metaphors with you when you GTFO.
Name:
Anonymous2007-02-20 23:38 ID:TfbhBCm1
Infinity is not a real number.
Name:
Anonymous2007-02-21 3:24 ID:D4Pcez+l
.99999 repeating is one there are proof's of it and even a wiki
my proof 1 is the same as 3/3 correct?
and 1/3 = .33333333333
and if we ad another 1/3 to get 2/3 that =.6666666 correct
so if we ad 1/3 or .3333repeating to 2/3 or .6666repeating we get both 1 and .99999 repeating
Name:
Anonymous2007-02-21 3:29 ID:D4Pcez+l
you think your so smart don't you, you dirty fucks with all you "fancy" words like "Calculus" "Maths" and "2"
QED. You guys fail to see that 0.999...∞ still misses the infinitely tiny fraction ...∞ to make it 1.
More evidence:
I have 0.999...∞, let me take off an infinitely small fraction ...∞
Now I have 0.999...∞ - ...∞ = 0.999...∞
As you can see, I have taken a number with an infinitely small fraction less than 1, took off an infinitely small fraction and still have the same number with an infinitely small fraction smaller than 1! Amazing, isn't it!
Like Hotel Infinity should make clear to you, you can't use infinity in regular arithmetic. 0.999...∞ repeating is a CONCEPT and the only thing you can derive from it is that it is smaller than 1. An infinitely small fraction smaller than 1, which is why we can only use it as a concept, and not in mathematical formulas.
>>30
Why should >>29 bother posting an argument? People have already presented dozens of arguments (which, unlike yours, are flawless and actually written by someone who has finished junior high) and you ignored all of them.
Name:
Anonymous2007-02-21 14:34 ID:tl5dVrYu
>>31
Flawless how? I pointed out exactly what flaw was made and provided proof for this reasoning. You disprove it, if you can. If this simply is not enough to explain things to you, I don't think you are capable of understanding the concept matter of infinity.
The problem is you are trying to put a finite point in your mind to an infinitely tiny fraction, which makes it not infinitely tiny anymore. As I explained, infinity is a concept, and if you don't understand that concept well you will make mistakes like "0.999... = 1"
>>32
You didn't point out a flaw, you pointed out that you have ABSOLUTELY NO FORMAL TRAINING IN MATH.
Name:
Anonymous2007-02-21 15:56 ID:yXM2Rzjs
>>No sir, it is you who does not understand the concept of infinity.
>Now I have 0.999...∞ - ...∞ = 0.999...∞
That is a flawed argument. You can't take sigma(9/10^n) from {n = 0 to ∞} and subtract 9/10^∞. hell, 9/10^∞ isn't even a number. What you can do though is take the limit as n approaches infinity of 9/10^n and subtract that.
sigma(9/10^n) from {n = 0 to ∞} - limit(9/10^n) as {n approaches ∞}
Unfortunately for you the limit as n approaches infinity of 9/10^n is ZERO, ZEROZEROZEROZEROZEROZEROZEROZEROZEROZERO!
In Effect:
sigma(9/10^n) from {n = 0 to ∞} - 0 = sigma(9/10^n) from {n = 0 to ∞}
Nice try dumbshit.
Name:
Anonymous2007-02-21 17:10 ID:C+qsu6k9
people should seriously start sdudying maths, calculus and analisys before demonstrating shitty things
Name:
Anonymous2007-02-21 17:40 ID:5QdPT5h7
okay
it's like with the graph of a limit
it touches one, but it isn't one. it's so close it doesn't matter and it's not worth arguing about. If you physically produced something with that accuracy, chances are it would get quantum particles touching or something, and theyd jump around and switch places as particles will do, and it would be the same fucking thing anyhow.
>>37
It's not "close to one", it IS 1. End of story. If you don't like it, go post in a fucking philosophy board you useless asshole. The real numbers are defined as the set of limits of cauchy sequences of rational numbers, and the cauchy sequence of rational numbers {0.9, 0.99, 0.999, ... } converges both to 0.999... and 1. Hence, they are equal.
thats just plain wrong.. it clearly is 0.999.....
3+6 = 9
this repeats infinitely.. its not even hard to understand. Youre talking as if 0.333.. repeating infinitely is 0.333... + some fraction.
You have totally misunderstood infinity and math, even though you claim that youre the one who claims to understand it. Just gtfo.
1 - 0.99... = 0.000000000000...
1 = 0.99...
Name:
Anonymous2007-02-21 19:25 ID:xvBnJwcw
Does 0.000... = 0? 1 - 0.999... = 0.000..., and since 0.999... ≠ 1, 0.000... must be infinitesimally greater than 0. Probably represented as 0.000...1.