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CandyFab.org

0.999... equals one.

But you knew that, right? Check out the Wikipedia article about different proofs that 0.999... equals 1. It's really quite interesting to see the wildly different methods that can be used to arrive at the same conclusion. While I'm partial to the Cauchy sequence version, the first of the proofs is my favorite, beautiful in its simplicity:

0.333.... = 1/3

3* 0.333... = 3 * (1/3)

0.999... = (3*1)/3 = 3/3 = 1

Q.E.D. =)

Contributed by Windell on Tuesday, March 13 2007 @ 09:23 PM PDT, in Everything Else
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Here's what others have to say about '0.999... equals one.':
0,999... uguale 1
Tracked on Saturday, March 22 2008 @ 08:39 AM PDT
0.999... equals one. | 60 comments | Create New Account
The following comments are owned by whomever posted them. This site is not responsible for what they say.
0.999... equals one.
From: Anonymous on Wednesday, August 22 2007 @ 01:11 PM PDT
But 0.333.... itself does not equal one-third; it is an approximation, just as saying 0.999.... equals one. Technically, 0.3 = 3/10, 0.33 = 33/100, 0.333 = 333/1000, etc. etc. 1/3 = 0.33333...... This is an infinitely repeating decimal. You can't use this arguement to "prove" 0.999... equals one because saying 0.333.... equals exactly 1/3 is already using an assumption. I don't know if this makes sense, but it does in my head! :-D
0.999... equals one.
From: Anonymous on Tuesday, March 25 2008 @ 03:35 PM PDT
Your proof is wrong! 1/3 is actually not 0.333 but 0.3(3 occuring infinite times!). Simply calculate 1/3 in your scientific calculator and try change it into a fraction.. You will see what I mean.
0.999... equals one.
From: Anonymous on Thursday, March 27 2008 @ 01:00 AM PDT
The "..." in this case means infinitely continuing. Same as writing .9 with a line over the 9. It's just impossible to type it that way.
0.999... equals one.
From: Anonymous on Thursday, March 27 2008 @ 04:03 AM PDT
it does make sense
.333 equaling 1/3 is just a close estimation, or theory, and not a actual proven fact
Im not saying its right or wrong just that it is possible
'theory' and 'fact' do not necessarily stand in opposition of each other and the .999... theory equaling 1 is just a theory
0.999... equals one.
From: Anonymous on Thursday, March 27 2008 @ 11:05 AM PDT
The proof is wrong - at least in my opinion, here's why :

0,333333333.... no matter how many 3's you add to the number it is not equal to 1/3 it is just nearly there(forgive the language, but i'm no mathematician) :D

(0,333...)*3 /= 3*(1/3)
0,333...=333.../1000... right? .Yes
so (333...../1000.....)*3 /= 1 - thus does not equal 3*(1/3)
than : 0,(3)*3 = 0,(9)
thank you and good night :)))) Tankowiec signing off
0.999... equals one.
From: Anonymous on Thursday, March 27 2008 @ 01:16 PM PDT
This is mathematically proven to be true.

Another proof is:

Let x = 0.9999....

Multiplying both sides by 10, we get:

10x = 9.999....


Subtracting the first equation from the second:

9x = 9

=> x = 1

Therefore, 0.9999.... = 1


<Trumpet call>

End of proof! ;)
0.999... does not equal one.
From: Anonymous on Friday, March 28 2008 @ 11:34 AM PDT
0.999 does not equal 1. When you represent 1/3 as 0.333 you are using a format that approximates the value (less precision). The reason that your calculator outputs 0.333 when you input 1/3 is because 1/3 can not be represented exactly due to the way numbers are stored in a computer. Therefore it uses an approximation. In the actual world however, there are no need for approximations as we are not limited by having to store numbers using a binary format so instead we use fractions. The Calculator provided in Microsoft Windows XP also does this. From the help section:
"Understanding Extended Precision - Extended Precision, a feature of Calculator, means that all operations are accurate to at least 32 digits. Calculator also stores rational numbers as fractions to retain accuracy. For example, 1/3 is stored as 1/3, rather than .333."
0.9999 can never equal one. It will always be less due to it being an approximation.
0.999... equals one.
From: Anonymous on Sunday, March 30 2008 @ 06:56 PM PDT
A way to think about this in words and logic instead of equations is that given the fact that .9999... has an infinite number of 9s at the end of it there is no number you can add to .999... to equal 1. Try it if you don't believe. .999...+.001= 1.000999... etc. Since there is no number between .999... and 1 you can conclude it is the same number. It sounds weird when you first hear it, but then again, no one ever guaranteed that common sense is always right.
0.999... equals one.
From: Anonymous on Thursday, April 03 2008 @ 10:56 PM PDT
Uhh... I'm just a code monkey, not some math genius, but...

1=0.999...
1-0.999...=0
but actually executing that operation gives
1-0.999...=0.000...1
so 0.000...1=0

for f(x)=1/n; f(0.1)=10, f(0.01)=100, f(0.001)=1000, etc. to f(0.000...1)=1000...0.0

but from http://en.wikipedia.org/wiki/Division_by_zero:

1/0.000...1= +inf
1/0=fail
as +inf!=fail, 0.000...1!=0, so 1!=0.999...

Uhh... WTFBBQ???

I'm wondering if mixing open-form ideas like 0.999... and 0.000...1 with closed form ones like equals is fully valid? Or maybe I'll just grab another banana and go back to my nice, simple, easy chars, ints and longs...
0.999... equals one.
From: Anonymous on Thursday, April 17 2008 @ 03:14 PM PDT
When one constructs the real numbers, either from cauchy sequences or using supremums of sets, you actually create equivalence classes. Just as when one constructs the rational numbers. That is why 1/2 = 2/4, it is because they belong to the same equivalence class. Likewise you see when constructing the reals that 1 = .99999...

P.S.
1/3 = .333...
It is not an approximation.
0.999... equals one.
From: alphachapmtl on Saturday, April 19 2008 @ 08:18 AM PDT
A simple proof:

x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x=1
0.999... equals one.
From: Anonymous on Wednesday, April 23 2008 @ 11:22 AM PDT
Jesus..
"Arguing" about this is so silly. People who don't "believe" it don't grasp that the ... means infinite repeating decimals. That simple. They think they're somehow clever by noticing that 0.333 isn't 1/3. Well, golly, really? 0.333... isn't the same thing as 0.333, and there is no number such as "0.000...1".

EOD

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