

Take your pick: How Can 0.999... = 1?  Purplemath[^]
For example:
x = 0.999...
10x = 9.999...
10x  x = 9.000...
9x = 9
x = 1
EDIT: As Nagy said[^], 2+ hours ago.
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
 Homer





1/6 = 0,166666
>*6
0,9996
x = 0,9996
10x = 9,996
10x x = 8,9964
9x = 8,9964
x = 0,9996
?
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HobbyProggy wrote: 1/6 = 0,166666
>*6
0,9996
Nope  6 × 0.1666... = 0.9999...
It's simple multiplication:
1 x 6 = 6 (0.6)
6 x 6 = 36 = add 3 to the column on the left (6 + 3 = 9), and put 6 in this column (0.96)
6 x 6 = 36 = add 3 to the column on the left (6 + 3 = 9), and put 6 in this column (0.996)
6 x 6 = 36 = add 3 to the column on the left (6 + 3 = 9), and put 6 in this column (0.9996)
6 x 6 = 36 = add 3 to the column on the left (6 + 3 = 9), and put 6 in this column (0.99996)
etc.
Because you're repeating the operation an infinite number of times, there's no point where you stop and leave the last digit as 6 . There's always another digit to the right which needs to be multiplied by 6 , carrying the 3 into the current column.
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
 Homer





So this works only for 0.999999 ?
Because the 6 at the end is in this case important to show that it wont be a 1 in the end
But nevermind, this is all just playing with numbers
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HobbyProggy wrote: Because the 6 at the end is ...
But there isn't a 6 at the "end", because there isn't an "end"!
Think of it like this:
using System;
static class Program
{
static void Main()
{
while (true)
{
Console.WriteLine("9");
}
Console.WriteLine("Squirrel!");
}
}
If you run that program, how long will you have to wait before it prints "Squirrel"?
"These people looked deep within my soul and assigned me a number based on the order in which I joined."
 Homer





STACK OVERFLOW
nvm
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mmm
working by 9, not working by 6
what about 69? does it works?
Oh, wait... I think I stop now since I am about to break the KSS rule
I am still on the way getting my coat
M.D.V.
If something has a solution... Why do we have to worry about?. If it has no solution... For what reason do we have to worry about?
Help me to understand what I'm saying, and I'll explain it better to you
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Well...
0.999... is
0+(9/10) + (9/100) + (9/1000) +...
which is 0+9*((1/10)+(1/100)+(1/1000)+...)
which is 9* (sum(1/10^n))
where the sum runs over n for n=1 to infinity.
Now for the same sum, if n ran from 0 to infinity, there is a convenient formula, that says it is (I'll mark this with a capital S to distinguish it from the one we're trying to calculate):
Sum(1/10^n)=1+1/10+1/100+...=1/(1(1/10))=1/(9/10)=10/9
therefore the sum we want is
1/10+1/100+...=(10/9)1=1/9
therefore
0.999...=9*(1/9)=1
So it's math.
On the other hand, computer calculated numbers are approximations, and in the binary system at that, and this is why you get all those rounding errors.
But I get the feeling you already know all this, and that you are just toying with us obsessive compulsive types.
Φευ! Εδόμεθα υπό ρηννοσχήμων λύκων!
(Alas! We're devoured by lambguised wolves!)





if the numerator and denominator are equal then the value must be 1.
Then the second part is almost right, but you missed a very important point, a recurring value is only an approximation rather than the absolute. ^{1}/_{3} is absolute, but the decimal 0.3^{.} is only an approximation.
Proof, without bad maths of the 9's reccuring 
0.9^{.} == 1
Multiply by 10:
9.9^{.} = 10
Subtract the original:
9.9^{.}  0.9^{.} = 9
9 = 9
QED.
This is accepted as proper maths, the 1=0 using division by zero is, however, not.
veni bibi saltavi





Nice
But i guess as long as we have an approximation i'll go with its not exactly 1 :P
Nagy Vilmos wrote: This is accepted as proper maths, the 1=0 using division by zero is, however, not.
I already said that in an other statement that hes wrong
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Actually, yes 0.9999..... is in fact 1:
x = 0.99999....
10 x = 9.999999.... multiply by 10
10 x = 9 + 0.99999.... split right side into arithmetic expression
10 x = 9 + x replace 0.9999.... with "x"
9 x = 9 subtract "x" from both sides
x = 1 divide by 9.
without dividing by "0"





Am I the only one that doesn't see the issue that we are not dividing 1 by 3.... 1 = 3/3... If you're going to divide by 3 on the 1... wouldn't you need to also need to divide 3/3 by 3 also... 1/3 = (3/3)/3.... which would both be .03333...
??





HobbyProggy wrote: because 1/3 is 0,333
No. 1/3 does not equal 0.333. 1/3 is APPROXIMATELY 0.333.
and 0.99999 is not technically 1. It is APPROXIMATELY 1
There is even a different symbol for it.
1/3≈0.333 (1/3 is approximately equal to 0.333)
0.99999≈1 (0.99999 is approximately equal to 1)
Money makes the world go round ... but documentation moves the money.





0.9999 is indeed approximately 1
0.9999... (note the dots there) is EXACTLY 1.
Φευ! Εδόμεθα υπό ρηννοσχήμων λύκων!
(Alas! We're devoured by lambguised wolves!)





yiangos wrote: 0.9999... (note the dots there) is EXACTLY 1.
Unfortunately, no. Even as 9s extend on infinitely, it only approaches 1, but never actually gets there. It is not EXACTLY 1, but a very close approximation.
Money makes the world go round ... but documentation moves the money.





Please take a look at the thread. There is a number of mathematical proofs that an infinite series of 9s in the decimal part is actually EXACTLY equal to 1. Heck, even I offered a proof.
There's no approximation there. 0.999... is EXACTLY equal to 1. Again, the dots play an important role. No dots, no equality.
EDIT No I didn't offer a proof.
Here's one.
0.999...= 9*(0.111...)
0.999...=9*[(11)+0.111...)
0.999...=9*[(1+0.111...)1]
0.999...=9*[(1+0.1+0.01+0.001+...)1]
Now 1+0.1+0.001+0.0001+... = 1/(10.1)=1/0.9
This is the closed form for the sum of an infinite number of terms of aa geometric series. This is NOT an approximation. This is the actual final value of summing infinite terms of a geometric series. You can see e.g. at Geometric series  Wikipedia, the free encyclopedia[^] or you can look it up in wolfram (too tired to look it up myself right now).
Therefore
0.999...=9*[(1/0.9)1]
0.999...=(9/0.9) 9
0.999...=(90/9) 9
0.999...=109
0.999...=1
QED.
There's no pesky math there (such as dividing by zero, as the OP did).
Φευ! Εδόμεθα υπό ρηννοσχήμων λύκων!
(Alas! We're devoured by lambguised wolves!)





HobbyProggy wrote: because 1/3 is 0,333..
No it isn't.
Mathematics defines and recognizes rational numbers.
Mathematics defines and recognizes decimal numbers.
They are two different things.
Your equating them as the same does not remove certainly hundreds of years of mathematics and mathematicians that recognize and correctly differentiate the two.
Certainly when I took mathematics courses that taught mathematics as it was implemented in computers they emphasized the assumptions and limitations of finite math. Actually understanding those assumptions and limitations makes the difference clear.
Note clearly that in the above those were mathematics classes and not computer science classes. The difference is often where the former teaches the math and the later teaches how to use the computer. (Yes I took both.)





Actually 1.0 = 0.9999... I did the derivation of this proof as an exercise in a algebra class back in the early 80s.





1 === 3/3 === X/X
The thing is base 10 decimal (0.333) does not have a correct way to represent 3/3, except putting bar above the value that repeats infinitely
_
0.333





Quote: Go to ParentThe only thing i can agree with is that 1 != 3/3 (at least not exactly) because 1/3 is 0,333... and multiplied with 3 it is just 0,99999.... which is technically 1 but not 100%
This is nonsense ... .999... is exactly equal to 1, just as .333... is exactly equal to 1/3.





Quote: A high school teacher showed this to me some 10+ years back (feeling old now).
10+ years is making you feel old? I saw that in high school 35+ years old.....
(*) Even then, was able to figure out the flaw.
Truth,
James





Quote: Dividing by (xy), obtain... ...the silent reproach of a million tearstained eyes.





Dominic Burford wrote: since we started with y nonzero.
oh, the irony.



