The thing that most people have confusion about is the infinite number of \(9\)s after the decimal.They tend to think that \(.999999...\) approaches \(1\) but never quite reaches \(1\). First thing to realize is that \(.999...\) does not only reach one but it is exactly equal to \(1\),and when I say exactly,I really mean it.Now,computing the above expression is a simple matter of substituting \(1\) in place of \(.99999...\).

If you still aren't contented,then think about it this way.What do we mean when we say that two real numbers are different?It means that there is at least one more real number between them(In fact there are an infinite number of them). Now answer this:Can you find any other real number between \(1\) and \(.9999....\)?If you can't ,the only conclusion is that they are equal.

yes the answer is one. one of example why 0,9999..........=1.
look 10/3= 3,33333333333333..... then 10= 3times3.333333333.......................
10=9.9999999999999999. next 10= 9+0.99999999..............................subtract 10-9=0.9999999......... so 1=0.999999999999........

The value of the expression will be zero and not one. Since the value of \(X\) which is given approaches the value of one but it really does not become equal to it and will always remain less than it and thus \([X] = 0\).

This is basically asking the greatest integer that is less than or equal to the limit of \(1-{10}^{-n}\) as \(n\) approaches infinity. Since the limit is \(1\), so is the final answer. The order matters.

@Keshav Tiwari
–
You have computed the floor of values that only have a finite number of \(9\)'s after the decimal.A brilliant way to think about this is :Can you find any other number between \(1\) and \(.9999....\)?If you can't the only conclusion is that they are equal.

The problem is that you have computed a finite number of 9's, as stated by @Rahul Saha . The floor function of 0.99999 is 0, but the floor of 0.9... (with an infinite number of 9's) is 1. That's because this number is equal to 1, as proved in many different ways by @Ossama Ismail , @Mursalin Habib and Rahul Saha

AS THIS IS AN INFINITE SUM I.E 0+9/10+9/100...................... SO ON AND SO FOURTH, WE SEE THAT ITS VALUE APPROCHES 1. WELL I DONT UNDERSTAND THE STIGMA/CONUNDRUM HERE.
THIS FITS RATHER PERFECTLY INTO THE DEFINITION OF GIF THAT GIF OF ALL NUMBERS LYING BETWEEN 2 INTEGERS IS EQUAL TO THE INTEGER LESS THAN OR EQUAL TO IT.
HERE 0.9999999999999999999999........... CAN BE TREATED AS THE LAST REAL NUMBER B/W 0 AND 1 AS IT NEVER REACHES ZERO.
SOME WILL ARGUE THAT TO MAINTAIN CONTINUITY WE WILL REGARD IT AS ONE BUT THINK AGAIN.

INFINITE IS NOT DEFINED OR RESTRICTED HENCE IT CAN BE DIFFERENT FOR YOU AND ME BUT WIL STILL BE THE SAME.

AND AS FAR AS CONTINUITY IS CONCERNED IF YOU HAVE A LINE MADE UP OF INFINTELY SMALL POINTS , REMOVING SOME WILL NOT MAKE IT DSICONTINOUS.
HENCE IT IS ZERO.

I don't know much about continuity but I think an infinite geometric series as defined as the limit of the geometric series as it approaches infinity. Thus \(.999\cdots =1\) follows by applying the definition.

Btw, what do you mean by the last real number between two numbers.? If there aren't an infinite number of real numbers between \(a\) and \(b\), then \(a=b\). A "last real number" can't exist in an open interval like \((0,1)\).

What I think is that the main question here is that whether 0.9999999...... is equal to 1 or not.and that I think it is.Therefore the answer should be 1.

AS THIS IS AN INFINITE SUM I.E 0+9/10+9/100...................... SO ON AND SO FOURTH, WE SEE THAT ITS VALUE APPROCHES 1. WELL I DONT UNDERSTAND THE STIGMA/CONUNDRUM HERE. THIS FITS RATHER PERFECTLY INTO THE DEFINITION OF GIF THAT GIF OF ALL NUMBERS LYING BETWEEN 2 INTEGERS IS EQUAL TO THE INTEGER LESS THAN OR EQUAL TO IT. HERE 0.9999999999999999999999........... CAN BE TREATED AS THE LAST REAL NUMBER B/W 0 AND 1 AS IT NEVER REACHES ZERO. SOME WILL ARGUE THAT TO MAINTAIN CONTINUITY WE WILL REGARD IT AS ONE BUT THINK AGAIN.

INFINITE IS NOT DEFINED OR RESTRICTED HENCE IT CAN BE DIFFERENT FOR YOU AND ME BUT WIL STILL BE THE SAME.

AND AS FAR AS CONTINUITY IS CONCERNED IF YOU HAVE A LINE MADE UP OF INFINTELY SMALL POINTS , REMOVING SOME WILL NOT MAKE IT DSICONTINOUS. HENCE IT IS ZERO.

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## Comments

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TopNewestThe thing that most people have confusion about is the infinite number of \(9\)s after the decimal.They tend to think that \(.999999...\) approaches \(1\) but never quite reaches \(1\). First thing to realize is that \(.999...\) does not only reach one but it is exactly equal to \(1\),and when I say exactly,I really mean it.Now,computing the above expression is a simple matter of substituting \(1\) in place of \(.99999...\).

If you still aren't contented,then think about it this way.What do we mean when we say that two real numbers are different?It means that there is at least one more real number between them(In fact there are an infinite number of them). Now answer this:Can you find any other real number between \(1\) and \(.9999....\)?If you can't ,the only conclusion is that they are equal.

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1

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The Value = 1

let X = 0.999999999999999999 ......

then 10 X = 9.999999999999999 ......

subtract (10 X - X) = 9 X = 9 =====> X =1

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yes the answer is one. one of example why 0,9999..........=1. look 10/3= 3,33333333333333..... then 10= 3times3.333333333....................... 10=9.9999999999999999. next 10= 9+0.99999999..............................subtract 10-9=0.9999999......... so 1=0.999999999999........

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The value of the expression will be zero and not one. Since the value of \(X\) which is given approaches the value of one but it really does not become equal to it and will always remain less than it and thus \([X] = 0\).

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The value

is\(1\).This is basically asking the greatest integer that is less than or equal to the limit of \(1-{10}^{-n}\) as \(n\) approaches infinity. Since the limit is \(1\), so is the final answer. The order matters.

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http://m.wolframalpha.com/input/?i=floor+0.999...&x=0&y=0

The problem is that you have computed a finite number of 9's, as stated by @Rahul Saha . The floor function of 0.99999 is 0, but the floor of 0.9... (with an infinite number of 9's) is 1. That's because this number is equal to 1, as proved in many different ways by @Ossama Ismail , @Mursalin Habib and Rahul Saha

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Comment deleted Feb 07, 2015

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AS THIS IS AN INFINITE SUM I.E 0+9/10+9/100...................... SO ON AND SO FOURTH, WE SEE THAT ITS VALUE APPROCHES 1. WELL I DONT UNDERSTAND THE STIGMA/CONUNDRUM HERE. THIS FITS RATHER PERFECTLY INTO THE DEFINITION OF GIF THAT GIF OF ALL NUMBERS LYING BETWEEN 2 INTEGERS IS EQUAL TO THE INTEGER LESS THAN OR EQUAL TO IT. HERE 0.9999999999999999999999........... CAN BE TREATED AS THE LAST REAL NUMBER B/W 0 AND 1 AS IT NEVER REACHES ZERO. SOME WILL ARGUE THAT TO MAINTAIN CONTINUITY WE WILL REGARD IT AS ONE BUT THINK AGAIN.

INFINITE IS NOT DEFINED OR RESTRICTED HENCE IT CAN BE DIFFERENT FOR YOU AND ME BUT WIL STILL BE THE SAME.

AND AS FAR AS CONTINUITY IS CONCERNED IF YOU HAVE A LINE MADE UP OF INFINTELY SMALL POINTS , REMOVING SOME WILL NOT MAKE IT DSICONTINOUS. HENCE IT IS ZERO.

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I don't know much about continuity but I think an infinite geometric series as

definedas the limit of the geometric series as it approaches infinity. Thus \(.999\cdots =1\) follows by applying the definition.Btw, what do you mean by the last real number between two numbers.? If there aren't an infinite number of real numbers between \(a\) and \(b\), then \(a=b\). A "last real number" can't exist in an open interval like \((0,1)\).

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Is 0.9999999999999999999.... an integer?

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Yes, it is.

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What I think is that the main question here is that whether 0.9999999...... is equal to 1 or not.and that I think it is.Therefore the answer should be 1.

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agreed.

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AS THIS IS AN INFINITE SUM I.E 0+9/10+9/100...................... SO ON AND SO FOURTH, WE SEE THAT ITS VALUE APPROCHES 1. WELL I DONT UNDERSTAND THE STIGMA/CONUNDRUM HERE. THIS FITS RATHER PERFECTLY INTO THE DEFINITION OF GIF THAT GIF OF ALL NUMBERS LYING BETWEEN 2 INTEGERS IS EQUAL TO THE INTEGER LESS THAN OR EQUAL TO IT. HERE 0.9999999999999999999999........... CAN BE TREATED AS THE LAST REAL NUMBER B/W 0 AND 1 AS IT NEVER REACHES ZERO. SOME WILL ARGUE THAT TO MAINTAIN CONTINUITY WE WILL REGARD IT AS ONE BUT THINK AGAIN.

INFINITE IS NOT DEFINED OR RESTRICTED HENCE IT CAN BE DIFFERENT FOR YOU AND ME BUT WIL STILL BE THE SAME.

AND AS FAR AS CONTINUITY IS CONCERNED IF YOU HAVE A LINE MADE UP OF INFINTELY SMALL POINTS , REMOVING SOME WILL NOT MAKE IT DSICONTINOUS. HENCE IT IS ZERO.

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