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05042006  #1 
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Infinite 0.9 Not Equal To 1
I have to speak about this!
Its impossible for infinite 0.9 to ever equal 1 the reason for this is,there will always be a difference of infinite 0.1 A.R.B 
05042006  #2 
Join Date: Mar 2005
Posts: 10,653

In general, mathematicians will say that 0.999... is equal to 1. One argument is that 0.999... is very, very, very close to 1. The difference is infinitesimal and therefore irrelevant.
And the other "proof": x = 0.999... 10x = 9.999... 10x = 9.999...  x = 0.999... 9x = 9 x = 1
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Do Math and you can do Anything! Last edited by Mr. Hui; 05052006 at 03:30 PM.. 
05052006  #3 
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More info!
LETS LOOK AT THE TWO ARGUMENTS!
Argument (A) Infinite 0.9 = 1 Argument (B) its impossible for Infinite 0.9 to ever Equal 1 NOW LETS GIVE A PROOF SCORE VALUE FOR THE ARGUMENTS PUT FORWARD SO FAR!  Argument (A) Infinite 0.9 = 1 because it does? Evidence put forward = 0.9999999999999........(no calculations ever shown by anyone! as to what happens next)......999999...(and then 1)? Proof score value = 0  Argument (B) Infinite 0.9 will never = 1 because its an Infinite number! Evidence put forward = 1  (Infinite 0.9 stage one) = 0.1 This difference in the value of Infinite 0.9 and 1 never goes away! no matter how long the Example! or the stage Infinite 0.9 is at! So we start with a 100% proof calculation! showing the difference between the two numbers! and because we can always show this difference we end up with! Proof score value = 99% Everyone may now say why 99% and not 100% the reason for this is no one can prove the end of Infinity! But that's not what I am trying to do! as long as I can prove Infinite 0.9 is 99% of the time smaller than 1 it is far more Evidence than Argument (A). A.R.B 
05052006  #4 
Join Date: Mar 2006
Posts: 12

Another Solution
I asked a teacher about this today and she pointed me in the direction of using a geometric progression to solve / prove this. (Although I not sure how rigorous this is poof wise)
S(n) represents S with an n subscript, but I cannot get subscripts to work. Let S(n) = Sum of the first n terms of a geometric progression. a is the first term and r is the common ratio. S(n) = a + ar + ar^{2} + ar^{3} + ... + ar^{n1} Multiply everything by r r . S(n) = ar + ar^{2} + ar^{3} + ... + ar^{n1} + ar^{n} Subtract this from the original expression, apart from the first term in one expression and the last in the other all of the terms cancel. r . S(n)  S(n) = ar^{n}  a Simplify S(n)(r1) = a (r^{n}  1) S(n) = a(r^{n}  1) / (r  1) S(n) = a(1  r ^{n}) / (1  r) The reason I have just done all of this (I did have a reason ) , is because 0.9 recurring can be written as a geometric progression of nine tenths, plus 9 hundreths, plus nine thousandths etc. So S(n) = 9/10 + 9/100 + 9/1000 + ... + 9/10^{n} common ratio r = 1/10 You can calculate this by comparing the ratio of two consecutive terms. e.g. 0.09 / 0.9 = 0.1 For r < 1 as n > infinity, r^{n} > 0 This should read for values of r between 1 and 1 (Otherwise the series doesn't converge to a value), as n tends towards (approaches) infinity r^{n} approaches zero. (You may need to look up limits, this is the tricky bit ) So a(1  r ^{n}) / (1 / r) becomes a/(1r) for the sum to infinity As a = 9/10 and r = 1/10 S(infinity) = a/(1r) = (9/10) / (1  1/10) S(infinity) would be written S with the infinity symbol as a subscript. = (9/10) / (9/10) = 1 So point 9 recurring = 1. Isn't Mr Hui's way simplier.  Shura
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"Mathematics is the gate and key to the sciences."  Roger Bacon 
05052006  #5 
Join Date: Mar 2005
Posts: 10,653

A.R.B,
If you believe that 1  (0.999...) is not equal to zero, consider Zeno's Paradox. To get from Point A to Point B, one must reach half the distance. Then you must again reach half the distance, ad infinitum. In this sense, you will never be able to reach your goal because you will always have an infinitely small distance left to travel. Therefore, mathematicians accept that an infinitely small number to be equal to zero. __________________________________________________ _ Shura, Interesting proof. I will need time to study it.
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Do Math and you can do Anything! Last edited by Mr. Hui; 05052006 at 03:35 PM.. 
05092006  #6 
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More Info!
BELOW MORE EXAMPLE INFO! PLUS THE MISTAKES EVERYONE IS MAKING!.
(A) INFINITE Versus RECURRING! It is possible to have an Infinite number,the definition Recurring does not explain the number in the correct way! INFINITE = without any limit or end. RECURRING = happen again,repetition. Infinite 0.9 the .9's are not repeating themselves,every .9 is new as part of a continuos stream with no end! Recurring 0.9 the .9 is repeating itself!and is in a way going round in a circle,back on itself. (B) ITS IMPOSSIBLE TO CHANGE AN INFINITE NUMBER! Its impossible to add,multiply,subtract or do any math calculations to an Infinite number! The reason for this is because,its impossible to prove the length of an Infinite number! You need to know how long a number is before you can Calculate on it,otherwise all results are a guess!. © THE TEN TIMES CALCULATION MYTH! The Calculation 10 * 0.9 can only be done on the normal number 0.9 Its impossible to Calculate 10 * Infinite 0.9 because of (B) The Ten Times Calculation is often used to round the number up! Into a number that can be used in other calculations,which end up with a false result! The Decimal point can never be removed! If it was possible to Calculate 10 * Infinite 0.9 Stage one,(One Decimal place) Then the result is what everyone uses 10 * 0.9 = 9 which is convenient for everyone as this is now a whole number! But because we are dealing with an Infinite number, the results as to what happens if we could calculate further,are quite different! 10 * Infinite 0.9 Stage two,(Two Decimal places) = 10 * 0.99 = 9.9 we now have the Decimal point back! And is not so easy to work with,as in trying to make Infinite 0.9 ever Equal 1 This fluctuation in the Imaginary End Result! Goes on forever depending on how far you want to Calculate,there can be no End to this as its an Infinite number! (D) THE FALSE BELIEF THAT AS INFINITE NUMBERS GET LONGER THEY GET LARGER! I have seen many Examples of Infinite 0.9 = 1 Because,as the .9's get longer they are getting nearer to 1 In the world of Infinity there is no change in space! As the above trys to prove,and more important there is no change in Value! The first Infinite 0.9 Stage one,(One Decimal place) is the same Size, and Value as every other .9 produced! As I stated at the start of my Thread the 0.1 Difference between Infinite 0.9 and 1 is Infinite! Every Stage Infinite 0.9 is Looked at Running beside it is the 0.1 Difference for Ever! as in the Example Below, 0.99999999999999999999999999999999999999.......... ... 0.11111111111111111111111111111111111111.......... ... The only way the two can ever Equal the number 1,is if it was possible to ADD them together! But this would End the number Infinite 0.9 being Infinite! and result in a Contradiction! A.R.B 
05172006  #7 
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The Reason Infinite 0.9 <> 1
Hi!
The reason INFINITE 0.9 IS <> 1 IS BECAUSE!....... N = 1/3 = O.33333333333...... This calculation results in an INFINITE stream of 0.3's But more important is the fact that its a perfect Calculation! there are no numbers! or fractions of numbers left over! what I mean by this is! if it was possible to look at the last .3 it would onlly be .3 there are no other numbers Hiding! So now we want to find the true Value of INFINITE 0.9 Easy! N * 3 = 0.999999999......No numbers left over! Or as below! 0.3333333333333333333333333333333................. ...... + 0.3333333333333333333333333333333................. ...... + 0.3333333333333333333333333333333................. ......  0.9999999999999999999999999999999................. ...... =  A.R.B 
03272008  #8 
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I don't know how old this is, but here's my approach to the problem (apologies if bumping isn't allowed). Here, any number followed by an ellipsis ( … ) is recurring. Each line is numbered for the purpose of referring back to it (obviously)
1) 1/3 = 0.3… 2) 2/3 = 0.6… 3) 1/3 + 2/3 = 0.3… + 0.6… 4) 1/3 = 3/9 because 1/3 * 3/3 = 3/9 5) 2/3 = 6/9 because 2/3 * 3/3 = 6/9 6) 1/3 + 2/3 = 3/9 + 6/9 = 9/9 = 3/3 = 1/1 = 1 7) 0.3… + 0.6… = 0.9… = 1 because of line 3 8) 0.9… = 2/3 + 1/3 = 0.6… + 0.3… = 1 
03272008  #9 
Join Date: Mar 2005
Posts: 10,653

Interesting approach, Infinite9. I can't spot any fallacies at first glance.
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Do Math and you can do Anything! 
05022008  #10 
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