It is possible to show that 1 is exactly equal to 0.999999999 (recurring to infinite).
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Voting Style:  Open  Point System:  7 Point  
Started:  3/31/2008  Category:  Science  
Updated:  8 years ago  Status:  Voting Period  
Viewed:  4,047 times  Debate No:  3450 
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First, let me say that I do believe it is possible to show that 1 is exactly equal to 0.99999999999 (recurring to infinite), despite the counterintuitiveness of the statement. HOWEVER, I posted this debate because if I am wrong anywhere in my reasoning, I want to know about it. That said, I would like this debate to be taken on by a mathematicallyminded person who legitimately wants to debate the mathematical side of this topic. I ask please for no semanticallybased argument (ie. the definition of 'possible' etc.), so I can know for myself whether I'm correct or not.
Ok, to my argument... Let x = 1/3 y = 0.333333333333 (recurring to infinite) There is no doubt that 1/3 exactly equals 0.333333333333 (recurring to infinite), and we can therefore safely say that x=y. Therefore, it is obvious that 3x=3y, as both sides of the equation have merely been multiplied by 3. If we substitute the values of x and y into this equation we get: (3 x 1/3) = (3 * 0.333333333333 (recurring to infinite)) 3/3 = 0.9999999999 (recurring to infinite) 1 = 0.9999999999 (recurring to infinite) So that's my reasoning. Please, if you are going to accept this debate, take note of my opening paragraph. Cheers.
The error of your argument is simple. You assume that 1/3 is the equivalent of .3 repetend. While I admit that this idea is common in every mathematics because they are so infinitely close together it is nonetheless untrue. 1/3 is cited as the equivalent of .3 repetend in much the same way pi is often cited as 3.14. The numbers are very close, but not exact. Proof: Examples .3 = 3/10 .33 = 33/100 .333 = 333/1000 Let x be the number of digits present on the left side of the above equation. Rule: x = x/10^x Substitute infinity for x Infinity = infinity/10^infinity. While this is quite literally infinitely close to 1/3 it is not 1/3. Thus 1/3 d.n.e .3 repetend. 

Hi Yraelz, thankyou for accepting the debate.
You said:  Examples .3 = 3/10 .33 = 33/100 .333 = 333/1000 Let x be the number of digits present on the left side of the above equation. Rule: x = x/10^x  This 'Rule' is incorrect for the above examples. I think you'll find that the correct 'Rule' would be: x = 3x/10^x Furthermore, and far more importantly, I don't think your reasoning is sound here. This is more of an inequation. For example, taking x=1, we get: 1 = 3/10 Which is clearly incorrect. While the left side of the equation does equal the number of digits in the righthand side's decimal value, this equation is clearly incorrect and cannot be used to prove or disprove anything. Also, you have gone on to let x = infinity. The concept of infinity should not be used in this manner (ie. let x =), unless used in regards to limits (ie. as x approaches infinity). It is an absolute fact that 1/3 does in fact exactly equal 0.333333 (recurring to infinite). They are two different ways to describe the exact same number. Now, while researching for this debate, I found several websites that deal with the very issue: http://polymathematics.typepad.com... I urge you to check this site out, as it provides a large number of various proofs for my suggestion that 1 exactly equals 0.999999999 (recurring to infinite). For example: 2/7 = 0.285714285714.... (to infinity) +5/7 = 0.714285714285.... (to infinity)  7/7 = 0.999999999999.... (to infinity) If you can provide any basis for why this is incorrect, please do so. Also, check out: http://en.wikipedia.org... "In mathematics, the recurring decimal 0.999… , which is also written as or , denotes a real number equal to 1. In other words, the notations "0.999…" and "1" represent the same number. The equality has long been accepted by professional mathematicians and taught in textbooks. Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience." My initial proof from Round 1 is also still unchallenged, as I have shown in a number of ways how the mathematics was flawed in your first argument. Cheers for the debate, it was fun.
Sorry, the equation I presented was most definitely an inequation based on pattern seen in the above example. Let me rewrite the rule. Proof: Rule: N(1) = 3/10 N(x) = N(x1) + 3/10^x Therefor: N(1) = 3/10 = .3 N(2) = 33/100 = .33 N(3) = 333/1000 = .333 N(4) = 3333/10000 = .3333 N(5) = 33333/100000 = .33333 And so on.... No matter how far out you go with this pattern the denominator will always be a multiple of 10, the numerator will always be a multiple of 3. However this is not true of 1/3. The fraction 1/3 will always have a numerator a multiple of 1 and the denominator a multiple of 3. For example: 3/10 is not equal to 3/9 33/100 is not equal to 33/99 333/1000 is not equal to 333/999 3333/10000 is not equal to 3333/9999 33333/100000 is not equal to 33333/99999 No matter the magnitude of this pattern the two fractions will never be exactly equal, there will always be a minuscule difference. As x approaches infinity the limit of the function x approaches 1/3 BUT is NOT "exactly" 1/3. It will become infinitely close, but it will never be exactly equal. So while yes, for all practical purposes you can go ahead and substitute .9 repetend for 1 as it will doubtlessly not effect whatever math problem you happen to be doing but this does not change the fact that in reality .9 repetend is not "exactly" equal to 1. 
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So Latina, are you actually gonna be debating on this site, or just commenting on how poorly everyone else did in their debates?
Think of asymptotes on graphs: The graphed function gets very, very close to the asymptote, but never quite touches it.
Or, think about it this way: 0 is not equal to (1/infinity), even though practically they are the same thing. One is absolutely zero, nada, nil. The other is very, very close to zero, it's 0.000 repeating 1, but it's not quite zero.
this is now 25 characters
Simply stated, if a number is infinitely close to another, then that number is, in essence, the other number  that is, there is no discrete difference between the two quantities, because, in order to achieve a value for the difference, one would have to cease the infinite regress, obviously defying the "infinite" part of the regress.
To put it arithmetically, we must accept that the difference between 1 and 0.(9) is smaller than any positive quantity; and of course, as per the Archimedean property, the only real number with the quality of being smaller than any positive quantity is zero. Therefore, the two values are equal.
Whether you elect to use the rule of converging geometric series (if abs(r)<1, then ar + ar^2 + ar^3 ... = (ar) / (1  r)), and conclude that 0.(9) = 9(1/10) + 9(1/10)^2 ... = (9(1/10)) / (1  (1/10)) = 1; or you elect to use the easier solution suggested previously, one must either accept this anomaly (or rather, peculiarity) to be true, or reject much of calculus, and many other branches of mathmatics.
ill use 0.(9) to show 0.999(to infinity)
x = 0.(9)
10x = 9.(9)
10x  x = 9.(9)  0.(9)
9X = 9
x = 1
amirite?
Also, what # comes in between 0.(9) and 1?
"In other words, the notations "0.999…" and "1" represent the same number. The equality has long been accepted by professional mathematicians and taught in textbooks. Various proofs of this identity have been formulated with varying rigour, preferred development of the real numbers, background assumptions, historical context, and target audience."
Thanks for the interest in my debate, I'm glad to know it has gotten you thinking. Enjoy.
I didn't see fifty, maybe wrong link...
I saw a few, those among them that weren't circular contained premises that would essentially render invalid the concept of .999..... even being a number, and something that isn't a number can't be 1.
There is no doubt that 1/3 exactly equals 0.333333333333 (recurring to infinite), and we can therefore safely say that x=y.
"
I have doubt as to it. Therefore find a new premise.
Indeed, I would surmise that there is EXACTLY THE SAME degree of doubt about 1/3=.3333.... as 1=.9999999999. they are essentially equivalent premises. Using one to prove another is essentially a circular argument.
E.g.
.11111...= 1/9
.22222...= 2/9
.33333...= 3/9
when we get to .9999...., we can logically conclude that it is equal to 9/9 which is indeed equal to 1.