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# 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: 9 years ago Status: Voting Period Viewed: 4,268 times Debate No: 3450
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18 comments have been posted on this debate. Showing 1 through 10 records.
Posted by leethal 9 years ago
Errm, ok this is the last time I will say this Latina... In mathematics, which was invented by humans, 1/3 does EXACTLY equal 0.3333 (recurring). It is empirical fact. That is how the laws of mathematics were created.
So Latina, are you actually gonna be debating on this site, or just commenting on how poorly everyone else did in their debates?
Posted by SexyLatina 9 years ago
Eh, this old debate again. I actually believe that Yraelz has a point: (1/3) is not equal to 0.333 repeating. It's close, sure, infinitessimally close, but not exactly the same.
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.
Posted by Krad 9 years ago
Northsider wins
this is now 25 characters
Posted by NorthSider 9 years ago
This is nothing more than an alternative method to propose Zeno's Paradox. There are many different versions of this seemingly paradoxical predicament, my favorite being Zeno's second paradox, which states that, in order to get half way to one's destination, one must first travel one quarter of the way; and, accordingly, to get one quarter of the way to one's destination, one must first travel one eighth of the way; ad infinitum. Of course, this paradox brings into question motion in its entirety and would suggest that motion was nothing more than an illusion were a number infinitely close to zero not equal to zero.

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.
Posted by Krad 9 years ago
srsly guise?
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?
Posted by leethal 9 years ago
50 was an obvious exageration. You've misread the link and I suggest going back and reading it again (with an open mind, if possible). Need further proof? Check out the Wikipedia article I also provided a link for, which explicitely states that
"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.
Posted by Ragnar_Rahl 9 years ago
50?
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.
Posted by leethal 9 years ago
Well, if you don't like my circular arguments to prove the point, perhaps one of the 50, non-circular proofs I provided in my link will help sway you? You will not find a mathematically respected person to agree with the statement "1/3 does not exactly equal 0.3333 (recurring to infinite)".
Posted by Ragnar_Rahl 9 years ago
"
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.
Posted by GaryBacon 9 years ago
In addition to the mathematical proofs, .99999(recurring to infinite) equals 1 conceptually when thinking of other recurring digits in fraction form.

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.
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