The Instigator
leethal
Pro (for)
Losing
27 Points
The Contender
Yraelz
Con (against)
Winning
41 Points

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
Debate Rounds (2)
Comments (18)
Votes (20)

 

leethal

Pro

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 counter-intuitiveness 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 mathematically-minded person who legitimately wants to debate the mathematical side of this topic. I ask please for no semantically-based 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.
Yraelz

Con

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 none-the-less 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.
Debate Round No. 1
leethal

Pro

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 right-hand 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.
Yraelz

Con

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(x-1) + 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.
Debate Round No. 2
18 comments have been posted on this debate. Showing 1 through 10 records.
Posted by leethal 9 years ago
leethal
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
SexyLatina
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
Krad
Northsider wins
this is now 25 characters
Posted by NorthSider 9 years ago
NorthSider
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
Krad
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
leethal
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
Ragnar_Rahl
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
leethal
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
Ragnar_Rahl
"
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
GaryBacon
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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