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Geek Culture / Is 10 = 9.9999999... ??

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RedneckRambo
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Posted: 25th Aug 2008 03:36
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How about we just say that .999.... does and doesn't = 1?

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Geryon
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Posted: 25th Aug 2008 03:40
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Quote: "0.999~ can be used to REPRESENT 1.0, but it's not."


Yes, just like much in the same manner, 2+2 is not 4, but represents it, because they are written differently.

OH WAIT
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Mahoney
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Posted: 25th Aug 2008 03:55
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Quote: "Yes, just like much in the same manner, 2+2 is not 4, but represents it, because they are written differently.

OH WAIT "


You are slowly becoming one of my favorite people here. XD

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tha_rami
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Posted: 25th Aug 2008 04:06
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Aaron Miller is making a lot of sense too, lol.


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ionstream
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Posted: 25th Aug 2008 04:07
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Quote: "0.999... is infinitely close to 1"


Infinitely close is zero, because the limit as k approaches infinity of 1/k is zero (this is calculus).

Quote: "Infinity doesn't exist, how can you place the last 9?"


There is no last nine!

Quote: "and no one is accepting any one's point of view except theirs"


This is not a point of view, this is math. We have several conclusive proofs, and all you have is "it doesn't seem right." Nobody here came to the conclusion that 0.9~ = 1 themselves, we all learned it from class or otherwise.

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Mahoney
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Posted: 25th Aug 2008 04:29
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Quote: "Nobody here came to the conclusion that 0.9~ = 1 themselves, we all learned it from class or otherwise."


And the first to claim it discovered it. They didn't make it up.

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Agent Dink
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Posted: 25th Aug 2008 05:08
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Quote: "A bullet travels at, say, 5000 cm per second. In the next 1/2 second it travels 2500 cm, in the next 1/4 sec it travels 1250 cm, and so on. Using your argument it couldn't possibly travel more than 10000 cm (= limit of 5000 + 2500 + 1250 + 625 + (etc ad infinitum) - which is absolute nonsense of course. It takes 2 secs to reach 10000 cm and the bullet continues (unless it hits a concrete wall of misunderstanding ). "


This bullet you speak of has a constant speed. Our frog doesn't. He's just taking his merry old time, hopping along whenever he feels like it.

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tha_rami
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Posted: 25th Aug 2008 05:20 Edited at: 25th Aug 2008 05:20
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Quote: "This is not a point of view, this is math. We have several conclusive proofs, and all you have is "it doesn't seem right." Nobody here came to the conclusion that 0.9~ = 1 themselves, we all learned it from class or otherwise."

As soon as humanity can calculate infinity, we will know for sure. Until then, it's a theory and I'm not buying it. Something is not something else.

There is this one term I've been digging through my mind for and I just remembered it: asymptotes. With this, how would one explain asymptotes? It doesn't make sense: "A linear asymptote is essentially a straight line that a graphed curve becomes closer and closer but does not become identical to."

An asymptote, as you all know runs into infinity, or in other words, into 0.999..., however, it NEVER becomes equal to 1.


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ionstream
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Posted: 25th Aug 2008 05:49
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Quote: "As soon as humanity can calculate infinity, we will know for sure."


This is it, this is how humanity calculates infinity.

Quote: "Something is not something else."


They're the same thing! 0.999~ is 1! You can write 1 as 0.9~ and write 0.9~ because they are the same value!

Quote: ""A linear asymptote is essentially a straight line that a graphed curve becomes closer and closer but does not become identical to.""


It won't be identical with really big values, but it will at infinity, which is why infinity is just so dern special.

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Agent Dink
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Posted: 25th Aug 2008 05:52 Edited at: 25th Aug 2008 05:53
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But if a human can't even comprehend infinity where is the guarantee that 0.9 repeating is equal to 1. No. 1 = 1. 0.9~ does not = 1.

It's a flaw with the decimal system, just as 1/3 and 2/3 are.

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tha_rami
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Posted: 25th Aug 2008 05:56 Edited at: 25th Aug 2008 05:57
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An asymptote actually runs into infinity and yet mathematicians claim that the function to which the asymptote belongs does never equal the asymptote itself. (a hyperbole with asymptote x = 1 will NEVER reach 1).

Quote: "An Asymptote of a real-valued function y = f(x) is a curve which describes the behavior of f as either x or y go to infinity.

In other words, as one moves along the graph of f(x) in some direction, the distance between it and the asymptote eventually becomes smaller than any distance that one may specify.

A linear asymptote is essentially a straight line that a graphed curve becomes closer and closer but does not become identical to."

Notice that they say 'to infinity', 'the distance between it and the asymptote eventually becomes smaller than any distance that one may specify' and 'does not become identical to'.


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Peter H
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Posted: 25th Aug 2008 06:05
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yeah, functions are NEVER equal to their asymptotes.

usually because if they were they would be dividing by zero or some other "illegal" operation.

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ionstream
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Posted: 25th Aug 2008 06:06
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I think that means for real values of x because later it says:



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Mahoney
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Posted: 25th Aug 2008 06:08 Edited at: 25th Aug 2008 06:08
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Quote: ""


/asymptote argument

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tha_rami
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Posted: 25th Aug 2008 06:19 Edited at: 25th Aug 2008 06:27
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Okay, I'm convinced. 0.999... = 1.

This discussion convinced me that the asymptote argument, too, was false. It starts to get interesting on page 3.

Do note that pancakes are better than waffles in the same way 0.999... equals 1.


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ionstream
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Posted: 25th Aug 2008 06:51
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Welcome to the other side, brother. Now drink this ram's blood and your journey will be complete!

Quote: "where is the guarantee that 0.9 repeating is equal to 1."


It's in the proofs we just gave you!

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mamaji4
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Posted: 25th Aug 2008 11:19 Edited at: 25th Aug 2008 12:45
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The CONTRADICTION! The reason I started this thread.

By mathematical Induction:
--------------------------
For a = 9.9 accurate upto 1 decimal place
10a - a = 99.0 - 9.9
< 99.9 - 9.9
Therefore 9a < 90
a < 10
i.e. 9.9 < 10

For a = 9.99 accurate upto 2 decimal places
10a - a = 99.90 - 9.99
< 99.99 - 9.99
Therefore 9a < 90
a < 10
i.e. 9.99 < 10

For a = 9.999 accurate upto 3 decimal places
10a - a = 99.990 - 9.999
< 99.999 - 9.999
Therefore 9a < 90
a < 10
i.e. 9.999 < 10

Therefore, by induction
9.999... < 10 --- EQUN. 1


By limit of an infinite geometric series:
-----------------------------------------
We can represent the number
9.999... = 9 + 0.999...
0.999... can be represented by the infinite geometric series
0.999... = 0.9 + 0.09 + 0.009 + ...
= 9/10 + 9/100 + 9/1000 + ...
= 9/(10^1) + 9/(10^2) + 9/(10^3) + ...

oo
= Sum 9/(10^n) where oo is the symbol for infinity
n=1

But we have for an infinite geometric series the following result
a + a*r + a*(r^2) + ... where a and r are constants
oo
= Sum a*(r^(n-1)) = a/(1-r) where |r| < 1
n=1

Using the above result we see that our infinite geometric series has
a = 9/10 and r = 1/10
Therefore
oo
Sum 9/(10^n) = a/(1-r)
n=1
= (9/10)/(1-(1/10))
= (9/10)/(9/10)
= 1

i.e. Limit 0.999...n = 1
n->oo

i.e. Limit 9.999...n = 9+1 = 10
n->oo


Now comes the CONTRADICTION:
----------------------------
From EQUN. 1
10 > 9.999...
From EQUN. 2
9.999... = 10
Therefore,
10 > 10

You can imagine the implications.
1)If we accept this mathematical contradiction, math is not a perfect science.
2)If we assume the Theory of Induction is flawed and therefore EQUN. 1 is not accurate , math is not a perfect science
3)If we assume the Theory of Limits is flawed and therefore EQUN. 2 is not accurate, math is not a perfect science

SIDE NOTE:
----------
There seems to be some confusion where people think that the finite difference always termintes in a finite numberof zeros followed by a 1, that the same case will hold true for an infinite number of zeros.

i.e Although the follwing would hold true for finite number of
zeros,
1 - 0.9 = 0.1
1 - 0.99 = 0.01
1 - 0.999 = 0.001
1 - 0.9999 = 0.0001
1 - 0.99999 = 0.00001
1 - 0.999999 = 0.000001
1 - 0.9999999 = 0.0000001
1 - 0.99999999 = 0.00000001
.
.
.
1 - 0.999...n = 0.000...(n-1) 1

For an infinite number of zeros we can show mathematically:

Let us assume that the (n)th digit, after the (n-1)th zero, is 1
0.000...(n-1) 1 = 0/10 + 0/100 + 0/1000 + ... + (n-1)/10^(n-1) + 1/10^n

If we evaluate the first n-1 terms of this series
Limit 0.000...(n-1) = 0
n->oo

If we have to evaluate the last term of this series
Limit 1/10^n = 0
n->oo

Clearly at the end of an infinite number of zeros the digit 1 evaluates to 0

Therefore,
Limit 0.000...(n-1) 1 = 0
n->oo

Therefore,
1 - 0.999...n = 0.000...(n-1) 1
= 0
Therefore,
1 = 0.999...

Like I said earlier it is not so much a question of whether we can prove whether 1 = 0.999... ,but whether the Theory of Limits is inaccurate and math is not a perfect science
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ionstream
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Posted: 25th Aug 2008 11:53
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Induction only works with natural numbers, so it would fall apart when you try to stick infinity in there. The rest is true though.

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Leadwerks
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Posted: 25th Aug 2008 12:13 Edited at: 25th Aug 2008 12:18
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0.9...=1. It's not "inaccurate", it is just another way of writing it. That fact that the 9's never terminate means, by definition, it is 1.0. Not "equal to one", it IS one. The "contradiction" above doesn't make any sense, because it assumes the 9's will eventually terminate, which they won't.
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mamaji4
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Posted: 25th Aug 2008 12:24 Edited at: 25th Aug 2008 12:25
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Quote: "Induction only works with natural numbers, so it would fall apart when you try to stick infinity in there."


Mathematical induction:
If a relation holds true for n = 1,2,3...m where m,n are natural numbers then it holds true for m+1
As far as I know the set of natural numbers extend to infinity.
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Green Gandalf
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Posted: 25th Aug 2008 12:47 Edited at: 25th Aug 2008 12:48
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There is no contradiction - just an error of logic.

Here's the error:

Quote: "Therefore, by induction
9.999... < 10 --- EQUN. 1"




All induction can show is that each number in the sequence is less than 10 (undisputed) - it tells you nothing about the limiting value of the sequence. Which, believe it or not, is

wait for it ...

just a bit longer ...







Several people here seem to be recruiting for the Flat Earth Society.

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mamaji4
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Posted: 25th Aug 2008 12:54 Edited at: 25th Aug 2008 13:09
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If we assume the result 10 > 10 to be valid,
it is trivial then to show that n > n for any positive integer

But, if n > n for any positve integer n
m > n for any positive integer, m = n
Since m > n
m+x = n
x = n - m
= 0 since m = n
Since x = 0 , m = n
Therefore m CANNOT be greater than n , for m = n
i.e. n CANNOT be greater than n, for any positive integer n

CONCLUSION:
Anything can be proved and disproved using the different laws of mathematics, to suit yourself as you please.
MATH IS NOT A PERFECT SCIENCE!!!

SIDE NOTE:
----------
On an earlier thread I had proved(though not very rigorously) that an infinitely large number is greater than itself.
From the above proof, for any positive integer n, I showed that
n CANNOT be greater than n
Since natural numbers extend to infinity,
an infinitely large number CANNOT be greater than itself

I have disproved my original proof.

I repeat once again,
MATHEMATICS IS THE MANIPULATION OF EXISTING THEOREMS TO SUIT YOURSELF. IT IS NOT THE RIGOROUS SCIENCE IT IS MADE OUT TO BE.
Especially when no two mathematicians can generally agree upon the same thing.
And they are the ones who made the laws in the first place.
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mamaji4
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Posted: 25th Aug 2008 13:23 Edited at: 25th Aug 2008 13:56
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Quote: "Several people here seem to be recruiting for the Flat Earth Society."


It can easily be shown that the earth is flat.

A circle consists of a series of points linked to each other, each point being infinitesimally small.
An infinitesimally small point can only be defined by the slope at that point(Differential Calculus)
Lim dy/dx
dx->0

Clearly the slope is a straight line in the xy plane
i.e. any point on a circle is essentially "flat"

Any point on any Line of latitude or longitude on the earths surface is flat. (The infinitesimally small tile you are standing or sitting on right now is flat, to cite a practical example)
Since a circle is essentially a straight line that has no end,
the circle is a straight line.
Therefore by definition, the earth which is an infinitely large set of straight lines of latitude and longitude is flat.
(In case you are wondering why I consider the set of lines of latitude and longitude to be infinitely large, it because any point on any line is infinitesimally small)
The earth just appears to be spherical when looked at by the naked eye.(Maybe because our eyeballs are spherical and they distort all images that fall on the retina)
If you look at the earth with a mathematical eye it is clearly flat.
First impressions, by the naked eye can be deceiving, as you must have noticed after one month with your new date. Don't trust your naked eyes. Mathemtical proofs are more accurate.

You can manipualte any mathematical fact to prove anything.

And History is testament to the fact that math arguments can extend to the end of time, and even beyond, to infinity.

MORAL OF THE STORY:
Whenever you read a mathematical statement, question everything, take nothing at face value, while trying to hang on to your sanity.
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Green Gandalf
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Posted: 25th Aug 2008 14:10
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I've notified the men in white coats for you ...
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mamaji4
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Posted: 25th Aug 2008 15:34
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Quote: "I've notified the men in white coats for you ... "


Don't change the topic. Stand up and fight like a man.
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mamaji4
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Posted: 25th Aug 2008 19:05
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I'd like to apologise if I hurt any sentiments during the course of the discussion.
It's just that I get all fired up when the topic is math.
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Geryon
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Posted: 25th Aug 2008 20:11
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I hate this thread. When it comes to threads like this, I find smarter people at http://www.gamefaqs.com/boards/gentopic.php?board=400 (the current thread on that board is an exception). I can't tell you how hard it is not to just rant and flame everyone who doesn't accept mathematical proof as valid.

C'mon! Aren't we on this board a bunch of programmers? How is it that people can't accept concrete proof, when multiple versions reach the same conclusion?

Hey, Mahoney, you said I was becoming one of your favorite posters on the board. Is that favorite as in you like me? Or favorite as in you love to laugh at the ridiculousness you see in my posts?
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Mahoney
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Posted: 25th Aug 2008 21:19
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Quote: "Hey, Mahoney, you said I was becoming one of your favorite posters on the board. Is that favorite as in you like me? Or favorite as in you love to laugh at the ridiculousness you see in my posts? "


Things like "OH WAIT" right at the end. I find it awesome.

Now, about infinite series:

I may be explaining this incorrectly, but I'm going to try.

The layout for infinite series:



If |r| < 1 (r being the ratio), then



With the common ratio of 1/10, the equation works out simply:



Please correct any errors I made.

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ionstream
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Posted: 25th Aug 2008 21:20 Edited at: 25th Aug 2008 21:22
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Quote: "As far as I know the set of natural numbers extend to infinity. "


Yes but they don't include infinity, because infinity is certainly not an integer. It's similar to the fact that there are an infinite amount of numbers between 4 and 5.

Induction is meant to work with any natural number of n, given that n = 1, 2, and 3 are true, so you can pick any integer no matter how large. You cannot use infinity.

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RedFlames
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Posted: 25th Aug 2008 22:34 Edited at: 25th Aug 2008 22:34
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I have not read all of this thread (1. page ) because it would only result in a headache, so i dont know if 'my' idea has already been mentioned...

Isnt it as simple as, that the gap between 0,999... and 1.0 will be decreased to infinitly small size with every single one of the infinite '9's ? So basically there IS a difference but its INFINITLY small??

Just my 2 cents that prolly were proven wrong on page 2


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Mahoney
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Posted: 25th Aug 2008 22:37
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Quote: "Isnt it as simple as, that the gap between 0,999... and 1.0 will be decreased to infinitly small size with every single one of the infinite '9's ? So basically there IS a difference but its INFINITLY small??

Just my 2 cents that prolly were proven wrong on page 2 "


We did discuss this.

As much as instinct tells you that there would be that infinitesimal difference, you have to remember that we are talking about infinite 9's. It doesn't get to a point where there is a 0.<infinite 0's>1 difference, due to the fact that the infinite 0's keep the 1 from ever being reached at all!

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RedFlames
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Posted: 25th Aug 2008 22:41
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Quote: "We did discuss this.

As much as instinct tells you that there would be that infinitesimal difference, you have to remember that we are talking about infinite 9's. It doesn't get to a point where there is a 0.<infinite 0's>1 difference, due to the fact that the infinite 0's keep the 1 from ever being reached at all!"

Awww c'mon somewhen we'll find the 1

Well ill eventually read the other pages... but that will only confuse me a lot, where is that infinite amount of confusing threads on geek culture coming from

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Mahoney
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Posted: 25th Aug 2008 22:46
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Quote: "where is that infinite amount of confusing threads on geek culture coming from"


People love to argue about this, for some reason. But, once you give all the proofs, and do your best to answer the questions of the confused, you can't do anymore than let them listen to what they want to listen to.

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mamaji4
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Posted: 25th Aug 2008 23:01
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Quote: "Induction is meant to work with any natural number of n, given that n = 1, 2, and 3 are true, so you can pick any integer no matter how large. You cannot use infinity.
"


Please refer to the following link to clear any misconceptions about how inductin works.
http://en.wikipedia.org/wiki/Mathematical_induction

I do believe I have provided a proof by mathematical induction that
10 > 9.999...
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ionstream
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Posted: 25th Aug 2008 23:09
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Quote: "Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one."


Natural numbers, that excludes infinity.

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mamaji4
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Posted: 25th Aug 2008 23:11
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Quote: "Natural numbers, that excludes infinity"


Are you implying that the set of natural numbers is not infinitely large?
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Mahoney
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Posted: 25th Aug 2008 23:17
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Quote: "Are you implying that the set of natural numbers is not infinitely large? "


It's infinitely large because of the fact that it's limit (infinity) is never reached. That's elementary logic.

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ionstream
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Posted: 25th Aug 2008 23:20
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Mahoney's got it. It's infinitely large, but it does not include infinity. Infinity is not an integer. I think my calc book wrote out the range of integers to be something like:

-infinity < integers < infinity

So less than (or greater than) but not equal to.

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Mahoney
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Posted: 25th Aug 2008 23:25 Edited at: 26th Aug 2008 00:01
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Quote: "Mahoney's got it."


I find it an honor to have someone that I believe has been through/is in college say that (especially when discussing topics such as infinity), since I'm only in 11th grade. So, you just made my day.

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ionstream
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Posted: 25th Aug 2008 23:34
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I don't know if you're being sarcastic but I'm only a freshman in college!

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mamaji4
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Posted: 25th Aug 2008 23:43
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Quote: "Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one."


To show that 10 > 9.999... by mathematical induction, rigorously following the above definition, by extending it to the infinite sequence 9.999...

For n =1
a = 9.9 accurate upto 1 decimal place
10a - a = 99.0 - 9.9
< 99.9 - 9.9
Therefore 9a < 90
a < 10
i.e. 9.9 < 10
WE HAVE PROVED THAT THE FIRST STATEMENT OF THE INFINITE SEQUENCE IS TRUE.

For n = 2
a = 9.99 accurate upto 2 decimal places
10a - a = 99.90 - 9.99
< 99.99 - 9.99
Therefore 9a < 90
a < 10
i.e. 9.99 < 10

For n = 3
a = 9.999 accurate upto 3 decimal places
10a - a = 99.990 - 9.999
< 99.999 - 9.999
Therefore 9a < 90
a < 10
i.e. 9.999 < 10

For n = m, where m is any natural number in the infinite sequence of natural numbers
9.999...m < 10

For n = m+1
9.999...m+1 < 10

FOR ANY ONE STATEMENT IN THE INFINITE SEQUENCE OF STATEMENTS IT IS TRUE. THEREFORE, IT IS TRUE FOR THE NEXT STATEMENT.

Therefore, by induction
9.999... < 10
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ionstream
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Posted: 25th Aug 2008 23:50
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Again, the infinite sequence of natural numbers, but not infinity itself! The number of 9's must be an integer for your proof to work via induction!

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Mahoney
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Posted: 25th Aug 2008 23:51 Edited at: 25th Aug 2008 23:53
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Quote: "FOR ANY ONE STATEMENT IN THE INFINITE SEQUENCE OF STATEMENTS IT IS TRUE. THEREFORE, IT IS TRUE FOR THE NEXT STATEMENT.

Therefore, by induction
9.999... < 10"


But, as others have already pointed out and I have supported, induction only applies to values of n that are natural numbers. Infinity is excluded.

Quote: "I don't know if you're being sarcastic but I'm only a freshman in college!"


Although I just realized that it came off sarcastic, I was serious. You seem to know a lot more about mathematics than I. And, since I plan to go pretty deep into mathematics in college, I found it pretty awesome to actually have you agree with my argument about infinity.

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ionstream
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Posted: 26th Aug 2008 00:00
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Quote: "Although I just realized that it came off sarcastic, I was serious. "


Heh, it was the rolling-eyes face at the end that made me wonder . At my high school we had a one college level calculus course that I took, and one part of it taught all about the nature of infinity and limits and sequences and stuff. I wholeheartedly recommend that class if you're school has it! It had a lot of parts where everyone in the class would be like "no freakin way! how is that possible!" which were really fun and amazing once you understand it!

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mamaji4
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Posted: 26th Aug 2008 00:01
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Quote: " It's infinitely large, but it does not include infinity. Infinity is not an integer. I think my calc book wrote out the range of integers to be something like:

-infinity < integers < infinity

So less than (or greater than) but not equal to."


The reason you don't have the "=" is because infinity is by definition an undefined number

Quote: "But, as others have already pointed out and I have supported, induction only applies to values of n that are natural numbers. Infinity is excluded.
"

The definition of induction clearly states that it is valid for an infinite sequence, and still you insist that infinity is excluded.


Quote: "Again, the infinite sequence of natural numbers, but not infinity itself!"

I have no idea how to convince you that the set of natural numbers is an infinitely large set consisting of infinitely large numbers.
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Mahoney
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Posted: 26th Aug 2008 00:06
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Quote: "Heh, it was the rolling-eyes face at the end that made me wonder . At my high school we had a one college level calculus course that I took, and one part of it taught all about the nature of infinity and limits and sequences and stuff. I wholeheartedly recommend that class if you're school has it! It had a lot of parts where everyone in the class would be like "no freakin way! how is that possible!" which were really fun and amazing once you understand it!"


Oops. I used the incorrect smiley. XD

I'd love to take that class. I'm never as excited about learning something as I am when I read about things like 0.999... = 1, for instance.

But, I'm already taking Trigonometry for my senior year, and I'm not sure that I'd want to cram another class in. But, I plan on taking plenty of similar classes in college. So, it's all good.

Quote: "I have no idea how to convince you that the set of natural numbers is an infinitely large set consisting of infinitely large numbers."


I'm not denying the fact that it's infinitely large. What I'm saying is that the set of natural numbers includes infinitely large numbers due to that fact that it's limit is infinity. That limit will never be reached, because infinity isn't a number, nor is it an integer!

When infinity is thrown into the equation, things are very different.

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mamaji4
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Posted: 26th Aug 2008 00:11
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Quote: "because infinity isn't a number, nor is it an integer!"


I am at a total loss for words.
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ionstream
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Posted: 26th Aug 2008 00:12
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Quote: "The reason you don't have the "=" is because infinity is by definition an undefined number"


There are plenty of cases where a set includes infinity.

Quote: "I have no idea how to convince you that the set of natural numbers is an infinitely large set consisting of infinitely large numbers."


Think about it like this: There are an infinite amount of numbers between 1 and 2. The set between these two numbers is infinitely large, yet it does not contain 1 and 2 nor does it contain infinity. I believe my teacher called it "local" or "relative" infinity (something infinity anyways). The set of integers is the same case. For every integer there is one integer after it, creating an infinite set if integers. It will never contain positive or negative infinity.

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mamaji4
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Posted: 26th Aug 2008 00:18
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Quote: "Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one."


I think what you are in essence saying, is that the above definition of Mathematical Induction is flawed.

And you might even be right. Which is why I made this post in the first place. To show that math is manipulation to suit yourself.
At least you will have proved beyond a shadow of a doubt that I am right in that, even though my proof and the above definition of induction are flawed, according to you.
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Mahoney
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Posted: 26th Aug 2008 00:23
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Quote: "Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers. It is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if any one statement in the infinite sequence of statements is true, then so is the next one"


Quote: "I think what you are in essence saying, is that the above definition of Mathematical Induction is flawed."


Actually, I'm not. Notice the emphasis I added. I'm saying (and I think ionstream is saying) that Induction doesn't include infinity.

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