# .99999 does not equal 1

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#### narcberry

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##### Re: .99999 does not equal 1
« Reply #510 on: August 24, 2008, 07:42:58 PM »
The infinity digit of 0.9999... is not defined either, it actually doesn't exist.

#### Parsifal

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##### Re: .99999 does not equal 1
« Reply #511 on: August 24, 2008, 07:44:47 PM »
You cannot know the infinity digit or 0.999... but require it to be 9 for 0.999... to equal 1.

I don't see why knowing the infinity digit is necessary. Consider this:

Take An to be the nth element in the sequence 9 * 10-1 + 9 * 10-2 + ... + 9 * 10-n, where n is a finite positive integer.

Then there is some number B such that An < B < 1.

But because limn→∞ An = 1, then there will be some finite positive integer k such that An+k > B.

Therefore, every number that is less than 1 can be exceeded by An simply by making n a large enough, but finite, positive integer.

Since to have an infinite number of 9s simply means that no matter how large the number of finite 9s you count, there will always be at least one more, 0.999... = 1.

QED
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#### Roundy the Truthinessist

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##### Re: .99999 does not equal 1
« Reply #512 on: August 24, 2008, 07:45:24 PM »
The infinity digit of 0.9999... is not defined either, it actually doesn't exist.

It doesn't exist in reality, but it is defined.  Number theory, narc.  Try to keep up.
Where did you educate the biology, in toulet?

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#### Oscar Wilde

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##### Re: .99999 does not equal 1
« Reply #513 on: August 24, 2008, 08:10:51 PM »
Couldn't we at least feed narcberry on a more entertaining subject? Another Floating Oceans thread, perhaps?

Besides, I want to expand on my seamill theory...

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #514 on: August 24, 2008, 10:09:39 PM »
You cannot know the infinity digit or 0.999... but require it to be 9 for 0.999... to equal 1.

I don't see why knowing the infinity digit is necessary. Consider this:

Take An to be the nth element in the sequence 9 * 10-1 + 9 * 10-2 + ... + 9 * 10-n, where n is a finite positive integer.

Then there is some number B such that An < B < 1.

But because limn→∞ An = 1, then there will be some finite positive integer k such that An+k > B.

Therefore, every number that is less than 1 can be exceeded by An simply by making n a large enough, but finite, positive integer.

Since to have an infinite number of 9s simply means that no matter how large the number of finite 9s you count, there will always be at least one more, 0.999... = 1.

QED

The problem is assuming the limit tells you anything about n = infinity. I've given a simple example where the limit of n as it approaches infinity is completely different than when n actually is infinity. Your limit implies 0.999... = 1 but it does not follow that 0.999... = 1.

#### Raist

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##### Re: .99999 does not equal 1
« Reply #515 on: August 24, 2008, 10:14:48 PM »
The limit says approaching infinity, because at infinity, the answer is undefined.

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #516 on: August 24, 2008, 10:16:53 PM »
The limit says approaching infinity, because at infinity, the answer is undefined.

Exactly, a paradox. Your formulae require that n(infinity) = 1, yet n(infinity) = undefined.

This is because the idea of 0.999... is mathematically stupid.

#### Parsifal

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##### Re: .99999 does not equal 1
« Reply #517 on: August 25, 2008, 12:59:08 AM »
The problem is assuming the limit tells you anything about n = infinity. I've given a simple example where the limit of n as it approaches infinity is completely different than when n actually is infinity. Your limit implies 0.999... = 1 but it does not follow that 0.999... = 1.

I made no such assumption. All I did was state that since An+1 > An (because An+1 = An + 9 * 10-(n+1)), and since limn→∞ An = 1, there is always some value k such that if An < B < 1, then An+k > B. Therefore, any number less than 1 can be exceeded by increasing n enough, while keeping it finite. Since putting an infinite number of 9s onto the end of An cannot decrease its value, 0.999... is greater than every number that is less than 1.
I'm going to side with the white supremacists.

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#### zeroply

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##### Re: .99999 does not equal 1
« Reply #518 on: August 25, 2008, 08:09:51 AM »
Since putting an infinite number of 9s onto the end of An cannot decrease its value, 0.999... is greater than every number that is less than 1.

Actually, if I'm reading it correctly you've shown that 0.999... is greater than or equal to every number less than 1. You are assuming that you're working in a group which is topologically continuous, but not showing it.

So suppose we defined a real number as a sequence (n,d1,d2,d3,...) where n is an integer and d1,d2,d3,... are integers such that 0<=di<=9. You can work out operations to define a+b, a*b, etc. and the zero of the group would be (0,0,0,...)

Define (a,n1,n2,n3,...)=(b,m1,m2,m3,...) iff a=b, n1=m1, n2=m2, etc.

In that case we have (0,9,9,9,...) and (1,0,0,0,...) as distinct elements.

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #519 on: August 25, 2008, 12:35:05 PM »
Exactly.
0.999... != 1

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#### mwahaha

##### Re: .99999 does not equal 1
« Reply #520 on: August 25, 2008, 02:52:46 PM »
I have no clue why I still look at this thread.

#### Parsifal

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##### Re: .99999 does not equal 1
« Reply #521 on: August 25, 2008, 05:36:05 PM »
Actually, if I'm reading it correctly you've shown that 0.999... is greater than or equal to every number less than 1. You are assuming that you're working in a group which is topologically continuous, but not showing it.

So suppose we defined a real number as a sequence (n,d1,d2,d3,...) where n is an integer and d1,d2,d3,... are integers such that 0<=di<=9. You can work out operations to define a+b, a*b, etc. and the zero of the group would be (0,0,0,...)

Define (a,n1,n2,n3,...)=(b,m1,m2,m3,...) iff a=b, n1=m1, n2=m2, etc.

In that case we have (0,9,9,9,...) and (1,0,0,0,...) as distinct elements.

But in that case, you would need to know the infinity digit of 0.999... to comment on its equality to 1. That system doesn't work with infinite series.
I'm going to side with the white supremacists.

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#### zeroply

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##### Re: .99999 does not equal 1
« Reply #522 on: August 25, 2008, 07:45:12 PM »
Actually, if I'm reading it correctly you've shown that 0.999... is greater than or equal to every number less than 1. You are assuming that you're working in a group which is topologically continuous, but not showing it.

So suppose we defined a real number as a sequence (n,d1,d2,d3,...) where n is an integer and d1,d2,d3,... are integers such that 0<=di<=9. You can work out operations to define a+b, a*b, etc. and the zero of the group would be (0,0,0,...)

Define (a,n1,n2,n3,...)=(b,m1,m2,m3,...) iff a=b, n1=m1, n2=m2, etc.

In that case we have (0,9,9,9,...) and (1,0,0,0,...) as distinct elements.

But in that case, you would need to know the infinity digit of 0.999... to comment on its equality to 1. That system doesn't work with infinite series.

You wouldn't need to any more than you would working with regular arithmetic sequences. The sequence (1,2,3,1,2,3,1,2,3,...) is well-defined but doesn't have any terminal digit. You don't even need a repeating pattern, consider (1,2,1,1,2,1,1,1,2,1,1,1,1,2,...) which is perfectly well defined.

Since I already know that the first few terms are different, I can immediately say that 1 != 0.999... since we have that iff in there. If you know that the 44th terms don't match, you don't need to compare any further.

Addition and subtraction would be easy to implement, and multiplication shouldn't be too difficult either if you want to write out the rules involved.

Obviously I realize that the resulting structure wouldn't actually match the real numbers and eventually you'd run into all sorts of problems with the topology. Still, it would be fun to work out how far you could go based on that construction. I believe that's the naive view that Narc has of the reals - and it's what would seem obvious to someone who hasn't taken elementary analysis.

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #523 on: August 27, 2008, 09:57:59 PM »
So if you agree that 0.999... != 1, what are you guys debating?

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#### mwahaha

##### Re: .99999 does not equal 1
« Reply #524 on: August 27, 2008, 10:10:48 PM »
I have no clue why I still look at this thread.

#### Roundy the Truthinessist

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##### Re: .99999 does not equal 1
« Reply #525 on: August 27, 2008, 10:22:20 PM »
I have no clue why I still look at this thread.

Then please, for the love of God, stop posting in it.
Where did you educate the biology, in toulet?

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #526 on: August 28, 2008, 01:14:31 PM »
Coming back from the raist/hara/mod derailment....

We left off at:
So if you agree that 0.999... != 1, what are you guys debating?

#### Trekky0623

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##### Re: .99999 does not equal 1
« Reply #527 on: August 28, 2008, 01:25:32 PM »
What's up with the factorial at the end?

0.999...! does equal one, but that's not what we're discussing.

...is it?

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #528 on: August 28, 2008, 01:47:46 PM »
0.999...           !=              1

!= is used to say not equal.

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#### mwahaha

##### Re: .99999 does not equal 1
« Reply #529 on: August 28, 2008, 01:50:56 PM »
No, hun, that's what this is for: ≠.

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #530 on: August 28, 2008, 01:52:08 PM »
!=

<>

Most of us are multilingual in mathematics and science, sorry Muffz, you're just an idiot.

#### Trekky0623

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##### Re: .99999 does not equal 1
« Reply #531 on: August 28, 2008, 01:57:14 PM »
Never seen that before... weird.

But, 0.(9) = 1

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#### mwahaha

##### Re: .99999 does not equal 1
« Reply #532 on: August 28, 2008, 02:06:04 PM »
HA even someone as genius as Trekky didn't know, so I am not an idiot!

#### cmdshft

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##### Re: .99999 does not equal 1
« Reply #533 on: August 28, 2008, 02:06:36 PM »
HA even someone as genius as Trekky didn't know, so I am not an idiot!

I must be even "more genius" than Trekky, since I knew about it.

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#### rodeoboy

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##### Re: .99999 does not equal 1
« Reply #534 on: August 28, 2008, 02:17:06 PM »
Why the fuck is this thread still going?

0.9999 (recurring) =1

end of.
All cats have four legs. My dog has four legs. Therefore my dog is a cat.

#### narcberry

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##### Re: .99999 does not equal 1
« Reply #535 on: August 28, 2008, 02:25:00 PM »
It is still going because people cant just give in to the mathematical facts.

0.999... requires an infinite amount of 9's

whereas 1 requires none

How are these identical?

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#### Sir_Drainsalot

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##### Re: .99999 does not equal 1
« Reply #536 on: August 28, 2008, 03:53:59 PM »
Why the fuck is this thread still going?

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#### zeroply

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##### Re: .99999 does not equal 1
« Reply #537 on: August 28, 2008, 04:42:17 PM »
Why the fuck is this thread still going?

0.9999 (recurring) =1

end of.

We're defining "Narcberry numbers" as sequences (a,d1,d2,d3,...) where a is an integer and d1,d2,d3,... are integers such that 0 <= di <= 9

Then we have (a,n1,n2,n3,...) = (b,m1,m2,m3,...) iff a=b, n1=m1, n2=m2, etc.

Arithmetic operations are based on carrying digits etc.

Since (0,9,9,9,...) != (1,0,0,0,...) the challenge is to prove that Narcberry numbers are fundamentally different from the reals and show how. Extra credit points if you can identify a number system that bijectively maps to Narcberry numbers.

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#### cbarnett97

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##### Re: .99999 does not equal 1
« Reply #538 on: August 28, 2008, 06:02:12 PM »
It is still going because people cant just give in to the mathematical facts.

0.999... requires an infinite amount of 9's

whereas 1 requires none

How are these identical?

f(x)=1/(1-x)

if i start at 0.9 and then I keep adding nines to the end of it at what point does the function become undefined?
Only 2 things are infinite the universe and human stupidity, but I am not sure about the former.

#### Parsifal

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##### Re: .99999 does not equal 1
« Reply #539 on: August 28, 2008, 06:46:50 PM »
HA even someone as genius as Trekky didn't know, so I am not an idiot!

I must be even "more genius" than Trekky, since I knew about it.

As did I. It's used in pretty much all decent programming languages. The ones that suck use <>, which just looks ugly.
I'm going to side with the white supremacists.