0.9_ = 1?

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twoshots
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30 Jan 2009, 2:51 pm

ruveyn wrote:

Death_of_Pathos wrote:

Do you deny that that that is 1?

Yes. It is pi squared over 6.

ruveyn

What he said.

This is a result of fourier series. We consider the fourier series for f(x)=x^2 on [-pi,pi] It can be shown that this is given by
f(x)= (π^2)/3+4Σ[(-1)^n]cos(nx)/n^2
f(π) = π^2 clearly, but the left hand side is
(π^2)/3+4Σ1/n^2 = π^2
Hence,
Σ1/n^2 = (2π^2)/12 = (π^2)/6

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Death_of_Pathos
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30 Jan 2009, 3:10 pm

twoshots wrote:

ruveyn wrote:

Death_of_Pathos wrote:

Do you deny that that that is 1?

Yes. It is pi squared over 6.

ruveyn

DOH!

Put up the wrong image. My original host refused to let me remote link, so I had to save and upload elsewhere. Mustve gotten mixed up. Soz.

All that times (6/pi^2) = 1, then.

robo37
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01 Feb 2009, 2:45 pm

Death_of_Pathos wrote:

A) whatever gave you the impression that 1.0_ isn't a repeating decimal?

But I could easily say that 0.9_ has an infinite number of nines in it and 1 doesn’t instead.

Death_of_Pathos wrote:

B) whatever gave you the impression that numbers are defined by their representations? (instead of the other way around)

If representations are defined by their numbers then numbers are defined by their representations.

Death_of_Pathos wrote:

C) whatever gave you the impression that each representation of number is unique?

Each representation represents a different thing.

Death_of_Pathos wrote:

D) Why would you cite Uncyclopedia as a reference, unless you were a troll? (in deed or intent, it doesn't really matter)

Because Uncyclopedia has a point, and I'm not trolling, a third of people do seem to think that 0.9_ doesn't equal 1. Are they trolling?

RockDrummer616 wrote:

there is no such number as 0.0_1, since there are infinite 0s, you will never get to the 1.

But you could say the same thing about 0.9_, Because there are infinite 9's you never get to the last 9 which makes the number 1.

Hector
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01 Feb 2009, 3:20 pm

robo37 wrote:

Death_of_Pathos wrote:

A) whatever gave you the impression that 1.0_ isn't a repeating decimal?

But I could easily say that 0.9_ has an infinite number of nines in it and 1 doesn’t instead.

1.0_ has an infinite amount of zeroes.

robo37 wrote:

Death_of_Pathos wrote:

B) whatever gave you the impression that numbers are defined by their representations? (instead of the other way around)

If representations are defined by their numbers then numbers are defined by their representations.

False

robo37 wrote:

Death_of_Pathos wrote:

C) whatever gave you the impression that each representation of number is unique?

Each representation represents a different thing.

False

robo37 wrote:

RockDrummer616 wrote:

there is no such number as 0.0_1, since there are infinite 0s, you will never get to the 1.

But you could say the same thing about 0.9_, Because there are infinite 9's you never get to the last 9 which makes the number 1.

Some "last 9" is not necessary, you only need the infinite series.

robo37
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01 Feb 2009, 3:41 pm

Hector wrote:

1.0_ has an infinite amount of zeroes.

But doesn't have an infinite amount of nines. 0.9_ has an infinite amount of nines so if 0.9_ is equal to 1 then you can say that 1 has an infinite amount of nines.

Last edited by robo37 on 01 Feb 2009, 3:48 pm, edited 1 time in total.

Hector
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01 Feb 2009, 3:46 pm

Not every number has a unique decimal representation. In the case of 1, you can also express it as 0.9_ (as a real number). So even though 0.9_ has an infinite amount of nines and 1 doesn't, there's still no problem with that.

nudel
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02 Feb 2009, 4:18 am

There seems to be confusion regarding the definition of decimal representation. I'll just post this one from Wikipedia:

For 0.9_, a(1) = 0 and a(n) = 9 for all n>1. So this is simply a geometric series, which converges to (9/(1-(1/10)))-9 = 1 using this formula:

Shiggily
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02 Feb 2009, 4:51 am

Hector wrote:

it seems that finite decimals do not have unique decimal representation.

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Hector
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02 Feb 2009, 5:52 am

Shiggily wrote:

Hector wrote:

it seems that finite decimals do not have unique decimal representation.

That is correct.

robo37
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02 Feb 2009, 10:55 am

Let me put it differently. The sentence '0.9_ has an infinite number of nines' is true. If 0.9_ is equal to 1 then the sentence '1 has an infinite number of nines' would be exactly the same as the sentence '0.9_ has an infinite number of nines'. If the sentence '0.9_ has an infinite number of nines' is true, then the sentence '1 has an infinite number of nines' (being the exact same sentence) is also true. As we all know 1 doesn't have any nines in it meaning that 0.9_ must not equal 1.

ruveyn
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02 Feb 2009, 11:09 am

Hector wrote:

Shiggily wrote:

Hector wrote:

it seems that finite decimals do not have unique decimal representation.

That is correct.

Any real number is the limit of an uncountable infinity of Cauchy sequences. There is no unique sequence that converges to a given real number as a limit. Infinite decimal expansions are shorthand for infinite series which in turn are sequences of partial finite sums that converge to a limit (that is what is meant by a convergent series).

ruveyn

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02 Feb 2009, 1:38 pm

robo37 wrote:

The number of 9s is irrelevant, they are two different ways to express the same number. How many 7s are in "Pi?" None, just a "P" and an "I."

Death_of_Pathos
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02 Feb 2009, 1:38 pm

robo37 wrote:

lol.

False syllogism is false.

azulene
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12 Aug 2009, 11:15 am

kalantir wrote:

1/3 = 0.33333_
2/3 = 0.66666_
1 = 3/3 = 0.99999_

Thats my take on it...

I don't know who can argue against the above, its pretty graphic and easy to understand. It also sneakily helps to avoid the notion of convergence, which I doubt is even relevant in this situation.

The rest of this thread is the natural confusion of people trying to comprehend infinity without accepting that to do so would make it finite and violate its own definition.

I had to dig this thread up because this problem made me lose interest in mathematics. Not due to the sort of argument that has gone on here, but because there are lots of valid proofs for it. At university I came up with one that wasn't in the text book. They looked at it, said I was right, but to do it the way I was told instead. Mathematics was actually pretty exciting for me until that very moment.

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ruveyn
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12 Aug 2009, 4:24 pm

robo37 wrote:

People seem to think that 0.9_ (0.9999999999...) = 1, but I'm not too shore. At http://en.wikipedia.org/wiki/0.999_%3D_1 Wikipedia says it is and brings up some evidence for it but Uncyclopedia has evidence to prove that 0.9_ doesn't equal one and proves Wikipedia's evidence wrong at http://uncyclopedia.wikia.com/wiki/0.999... . I know Uncyclopedia specialises in making stuff up but it has come up with some good evidence, like the fact that anything minus itself always equals 0, but 1-0.9_ equals 0.0_1, not 0. Do you think 0.9_ = 1?

Do you know what a convergent series is?

ruveyn

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12 Aug 2009, 10:58 pm

Ok, here's how I understood it. If 0.999... were NOT equal to 1, then there has got to be some other number which you could place between 0.999... and 1. But, since the nines after the decimal place go on forever, you cannot fit in any other number between 0.999... and 1. Therefore, 0.999... IS equal to 1.

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