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Geek Culture / Proofs that 0.999... = 1

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Zotoaster
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Posted: 5th May 2010 00:12
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Here's some interesting stuff I found out today:


First proof:
1/3 = 0.333...
3 * 1/3 = 0.999...
1 = 0.999...


Second proof:
1/9 = 0.111...
9 * 1/9 = 0.999...
1 = 0.999...


Third proof:
x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1



Fourth proof:
A geometric series can be expressed as:
a1 + a2*r + a3*r^2 + a4*r^3 + ... = ar / (1 - r)
So, for a real number, 0.999..., r = 1/10, so (9 * 1/10)/(1 - 1/10) = 9/10 / 9/10 = 1



Interesting eh?
Discuss.

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Shaun Of The Dead
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Posted: 5th May 2010 00:17
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What i dont understand is how your saying that 1 is a number that ISNT 1. 0.9999... is 0.9999... unless you round it up. It is still a seperate number.

Right?

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lazerus
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Posted: 5th May 2010 00:18
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http://lazerus-reborn.deviantart.com/
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David R
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Posted: 5th May 2010 00:20
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Quote: "3 * 1/3 = 0.999..."




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Zotoaster
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Posted: 5th May 2010 00:20
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Shaun of the dead,

There are recurring 9s (i.e., infinitely many). They are technically two ways of describing the same number.

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Math89
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Posted: 5th May 2010 00:20
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Well, the 'proof' is in the assumption that 1/3 = 0.333...
If you admit that 1/3 = 0.333..., then yes, 1 = 0.999... If you don't agree and say that an infinite accuracy can not represent a third, then it doesn't work.
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Zotoaster
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Posted: 5th May 2010 00:21 Edited at: 5th May 2010 00:21
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Then what is 1/3 as a real number?

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David R
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Posted: 5th May 2010 00:24 Edited at: 5th May 2010 00:24
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0.333 is a truncation of 1/3 it is not equal to it.

What you're saying is akin to pi = 3.14. If you base a proof off that, the lack of decimal accuracy completely destroys the point of it

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budokaiman
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Posted: 5th May 2010 00:24 Edited at: 5th May 2010 01:15
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I have a proof that x^-2 and 1/x^2 are the same, that I made when I was bored, I figure that it would fit here.

f(x)=x^-2
f'(x^-2)
f'(x)=-2x^-3

f(x)=1/x^2
f'(1/x^2)
=((x^2)f'(1) - f'(x^2)(1))/((x^2)^2)
=((x^2)(0) - (2x)(1))/((x^2)^2)
=-2x/((x^2)^2)
=-2x/x^4
f'(x)=-2x^-3

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Shaun Of The Dead
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Posted: 5th May 2010 00:25
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Yes. But 'technically' 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

Still isnt 1. It depends how you look at it, but it still isnt a whole number. Still a decimal.

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Zotoaster
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Posted: 5th May 2010 00:25
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David, the ... shows that it's recurring.

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David R
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Posted: 5th May 2010 00:37 Edited at: 5th May 2010 00:42
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OK, so 0.999 is actually officially =1 as far as the real numbers go. I did not know this

I still maintain the first proof is screwy/poor though. The value of 3*1/3 can be shown without reusing the literal value of a 1/3 shown before it. It's not the 'done thing'. But the logic of it equalling one makes sense

Quote: "David, the ... shows that it's recurring.
"


Yeah - because the second line shows it in a calculation I somehow managed to miss it/skim over it

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Rudolpho
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Posted: 5th May 2010 00:41 Edited at: 5th May 2010 00:45
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Quote: "f(x)=x^-2
f'(x)=x^-2

....

f(x)=1/x^2
f'(x)=1/x^2"

Those statements are invalid.


Otherwise I agree with Math89.
If 1/3 = 0.33... then 3 * (1/3) = 0.99...
But as 1/3 != 0.33... (no matter if the digits are infinite, it still isn't exactly correct) 0.99... != 1.

Edit:
Quote: "OK, so 0.999 is actually officially = 1 as far as the real numbers go. I did not know this"

Well, look at that

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DJ Almix
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Posted: 5th May 2010 00:42
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So is 2.99999-> 3?

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Rudolpho
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Posted: 5th May 2010 00:46
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No, but 2.999... is, apparently.

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General Jackson
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Posted: 5th May 2010 00:51
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Thats actually very interesting

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David R
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Posted: 5th May 2010 00:51
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Pretty decent quote I just found sums up quite nicely the arguments we've all made against it:

Quote: "The lower primate in us still resists, saying: .999~ doesn't really represent a number, then, but a process. To find a number we have to halt the process, at which point the .999~ = 1 thing falls apart. Nonsense"


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budokaiman
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Posted: 5th May 2010 00:58
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Quote: "Those statements are invalid."

Fixed.

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SikaSina Games
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Posted: 5th May 2010 01:02
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We have the same minds, Lazerus xD.

I partially understand theories 1-3 but I'm in GCSE so WTFH does the 4th one mean?!?!?! Lol, glad I'm not taking Maths for Sixth Form .

Basically, 0.999 would be 1 anyway when rounded up to the closest whole number, let alone the fact that it is a recurring decimal which signifies that it clearly is 1....

-SSG

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ionstream
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Posted: 5th May 2010 04:43 Edited at: 5th May 2010 04:44
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http://forum.thegamecreators.com/?m=forum_view&t=135800&b=2

This was discussed in great detail in that thread.

Math powers!

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OrzeL
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Posted: 5th May 2010 08:05 Edited at: 5th May 2010 08:06
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Quote: " "3 * 1/3 = 0.999..."
"


@DavidR i get what your saying because that really comes down to 3/3 which is 1 ... the first and second proof are really the same and I feel that you are forcing it to be 0.999... but the third proof seems most convincing and the fourth proof well i'm too lazy to get into that one.

The third proof is clever but if you really calculated it by determining how much accuracy you wanted it would equal back to w.e. number you started with as x as there is kind of unseen things happening with the last digit that is hard to account for.

Like say we used 0.999 and figured that accuracy satisfies me( mathematically )

then 10x = 9.99
then 10x - x = 9.99 - 0.999 (notice how they go to different decimal places and that would be impossible to account for if using ... to show reoccuring )
then 9x = 8.991 ( if using more accuracy like 0.9999 then 9x = 8.9991 )
then when you divide x = 0.999 (to whatever accuracy you chose)
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Rudolpho
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Posted: 5th May 2010 09:13
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Yes, but the deal is that there are infinitely recurring last digits.
Infinity + 1 = Infinity.
Infinity - 1 = Infinity.
So by that method, all the decimal nines actually would disappear.

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Virtual Nomad
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Posted: 5th May 2010 10:10
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i'll admit to not getting too far into math (calculus made me lose interest) but this kind of thing bugs me. i like black and white. ie, "it is or it isn't" and 0.999... isn't 1.

instead of declaring them equal, couldn't they have invented a new symbol, instead, which means "not equal but as close as you'll ever need to get" to distinguish the 2? i like to assume they avoided re-defining whatever this symbol might have stood for as we delve deeper and deeper and find the need for carrying the decimal to 1 shy of infinity

this topic makes me wonder how accurate do we ever need to be? ie, in whatever "micro engineering" scenarios we find our ways into, how many decimal places do we need to trudge through until we declare "this is close enough!"?

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Neuro Fuzzy
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Posted: 5th May 2010 11:51
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Quote: "and 0.999... isn't 1. "


the ... signifies repeating... it definitely is.

That's kind of like saying that the sum of
1/10+1/100+1/1000+1/10000+... 1/10^n
as n approaches infinity, isn't 1/9. Sure, you can't really calculate it out all the way... but it is what it is.

For fundamental truths in mathematics, one has to dispose of the idea of any true base number system. .333... in terms of thirds is 1. Fundamentally, .999... is 1. The problem arises when you want to calculate it all out into base 10. This is when you get into trouble with accuracy or whatever - but this is only when you want such a numerical result given a number.


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jrowe
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Posted: 5th May 2010 12:08
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"God made the integers; all else is the work of man"

For Fathers and Sons who enjoy wholy spirits.
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Dr Tank
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Posted: 5th May 2010 13:13
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Quote: "Yes. But 'technically' 0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

Still isnt 1."

That's because you need more 9s.
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Seppuku Arts
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Posted: 5th May 2010 13:49 Edited at: 5th May 2010 13:50
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0.999 is infinitely smaller than 1?

My limited maths knowledge would suggest this:

when you divide 1 by three, no value ought to be lost from the 'third' and for no value to be lost, there needs to be an infinite number, so you can never represent 1/3 of 1 accurately (hence we write 0.333...), so in appearance it looks like a value is lost, so when it comes to tripling it, it appears infinitely smaller than 1, but at no point should we consider there to be a value lost in the equation (where does that infinitely small number go? The answer is nowhere as the equation doesn't tell us to get rid of it) thus 0.999... = 1


Am I righteth?

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Shadowtroid
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Posted: 5th May 2010 13:57
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Quote: "Am I righteth?"


Yes.

I don't see why people don't get this. 0.999... if it was a finite number, would not work. But what we are saying is, it ISN'T a finite number. It goes on forever.

Is there no way to represent 1/3 as a number? According to some people's logic no. But there is, 0.333... And when that is .999... it IS 1!

So yes Sepp, you got it right.

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Red Eye
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Posted: 5th May 2010 14:50 Edited at: 5th May 2010 15:09
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Hi was making my maths homework:

And i saw this thread....

Here my idea about this...

As side note: Not saying it is or not... Just my opinon.



In the maths, to know her with certainty, with some certainty, we need to reinvestigate her basic assumpotions, u will need to start at the begin. If you are certain of ur propositions, then from there you could build ur infinitive certainty, till it ends somewhere so u need to reinvestigate ur assumptions again... lol paradox!!!

Anyway, i just had a new lamp burning inside my lamp, gonna edit this posts soon witha nother ppicture.

My edit:


Anotther edit:

Nice thread! Love this sort of things! Lets make a thread will all kind of this things... I would be in!

Other edit:

Think outside the box.

Later,

Red Eye

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mgarand
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Posted: 5th May 2010 15:06
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right... in other news...



Creativity is inventing, experimenting, growing, taking risks, breaking rules, making mistakes, and having fun.
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RedneckRambo
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Posted: 5th May 2010 19:38
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Didn't we already do this .9999... = 1 thing?

Signature's are stupid.
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David R
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Posted: 5th May 2010 19:46
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About two years ago, yes.

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bitJericho
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Posted: 5th May 2010 22:36 Edited at: 6th May 2010 02:06
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Quote: "Is there no way to represent 1/3 as a number? According to some people's logic no."


Sure... use base 3

base3(1 / 10) = base3(0.1) = base10(1 / 3) = base10(0.33333333...)

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Shadowtroid
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Posted: 5th May 2010 22:48 Edited at: 5th May 2010 22:48
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Quote: "Sure... use base 3"


Oh, awesome.

But yes! That can prove my point!

1/3 in base 3: 0.1
2/3 in base 3: 0.2
3/3 in base 3: 0.3 or 1

And 0.3 in base 3 is 0.99... in base 10.

Uhh...Seemed to work better in my head. Oh well.

I know that there is no 3 in base 3, but that's why these things cannot be used using normal numbers. It's all theory.

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Red Eye
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Posted: 6th May 2010 16:06
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And i dunno if anyone actually saw my post, but i think u cant say it anyway.

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Ocho Geek
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Posted: 6th May 2010 18:07 Edited at: 6th May 2010 18:08
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Quote: ""


that would be the most persuasive of your evidence, but I can only imagine it is a misconception of doing sums with 3 decimal points rather than true surds.

Either that or we have to scrap 1000's of years of mathmatic theorum and start from scratch

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Diggsey
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Posted: 6th May 2010 18:48
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People always get confused between real numbers, and our crude way of representing them using symbols.

The fact is, some numbers can be represented in multiple ways using our symbols.

ie. 0.9999... is the exact same number as 1, the only difference is the way we write it.

It's not like they are nearly the same but slightly different, they actually both mean the very same number.


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Red Eye
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Posted: 6th May 2010 19:38
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Quote: "People always get confused between real numbers, and our crude way of representing them using symbols.

The fact is, some numbers can be represented in multiple ways using our symbols.

ie. 0.9999... is the exact same number as 1, the only difference is the way we write it.

It's not like they are nearly the same but slightly different, they actually both mean the very same number.
"


So u mean:

1.00000 going on forever with .1 at the infinitive end is 1 as well?

Could u add a person to a hotel with inifitive rooms and infitive full.
Yes.

Means 1.00000000000000000 going for ever .1 isnt 1.

so is 0.99999 ..... not 1.

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Diggsey
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Posted: 6th May 2010 20:26 Edited at: 6th May 2010 20:29
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@Red eye
You can't have 1.000... with a 1 on the end, because if you have infinite zeroes, there is no end for the '1' to be
And yes, 1.0000... IS one. If you can understand that, then how is 0.9999... different?

Also your analogy with rooms doesn't work because infinity is not a number, it is a concept. You could theoretically have an infinite number of rooms, and an infinite number of people, but you then can't just add one person, because an 'infinite number' is not something you can add or subtract to or from.


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Red Eye
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Posted: 6th May 2010 21:16 Edited at: 6th May 2010 21:20
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Quote: "
Also your analogy with rooms doesn't work because infinity is not a number, it is a concept. You could theoretically have an infinite number of rooms, and an infinite number of people, but you then can't just add one person, because an 'infinite number' is not something you can add or subtract to or from."


Totally wrong.

Move all(infinitive) people one room (so people in first go to 2nd and 2nd to 3rd and so on, (infinitive). And 1fst room gets free.
There you go...
This is possible as there are infinitive rooms.

Quote: "You can't have 1.000... with a 1 on the end, because if you have infinite zeroes, there is no end for the '1' to be"


U can actually. I dunno if u know what domain is of a function, but it is a quite possible. Theoretical.

But lets say u cant, then u cant say 0.9999 is 1, as the above is not true. So it cant be said anyway.

Edit:

0.9999(infinitive) * 3 is not 3

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Indicium
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Posted: 6th May 2010 21:22
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Quote: "x = 0.999...
10x = 9.999...
10x - x = 9.999... - 0.999...
9x = 9
x = 1"


^ That pretty much proves it imo.

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Zotoaster
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Posted: 6th May 2010 21:29
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Quote: "U can actually. I dunno if u know what domain is of a function, but it is a quite possible. Theoretical."

What does this have to do with the domain of a function? If there are infinitely many 0s, there is no place for the 1, so yeah, the one simply doesn't exist here.


Quote: "But lets say u cant, then u cant say 0.9999 is 1, as the above is not true. So it cant be said anyway."

No, because you're essentially thinking, "somewhere, you have to take away 0.0000....0001 to get 0.999...", but this decimal place is infinitely many places along, so you never take it away, so it's essentially 1 - 0 = 1.

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David R
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Posted: 6th May 2010 21:42
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Quote: "I dunno if u know what domain is of a function, but it is a quite possible. Theoretical."


If we're talking set theory/tuples the domain of a function is composed of every element in the source/first part of each tuple... but what's that to do with anything?

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Diggsey
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Posted: 6th May 2010 21:46 Edited at: 6th May 2010 21:48
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Quote: "Totally wrong."


No it's not... http://en.wikipedia.org/wiki/Infinity

Quote: "Infinity is a concept in mathematics"


Quote: "Move all(infinitive) people one room (so people in first go to 2nd and 2nd to 3rd and so on, (infinitive). And 1fst room gets free.
There you go...
This is possible as there are infinitive rooms. "


That still doesn't mean that you can add and subtract from infinity in the same way as other numbers, because infinity is a concept.

Quote: "U can actually. I dunno if u know what domain is of a function, but it is a quite possible. Theoretical."


No you can't. These numbers are recurring decimals. Recurring means 'without end'. If a number has no end, then it is meaningless to say 'add a 1 on the end'.

To show that it wouldn't work if you said a recurring number had an end, try this:

x = 0.00000...1 (infinite 0s with a 1 at the end)
10x = 0.00000...1

10x - x = 0.00000...1 - 0.00000...1
9x = 0
x = 0

Therefore, the mysterious '1' at the end doesn't actually exist!


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Red Eye
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Posted: 6th May 2010 21:50 Edited at: 6th May 2010 21:51
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Quote: "If we're talking set theory/tuples the domain of a function is composed of every element in the source/first part of each tuple... but what's that to do with anything?"

Dunno. xd just dropped that idea. But nvm.

Quote: "That still doesn't mean that you can add and subtract from infinity, because infinity is a concept."


I didnt said that. I just said it is possible to add doesnt matter wich way, (outside the box).

O well. I will just stop here.

Quote: "No it's not..."


Sorry but i stand here and say it is wrong. Sorry, but Logics goes beyond maths. And wrong source anyway.

U guys should read Bertrand Russell his book, alot of this gets fully covered (infinitive covered).

Later,

Greets.

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Cyborg ART
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Posted: 6th May 2010 23:26
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Dividing 1 with 3 will cause serious memory leaks in universe. Please don't overdo it since it will steal to many resources...
Divide it once and you got 3s everywhere.



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Indicium
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Posted: 6th May 2010 23:36
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As far as DBpro is concerned

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Interplanetary Funk
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Posted: 7th May 2010 00:32 Edited at: 7th May 2010 00:33
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I'm no maths expert, but my understanding is that infinity + infinity = infinity while infinity-ANYTHING can be equal to anything from -infinity to infinity, as well as different mathematical formulas each give a different value for infinity, so as far as I understand it infinity is a variable number that can be equal to any number too large or too small to comprehend..?

ah! what do I know! =( plus I just fried my brain a little.. -.-

edit:
Also, anyone here heard of Grahams number?
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Shadowtroid
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Posted: 7th May 2010 01:18
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Ugh, I hate it when people do this...

INFINITY IS NOT A NUMBER! Just a concept! Anyone who does NOT believe that doesn't know what they are talking about.

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General Jackson
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Posted: 7th May 2010 01:43
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Quote: "Just a concept!"

Its not a concept its a hard cold fact. Numbers CANNOT end.

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