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

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mamaji4
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Posted: 23rd Aug 2008 23:36
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Proof:
Let a = 9.9999999...
10*a = 99.9999999...
10*a - a = 99.9999999... - 9.9999999...
9*a = 90
a = 10
Therefore, we have shown that
10 = 9.9999999...
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Math89
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Posted: 23rd Aug 2008 23:43
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It's easy to see where is the problem . Just don't take all those decimals :

Let a = 9.99
10*a = 99.9
10*a-a = 89.91 and not 90

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mamaji4
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Posted: 23rd Aug 2008 23:47
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Yep.
But 9.999...n where n -> infinity is assumed to be an infinitely recurring decimal.
All the laws of number theory get messed up at infinity.
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Cash Curtis II
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Posted: 23rd Aug 2008 23:49
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You can't prove something is not itself anyway. You should have known you made an error.


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Aaron Miller
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Posted: 23rd Aug 2008 23:49 Edited at: 23rd Aug 2008 23:55
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I assume you're using a calculator. Calculators, after a certain repetitive decimal, will round up. For example, 1/3 = .33333333333~ (One third). If you add two thirds to one third you'll get one whole. The problem is that when you use decimals to represent this .333333333333333 * 3 (The sum of one third and two thirds, or the product of one third and three) you'll get .999999999999999 if you write it all out by hand. So I believe computers (or maybe just the calculator's code) will round this value up so that it will be correct - 1 whole.

Quote: "let a = 9.99"

Reminds me of the original BASICs.

EDIT
Speaking of infinity, any number divided by zero logically equals infinity. Why? Lets look at it in this perspective: If you have a crate that can hold 15 cubic meters, and you have "magical" apples that take up absolutely no space, how many apples would fit into the crate? Well you would have to divide the size of the crate by the size of each apples, and if you were to say, have an infinite supply of these "magical" apples that take up no space, how many would fit into the crate? All of them, infinity. You keep on placing the apples into the crate, and each time you do you're not adding any space.

Cheers,

-naota

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mamaji4
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Posted: 23rd Aug 2008 23:53
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I didn't use a calculator because you can't represent an infinitely recurring decimal like 9.999999... on a calculator or on any finite data store.
It's a purely theoretical proof.
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Aaron Miller
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Posted: 23rd Aug 2008 23:56
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Well then you obviously made a mistake doing your math.

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bitJericho
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Posted: 23rd Aug 2008 23:56
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http://en.wikipedia.org/wiki/0.999...


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mamaji4
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Posted: 23rd Aug 2008 23:57
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By not using a calculator which would give me erroneous results, I made a mistake doing math?
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mamaji4
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Posted: 24th Aug 2008 00:02
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Jerico2day,

Yep, it questions whether the theory of Limits is an approximation or gives you the true value of an infinite geometric series.
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Aaron Miller
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Posted: 24th Aug 2008 00:05
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Those "proofs" don't mean anything. It just shows work arounds. Just as people work around bugs in software.

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-naota

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mamaji4
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Posted: 24th Aug 2008 00:13
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Aaron,

The Theory of Limits is well established. And it says that
0.999... = 1
If we don't agree to the above equality we are in essence saying that the Theory of Limits is 'inaccurate' and an approximation. That would be shaking up the very foundation of mathematics.
Its better then to accept that 0.999... = 1
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Aaron Miller
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Posted: 24th Aug 2008 00:16
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I do say it's inaccurate. 2 is not 3, why would 2.9 be 3? There are some other things which I'm not allowed (by the AUP) to talk about on these forums that can be compared to this as well. Any such belief that 0.999... = 1 is wrong. I say it's just a work around. An apple is not an orange, despite being made of atoms.

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-naota

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RalphY
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Posted: 24th Aug 2008 00:16
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Well surely 9.9... *9 would be 89.9...1 if you think about it.

9*9.9 = 89.1
9*9.99 = 89.91
9*9.999 = 89.991
9*9.9... = 89.9...1

Though I admit my maths is terrible.

You would get the same answer with 10*9.9...-9.9... as that would be 99.9...0 - 9.9...?

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mamaji4
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Posted: 24th Aug 2008 00:21
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Another way of demonstrating the proof.
Let a = 0.999...n , n -> infinity
1 - a = 1 - 0.999...n
= 0.000...n , n -> infinity

It can easily be shown that the infite series 0.000...n = 0
Therefore,
1 - a = 0
a = 1
0.999... = 1
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Aaron Miller
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Posted: 24th Aug 2008 00:24 Edited at: 24th Aug 2008 00:27
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Quote: "Let a = 0.999...n , n -> infinity
1 - a = 1 - 0.999...n
= 0.000...n , n -> infinity"

No,
= 0.000...n1.

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

EDIT
I have to go now - I'll continue talking in this thread tomorrow.

Cheers,

-naota

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mamaji4
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Posted: 24th Aug 2008 00:26
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Further compounding the problem is the following counterproof using induction. This would show that the Theory of Limits is an approximation and we could no longer call math a perfect science.
Whether my proof is valid or not is a moot point.

Here goes:
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. 9.99 < 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
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RedneckRambo
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Posted: 24th Aug 2008 00:28 Edited at: 24th Aug 2008 00:30
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Quote: "
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"

Game! Aaron wins. Lol jk.

I have no idea what the hell is going on.

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Mahoney
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Posted: 24th Aug 2008 00:32
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Quote: "I do say it's inaccurate. 2 is not 3, why would 2.9 be 3? There are some other things which I'm not allowed (by the AUP) to talk about on these forums that can be compared to this as well. Any such belief that 0.999... = 1 is wrong. I say it's just a work around. An apple is not an orange, despite being made of atoms."


First off, I'm curious how the AUP limits your discussion on mathematics.

Secondly, consider this:

1/3 = 0.33333...
3 x 1/3 = 1
3 x 0.33333... = 0.99999...
0.99999... = 1

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mamaji4
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Posted: 24th Aug 2008 00:33
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I don't know if you can represent the number as
0.000...n1 n -> infinity
because when n -> infinity you can only be sure of one thing that there will be an infinite number of zeros, without end.
By putting a 1 after n you are saying that the infinte number of zero's is FINITELY long and is then followed by a 1

Assuming that both your and my counterproof are accurate and
0.999... is not equal to 1
Does that mean that the Theory of Limits is inaccurate?
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mamaji4
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Posted: 24th Aug 2008 00:36
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If we don't want to shake up the foundations of mathematics, I personally feel that you have to accept that
0.999... = 1
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mamaji4
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Posted: 24th Aug 2008 00:41 Edited at: 24th Aug 2008 00:45
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But the inductive proof I gave shows 9.999... < 10
Does that mean that the Theory of mathematical induction is accurate and the Theory of Limits is inaccurate?
Things are really getting interesting.

Frankly, the only thing I am now sure of is that mathematics is not the perfect science that 2 + 2 = 4 makes it out to be.
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ionstream
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Posted: 24th Aug 2008 00:41
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0.9 bar is indeed equal to one. We had this problem in ninth grade, in which we were instructed to make a geometric series out of it:

0.9~ = 9/10 + 9/100 + 9/1000 ...

0.9 + 0.09 + 0.009 =

r = 0.1

So any term could be represented by:

g(n) = 0.9 * (0.1)^(n-1)

Then use the infinite sum of a geometric series:

S = (a1/( 1 - r ))
S = 0.9/(1 - 0.1)
S = 0.9/0.9
S = 1

Another way to think of it is that 7/9 = 0.7~, 8/9 = 0.8~, 9/9 = 0.9~ and anything over itself is 1.

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Mahoney
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Quote: "7/9 = 0.7~, 8/9 = 0.8~, 9/9 = 0.9~"


GENIUS! Congratulations.

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Cash Curtis II
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Posted: 24th Aug 2008 00:48
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@Jerico -
Thanks for the link.

Here's a better proof...




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mamaji4
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Posted: 24th Aug 2008 00:56
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Another way to look at it.
Just as Newton's Laws of classical mechanics are invalidated by the Law of Relativity at the limiting speed of light, I feel that the Laws of classical number theory are invalidated at the limits of infinity.
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tha_rami
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Posted: 24th Aug 2008 00:58
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I get the proof, but it doesn't make sense. A individual value does only equal itself, doesn't it? So, what would 1-0.999... be? Surely not 0? Like Aaron Miller said, 1-0.9 = 0.1, 1-0.09 = 0.01 ect. ect.


A mod has been erased by your signature because it was larger than 600x120
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Seppuku Arts
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Posted: 24th Aug 2008 01:06 Edited at: 24th Aug 2008 01:07
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Surely 10 is 10? We define it as being 10, thus it is 10, how can it be any number other than what we define it to be? Unless I'm missing something?

Is '1' not a complete number? Just as we define it? If I have 10 1s then what's missing from the value - say if I have 10 cat - is 1 cats only very slightly not a cat?

Or am I reaching a level of maths that contradicts any basic level of maths?

You sir have the moral ambivalence of a mutated shrimp!
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mamaji4
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Posted: 24th Aug 2008 01:15
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We have a funny paradox here:
In a math paper the question is,
Evaluate 0.999...
If you answer 0.999... = 1 is it correct, or if you answer 0.999... < 1 is it correct?
Or should the professor who is himself in doubt, toss a coing before marking it right or wrong?
Boy am I glad I'm out of High School.

Dad: You got that sum wrong, didn't you?
Boy: The prof is wrong?
Dad: How do you know the proof is wrong?
Boy: Because the other prof said it was right?
Dad: I thought you said the proof was wrong.
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RedneckRambo
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Posted: 24th Aug 2008 01:19
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Maybe our math is just wrong in general. Lol.

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Seppuku Arts
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Posted: 24th Aug 2008 01:25 Edited at: 24th Aug 2008 01:26
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Maths can't be very accurate if 1 = 0.999. Applying it to objects it appears illogical, how can my mouse be anyless than itself than itself? IE, 1 of itself.

This appears to be why some numbers cannot be accurately split up into 3rds or 6ths and is just a flaw in mathematics and it probably why in some equations it's better to use fractions than decimals.

You sir have the moral ambivalence of a mutated shrimp!
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mamaji4
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Posted: 24th Aug 2008 01:25 Edited at: 24th Aug 2008 01:28
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Jenkins,

You the only one on the forums that got it right. It's wrong.

I think I'm getting a splitting headache. Now where's that Paracetamol.
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Mahoney
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Quote: "Maths can't be very accurate if 1 = 0.999."

Not 0.999, 0.<infinite 9's>.

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Aaron Miller
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I managed to make it back to the internet today.

I have this to say: MATH IS INACCURATE. It's not perfect.

Also, if we limit the decimals to 3 places in 1/3, then we get 0.333. If we then multiply this number by 3 we get 0.999. If we increment the decimal places to 4 then 1/3 = 0.3333 and then multiply again we get 0.9999. Let me put this in different terms. We'll multiply 0.999 by 1000 and make 999. Now, we'll multiply 1.0 by 1000 and make 1000. 999 is not 1000 as I'm sure you can notice. For maths 0.333333 * 3 = 1.0 is 0.999999 rounded up.

I'll put this in other terms. Take this C program for example:


What do you think the output is? This:


If 0.99999~ is equal to 1.0 then why does this C program output two different values and not the same value? Because they're NOT the same. 0.9 is not 1.0. 0.99 is not 1.0. 0.999 is not 1.0. 0.9999 is not 1.0. A dragon is not a dragon fly, no matter how small a difference in the wording. A potato is not a tomato, no matter how small a difference in the wording. So why would a number be another number due to a small difference in their values? Since a decimal is only a scale then what if we move the scale up and ignore decimals? Is 99999999999 really just 100000000000? No, it is not. Multiply 2 by 2 and you get 4. Multiply 2 by 2.5 and you get 5.

Cheers,

-naota

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ionstream
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Posted: 24th Aug 2008 01:51 Edited at: 24th Aug 2008 01:52
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Aye, the ". . ." at the end implies infinite 9's.


0.9999 != 1
0.99999999999999999999999 != 1

0.9 with an infinite amount of 9's at the end is equal to 1. Math is not inaccurate.

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Aaron Miller
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@ionstream
Check the first 3 values of 0.999... and 1.0. 1.0 is really just 1 with infinite zeros at the end, so does that mean it's really 1.1?

Cheers,

-naota

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Mahoney
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Quote: "@ionstream
Check the first 3 values of 0.999... and 1.0. 1.0 is really just 1 with infinite zeros at the end, so does that mean it's really 1.1?

Cheers,

-naota"


You're completely missing the point of the theory.


Quote: "In mathematics, the recurring decimal 0.999… denotes a real number equal to 1."


We never said that the same applies with 0's.

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ionstream
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1.0 is not 1.1, as adding zeroes to a number won't increase it. Adding a 9 however causes the number to approach one, with it will only reach if there are an infinite amount.

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Benjamin
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0.999... isn't a real number.

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Alucard94
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Nothing that goes infinite is a real number.

Hey I have a challenge, write down 9,9999... and see how long it takes you (I know it proves nothing I just had to say it)


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Mahoney
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Quote: "Nothing that goes infinite is a real number. "


Quote: "In mathematics, the recurring decimal 0.999… denotes a real number equal to 1."


Emphasis added.

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Aaron Miller
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I'm aware adding 0 to a number doesn't increase it. But since you did say that:

2 + 0.1 = 2.1 + 0.01 = 2.11 + 0.001 = 2.111

How will that make 2 equal 3 if the number value continuously gets smaller? Yes, it gets closer, infinitely, but the number also gets smaller infinitely. Try to continuously add to that algorithm above and see if you can reach 3. You can't.

Cheers,

-naota

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Seppuku Arts
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Quote: "Quote: "Maths can't be very accurate if 1 = 0.999."
Not 0.999, 0.<infinite 9's>."


I know, 0.999 was just the format we were told to use to represent recurring numbers at school. Saves saying 'infinite <x>' all the times, especially when we know we're talking about recurring numbers.

So...again, is 1.0 the same as 0.999, because I'm seeing 2 different values.

The only way I can see it to make sense is to accept that mathematics is flawed.

You sir have the moral ambivalence of a mutated shrimp!
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ionstream
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You're right, adding one's won't work either but for a different reason. 0.1~ can be expressed as 1/9. 2 + 1/9 = 2.1~. Putting .1 as a1 into the geometric sequence proof would yield .1/.9, which is 1/9. As stated before, 0.9~ can be expressed as 9/9, which is one.

Guys, stop saying that mathematics is flawed, people much smarter than all of us combined have been studying this for literally thousands of years and these are the facts they came up with! Just because you don't understand a small part doesn't mean the entire system is broken.

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Benjamin
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0.999... isn't a tangible number.

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Aaron Miller
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Posted: 24th Aug 2008 02:55 Edited at: 24th Aug 2008 02:57
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Quote: "Guys, stop saying that mathematics is flawed, people much smarter than all of us combined have been studying this for literally thousands of years and these are the facts they came up with! Just because you don't understand a small part doesn't mean the entire system is broken."

Mathematics are flawed.

http://en.wikipedia.org/wiki/Weird_number

Cheers,

-naota

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dark coder
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Posted: 24th Aug 2008 03:06 Edited at: 24th Aug 2008 03:07
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Quote: "This appears to be why some numbers cannot be accurately split up into 3rds or 6ths"


They can be, look: 1/3 and 1/6. I just did it, the issue here comes when you try to represent either of these in decimal, but we can just say it's 0.333... and 0.1666... .

Quote: "If 0.99999~ is equal to 1.0 then why does this C program output two different values and not the same value? Because they're NOT the same. 0.9 is not 1.0."


Really? I just tried this now and both my 'float var1 = 0.99999999999999999999999f; double var2 = 0.99999999999999999999999;'
showed up up as 1.0 in the debugger. Besides what does C have to do this?

I notice when you write 0.999... you miss out the ellipsis, this possibly leads to confusion as you try to visualize the number as simply 0.999 hence why you wrote your series of sums using 0.9, 0.99 etc. But the fact is that you cannot show 0.999... as it's an infinite series of 9, you cannot show 0.999... in a C data type such as a float or double, it's merely a concept and not a real number. Because of this you can't say its like saying 0.999 = 1 or 0.9999 = 1 because it isn't, those are real numbers, 0.999... isn't. Besides if you read the page linked to earlier it has thousands of words on the proof and shows arguments like the one you're raising here.

Edit:

RE: [/href]http://en.wikipedia.org/wiki/Unsolved_problems_in_mathematics, just because something is unsolved doesn't mean it's flawed. Is science flawed because we don't know the exact cause of the big bang and everything governing it? I think not.

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Mahoney
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Quote: "Is science flawed because we don't know the exact cause of the big bang and everything governing it?"


I'm sorry, but I must make a correction. The question would correctly be stated "Is our theory flawed because we don't know the exact cause of the big bang?"

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draknir_
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Posted: 24th Aug 2008 03:12
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Mamaji4 and Ionstream are correct here. A number with a recurring decimal like 0.999999... is the same as 1.0
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dark coder
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Posted: 24th Aug 2008 03:16
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Quote: "I'm sorry, but I must make a correction. The question would correctly be stated "Is our theory flawed because we don't know the exact cause of the big bang?""


You're saying maths and science are both theories? Perhaps you should read the tops of this and this.

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