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.9999... = 1 ?!

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J.Murakami

Post subject: .9999... = 1 ?!

Posted: Apr 13, 2004 - 11:23 PM

Joined: Apr 13, 2004
Posts: 3
Location: Acton, CA

Blizzard Entertainment posted this on their site for April Fools. I have friends say that it's true and others say that it's false. What do you guys think?

Proof:

lim(m --> ‡) sum(n = 1)^m (9)/(10^n) = 1
0.9999... = 1

Thus x = 0.9999...
10x = 9.9999...
10x - x = 9.9999... - 0.9999...
9x = 9
x = 1

Thanks much!

-Joanna

P.S. I said that it's incorrect... (I hope I'm right...) (>_<)

mathmate

Post subject:

Posted: Apr 20, 2004 - 08:27 PM

Joined: Nov 09, 2003
Posts: 733

I actually went to look up their proof, and apparently they had a long discussion on the subject at their forum for the longest time. The President believes that this will solve the problem once and for all.
Please note that the first line should read as follows:

lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1

where the 'infinity' sign was not printed properly in Joanna's post.

I personally believe the proof is mathematically correct. Practically, we all know that infinity is not achievable, so we tend to have hiccups.

Computers, for example, can only display a value (or a variable) to a number of bits, usually 8, 16, 32, 64, or 128.

Calculations often involve inaccuracies and thus end up with 0.99999999999999478 (or something like that) instead of one. To convert it to an integer, some programs use the floor function, which will chop off everything after the decimal point, and return a value of 0.99999999999999478=0! (This is an exclamation mark, and not a factorial sign).

So Joanna, even though I believe the proof is mathematically correct, I understand your skeptism. I hope my answer does not make you lose an expensive bet, so I invite everyone to give their opinions.

Readers out there, please post to give your opinions!

mathwhiz007

Post subject: their right

Posted: Apr 22, 2004 - 02:13 AM

Guest

0.1111....... is 1/9
0.2222....... is 2/9
0.3333....... is 3/9 or 1/3 or 33 1/3%
0.4444....... is 4/9
0.5555....... is 5/9
0.6666....... is 6/9 or 2/9 or 66 2/3%
and so and so on...

until 0.9999....... is 9/9 or i whole. the reason for that is beause 0.9999 goes on forever.... 0.9999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
999999999999999999999999999999999999999999999999999999999999999999999999999999
9999999999999999999999999999999999999999999999999999999999999999999999999999......
and so on. it's basically one. so it is considered that. That's why 0.6666......is 66 2/3 that's 2/3 of 100 is 66.666666666666666666666666666666666666666666666666666666666666666666666...you get the point. Laughing

J.Murakami

Post subject:

Posted: Apr 22, 2004 - 01:42 PM

Joined: Apr 13, 2004
Posts: 3
Location: Acton, CA

Where I think the proof might have some error is in the fact that they state that x=.99999... and then they "prove" that x=1. Isn't that a bit shifty (couldn't think of a better word...)? Wouldn't that be similar to saying that x=3 and then proving that x=4? *shrugs*

mathmate

Post subject:

Posted: Apr 22, 2004 - 09:18 PM

Joined: Nov 09, 2003
Posts: 733

If you look at mathwhiz007's illustration, he wrote
0.1111..... is 1/9, and it is accepted!
It is because 0.1111.... is a mathematical notation to represent an infinite chain of one's after the decimal point, which IN THE LIMIT becomes 1/9.
Proceeding the same way, it follows that 0.9999.... is 9/9=1.
This is mathematically correct, as I stated earlier, as long as we accept the limit of the value as the chain's length goes on indefinitely.
If there is a finite limit on the length, as in computers, then the equality will no longer hold.

Guest

Post subject:

Posted: Apr 23, 2004 - 12:33 AM

But what about the proof?
I thought that you can't use Algebra with anything
concerning infinity. I remember one of my Professors at my
Uni. lecturing on how you can't use Algebra with infinity...
Are infinite fractional values not included in this?

Are you guys frustrated yet? LOL (^_^)

J.Murakami

Post subject:

Posted: Apr 23, 2004 - 12:36 AM

Joined: Apr 13, 2004
Posts: 3
Location: Acton, CA

That last post was by me BTW... (>_<)

mathmate

Post subject:

Posted: Apr 24, 2004 - 09:24 PM

Joined: Nov 09, 2003
Posts: 733

On the contrary, I am getting help to provide a convincing proof.
Stay tuned.

plasma

Post subject:

Posted: Aug 20, 2004 - 04:48 AM

Joined: Aug 19, 2004
Posts: 1

Hello!
Remember: 0.999…… is a rational number.

Let us assume that between 0.9999….. and 1 there is a numbers and let the number be x.

So, 0.999…..+x=1

x=1-0.9999999999…….

Let us subtract in a usual way.

Code:

1.000000000000…………….
-
0.999999999999……………..
----------------------------------------

Since we can’s subtract 9 from 0, 0 share 1. So, each 0 will become 9.

Code:

0.999999999999……………………..
-
0.999999999999………………………
----------------------------------------------------
0.000000000000000………

Look, 1 is moving to the right changing 0 to 9.

And look again,

0.0000000……… is the answer.
0.0000000…… + 0.9999……. =1

0.0000000000000…………… here the number moving in the infinite place is 1.

It will go on forever, but since we are given the limit,

0.000000………=0

So, 0+0.99999………=1

That is, 0.99999……. =1
And this contradicts with our previous assumption. Then, we will conclude that:
given the limit

Code:

lim(m --> ∞) sum(n = 1)^m (9)/(10^n) = 1 we will say 0.99999……=1

What happened to the screen? Laughing

bobimm

Post subject:

Posted: Aug 20, 2004 - 09:48 AM

Joined: Jul 30, 2004
Posts: 4

I do not believe this equation.
Consider this:
x = 0.9999999999999.......9
2x = 1.9999999999999.......8
3x = 2.9999999999999.......7
4x = 3.9999999999999.......6
and so on, where
8x = 7.9999999999999.......2
9x = 8.9999999999999.......1
therfore, 9x does not = 9
Math is one subject
Computer calculaton is something else, and can be inaccurate because of truncation
Bobimm

mathmate

Post subject:

Posted: Aug 20, 2004 - 12:44 PM

Joined: Nov 09, 2003
Posts: 733

For those who have worked with recurring decimals, we recall that:

0.11111...... = 1/9
0.22222..... = 2/9
0.33333..... = 3/9

0.99999.... = 9/9 = 1

Hope this helps.

JangosClone

Post subject:

Posted: Apr 16, 2005 - 12:46 AM

Joined: Mar 08, 2005
Posts: 170
Location: Denver, Colorado

You all said that;
0.111111...=1/9
You said this because it is accepted. It is only accepted because humans cannot comprehend infinity. In truth 0.111111... does not equal 1/9. It is merely infinitely close to 1/9. I agree with Math mate that this proof is mathmatically correct. Or, more accurately, it is correct within the confines of our understanding.
Anyway, that's what I think.

_________________
www.marshill.org

mathmate

Post subject:

Posted: Apr 16, 2005 - 04:33 PM

Joined: Nov 09, 2003
Posts: 733

As a matter of fact, we all agree that:

0.11111... = 1/9, where

... represents "one"'s ad infinitum.

We do not agree that 0.11111 = 1/9, simply because they are not equal.

In this particular case, I would say, on the contrary, everybody understands infinity, but in a different context from what you have mentioned elsewhere, namely the big number.

Let me know if our points of view converge.

clay

Post subject:

Posted: Jul 16, 2006 - 05:04 PM

Site Admin

Joined: Jul 11, 2003
Posts: 106
Location: Charlottesville, Virginia, USA

Here's a fun blog post that seeks to prove once and for all that $.\overline{9}=1$ :

No, I'm Sorry, It Does

Post subject:

Posted: Jul 16, 2006 - 11:34 PM

Joined: Jun 13, 2006
Posts: 27
Location: Denmark

Many people have huge problems realizing that 0.999... = 1. The way I have had must luck explaining it, is in base 2 (binary). If people realize that 1 = 0.111... in base 2, it is logical that 1=0.999... in base 10.

So:
0.111... (base 2) = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 ... 1/2^n (base 10)

Ergo is 0.111... (base 2) equal to the infinte sum:
$\Sigma ^{\infty}_{n=1} \frac{1}{2^n}$

And this sum is indeed proven to be 1...

I use this simpler infinte serie instead of
$\Sigma ^{\infty}_{n=1} \frac{9}{10^n}$
because it is easier to accept that the first sums to 1. People - where i am - have never even heard about infinte series, and the sentence "Infinite geometric series" will just put them asleep before i have finished saying it Wink

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