Is There a Shorter Proof for 0.999... = 1?

[EnumaElish]

...
1/3 = 0.333...
1.3 * 3 = 0.333... * 3
1 = 0.999... also works.
...

Doesn't this merely shift the punchline of the proof, so to speak? Doesn't one have to prove 1/3 = 0.333... ? How's that easier than proving 1 = 0.999... directly?

Either I am the smartest one in this bunch or I missed the point of a lot of the postings here.

 

[TenaliRaman]

Doesn't this merely shift the punchline of the proof, so to speak? Doesn't one have to prove 1/3 = 0.333... ? How's that easier than proving 1 = 0.999... directly?

Either I am the smartest one in this bunch or I missed the point of a lot of the postings here.

I am not going to justify the post made there, but many people find it easier to comprehend the fact that 1/3 = 0.3333... than 1 = 0.9999...
Why??
Divide 1 by 3 using long division and it seems apparent (to them) however there is no such process to get 0.99999... from 1, so obviously they think its an error. However naive this argument may sound, i believe it appeals to their thought process and hence 1=0.99999... becomes a much talked abt issue in many forums.

No wonder, matt wants it to be immortalised as an FAQ

-- AI

 

[EnumaElish]

... however there is no such process to get 0.99999... from 1, so obviously they think its an error. ... -- AI

The closest expression I can come up with is: 1 = 0.999... * 1.000...

Or 1/1.000... = 0.999...

It's sort of like saying "1 is the geometric average of 0.999... and 1.000..."

Example:

Suppose a population grew -0.000... in the first year and +0.000... in the second year; what is the cumulative growth rate (CGR) in two years?

1 + CGR = (1-0.000...)(1+0.000...)
1 + CGR = 0.999... * 1.000...

Intuitively there was zero growth, so CGR = 0.

1 = 0.999... * 1.000... There.

If one is prepared to accept the CGR = 0 intuition then it becomes possible to show 1 = 0.999... directly:

1 + CGR = (1-0.000...)(1+0.000...) = 1² - (0.000...)²

Intuitively CGR = 0 so,

1 = 1 - (0.000...)(0.000...) = 1 - 0.000... = 0.999... So there!

 

[boteet]

Hello enumaelish, if cgr=0 intuitivley, you will not be able to get
1 + CGR = 0.999... * 1.000...

 

[EnumaElish]

Hello enumaelish, if cgr=0 intuitivley, you will not be able to get
1 + CGR = 0.999... * 1.000...

Can you elaborate, even it's along the lines of "obviously you haven't had your 3rd cup of coffee this AM, because ..."?

 

[rachmaninoff]

Why is this a five page thread? What's wrong with the three-line epsilon proof:

$$\begin{align*} &\mbox{For } \epsilon > 0, \exists n > \log ( \epsilon^{-1} ) > 0 \mbox{ such that} \\ & |1 - \sum_{k=1}^{n}9*10^{-k} | = 10^{-n} < \epsilon \\ & \mbox{Therefore: } 0.99\overline{9}\equiv \lim_{n \rightarrow \infty} \sum_{k=1}^{n}9*10^{-k} = 1 \end{align}$$

Is anyone not happy?

 

[EnumaElish]

Why is this a five page thread? What's wrong with the three-line epsilon proof:
...
Is anyone not happy?

Not creative enough!

 

[Icebreaker]

Is anyone not happy?

Anyone who's math-literate enough to read that wouldn't need it in the first place.

 

[toocool_sashi]

This means
$$3(\frac{1}{3})=1$$
.999999999... is just one possible way to write 1 here are a few others

this is exactly what we have 2 prove isn't it!? i liked the epsilon proof...it involed no arithmetic that's why lol ;) but see...if u can convince urself that 3*0.3333333... = 0.99999999... then u mite as well convince urself that 1-0.9999999... = 0.00000000.. this wud become a shorter *proof*

 

[TenaliRaman]

1-0.9999999... = 0.00000000.. this wud become a shorter *proof*

Oh! But there will be a 1 at the end

-- AI

 

[Zurtex]

Oh! But there will be a 1 at the end

-- AI

Yes at the end of an infinite number of 0s

 

[TheGinkgoNut]

how about just this simple one with arguably less demands on the idea of "infinity":

$$\begin{align*} x &=& 0.999\overline{9} \\ 10x &=& 9.999\overline{9} \\ 10x-x &=& 9 \\ 9x &=& 9 \\ x &=& 1 \\ \end{align}$$

 

[matt grime]

and you've proven that arithmetic is well defined on strings of recurring (or any infinitely long) decimals have you?

 

[TheGinkgoNut]

i've opened a pandora's box haven't i...

 

[EnumaElish]

i've opened a pandora's box haven't i...

More like a Russian doll, IMO.

 

[mathwonk]

gingkonut, that was the arguemnt i was shown in 8th grade and i always remembered it and liked it. that's the right level for it, kids who don't need rigor, but enjoy being amazed.

 

[matt grime]

There is no harm in using white lies when explaining maths, often it is the best way of getting across information. And the multiplying by ten thing is a good example: we know what *ought* to happen when we do it, and as mathwonk says that is often good enough. the problem comes from the die hards who refuse to accept this *motivational* reason and then dismiss the full on proof with all the details and carefully laid out definitions (I would suggest because they do not understand the formality of mathematics). hence in this case (and this is only my opinion now because of the attitudes displayed repeatedly in this forum where the audience is of all levels) i tend to immediately go for the overkill approach since there is a good chance the reader is not a kid to be amazed.

 

[TOKAMAK]

Does anyone have any links to the original discussion from battle.net? Thanks in advance.

 

[bao_ho]

x = 0.9999
9x = 9
11x = 10.9999
2x = 1.9999?

 

[HallsofIvy]

x = 0.9999
9x = 9
11x = 10.9999
2x = 1.9999?

Starting from x= 0.9999, how do you get 9x= 9? According to my calculator 9x= 9(0.9999)= 8.9991! (And 11(0.9999)= 10.9989, not 10.9999.)

Anyway, why are you talking about 0.9999? Everyone else is talking about 0.99999... (which is easily shown to be equal to 1.)

 

[eNathan]

J33Z these threads are always poping up, well I think this thread was pulled up from a while ago but...

This is my conclusion
$$1 - .999~ = 000 \infty$$
To say that $$1 != .999\infty$$ is to say that $$x * 000\infty!= 0$$

Sorry I forgot the latex code for <> or !=, what is it again

 

[yourdadonapogostick]

$$\infty$$ is undefined, so i would have to say that is completely wrong

 

[rachmaninoff]

$$\infty$$ is undefined, so i would have to say that is completely wrong

Well, he is completely wrong, but $$\infty$$ is well-defined (you're wrong about that), but arithmetic on $$\infty$$ is not, so $$\infty \cdot 0$$ is what's undefined.

 

[Zurtex]

J33Z these threads are always poping up, well I think this thread was pulled up from a while ago but...

This is my conclusion
$$1 - .999~ = 000 \infty$$
To say that $$1 != .999\infty$$ is to say that $$x * 000\infty!= 0$$

Sorry I forgot the latex code for <> or !=, what is it again

Your looking for $\neq$

Would you like to make any mathematical sense of that? Or should I not bother asking?

 

[eNathan]

Your looking for $\neq$

Would you like to make any mathematical sense of that? Or should I not bother asking?

here we go again

if you were to perform the following operation

$$1 - .9$$ you would get $$.1$$

if you were to do
$$1 - .9999999999$$ you would get $$.0000000001$$

Now if you were to manually try to subtract $$.999 \infty$$ from $$1$$, you would get $$000000000$$ for infintiy! hence,
$$1 - .999 \infty = 000 \infty$$ Make sense? So to say that $$1 - .999 \infty \neq 0$$ would also mean that that the infinite row of 0's (that you get from 1 - .999~) actaully doesn't equal 0! Which is false.

Now why did I present it as 0x != 0? Because there are an infinite number of zero's, which I called variable 'x'. Correct me if it's not proper to present infiity as a x. What I wrote was

$$x * 000\infty \neq 0$$
When what I should have wrote was
$$000\infty \neq 0$$

By the way, is $$0 \cdot \infty$$ really undefined? news to me, I always assumed it was 0.

 

[Townsend]

By the way, is $$0 \cdot \infty$$ really undefined? news to me, I always assumed it was 0.

In calculus when evaluating limits you can get to a point like infinity times 0 and at those times it is considered indeterminate. Thus you invoke L.H...

But without knowing how you arrived at infinity times 0 I don't think anyone can really call it undefined. As far as ordinary arithmetic goes I don't see how this would be a legal operation.

Regards,

I am not an expert so please go easy if I made an error...

 

[rachmaninoff]

here we go again

if you were to perform the following operation

$$1 - .9$$ you would get $$.1$$

if you were to do
$$1 - .9999999999$$ you would get $$.0000000001$$

Now if you were to manually try to subtract $$.999 \infty$$ from $$1$$, you would get $$000000000$$ for infintiy! hence,
$$1 - .999 \infty = 000 \infty$$ Make sense? So to say that $$1 - .999 \infty \neq 0$$ would also mean that that the infinite row of 0's (that you get from 1 - .999~) actaully doesn't equal 0! Which is false.

I see, the problem is we were misreading your (incorrect) notation. It's 0.999... or $0.\overline{9}$, not $.999 \infty$ (which looks like an attempted multiplication).

By the way, is $$0 \cdot \infty$$ really undefined? news to me, I always assumed it was 0.

Yes, arithmetic on infinity is not defined. If you attempt to define it, you get contradictory results like
$$\lim_{x \rightarrow 0} \frac{1}{x} \cdot x = 1$$
$$\lim_{x \rightarrow 0} \frac{1}{x} \cdot 0 = 0$$

This probably already occurred a dozen times in this thread, I'm too lazy to look for precedents.

Will this thread ever die?

 

[yourdadonapogostick]

Well, he is completely wrong, but $$\infty$$ is well-defined (you're wrong about that), but arithmetic on $$\infty$$ is not, so $$\infty \cdot 0$$ is what's undefined.

then define $$\infty$$

 

[VietDao29]

then define $$\infty$$

The greatest number of all real numbers. Obviously, you won't be able to find what number actually infinity is.
----
@eNathan:
How come 1 - 0.99... = a row of 0s? Where is the little number 1.
I agree 0.99... = 1, but your proof does not make much sense to me.
Viet Dao,

 

[bao_ho]

1 / 9 = 0.1111recuring
0.9999recuring / 9 = 0.1111recuring

then 1 = 0.9999recuring

 

[eNathan]

rachmaninoff::
Hey thanks for pointing out my error, ill use $\overline{x}$ Instead of $x infty$ from now on ;) Atleast you understand my point and I wasnt called a wacko :lol:

nice proof bao_ho, btw.

 

[Zurtex]

rachmaninoff::
Hey thanks for pointing out my error, ill use $\overline{x}$ Instead of $x infty$ from now on ;) Atleast you understand my point and I wasnt called a wacko :lol:

nice proof bao_ho, btw.

Yeah, kind of my point, with all the infinities floating round I still can't really make sense what you are saying.

 

[EnumaElish]

1 / 9 = 0.1111recuring
0.9999recuring / 9 = 0.1111recuring

then 1 = 0.9999recuring

Your 1st step assumes your conclusion, when you replace 1 with "0.9999recuring".

Sorry I forgot the latex code for <> or !=, what is it again

\ne

 

[Zurtex]

Your 1st step assumes your conclusion, when you replace 1 with "0.9999recuring".

It's not a rigorous proof on many levels but it doesn't do that. It is showing that you get the same answer when you divide both of them by 9, hence they must be the same number.

 

[EnumaElish]

Okay, I got you.