## View Poll Results: Is 0.999... exactly equal to one?

Voters
86. You may not vote on this poll
• Yes, they are equal.

52 60.47%
• No, they are not equal.

34 39.53%

# Thread: Is 0.999... equal to one?

1. ## Re: The end of Cantor's flawed Diagonal Argument.

More thoughts on the infinite sum concept.

Given that and , it is very tempting to reason as follows:

Well, the limit of all the partial sums is , so why not just define it to be as Euler did?

1. Because the sum that matters is not a partial sum. It is an ill-defined sum without a last term.

2. Such definitions lead to nonsense such as 1/3 = 0.333... and 1 = 0.999...

2. ## Euler's Mistake in the Digit Proof

Euler's "blunder" revealed in rigorous detail:

Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"

First off, since when did become a variable that we needed to solve for ?? It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".

The huge blunder here is the combination of multiplying by 10 and subtracting a series from the resulting series, namely, 10 x 0.999... - 0.999...
There are other underlying assumptions too, but I addressed those earlier in the thread (specifically, multiplying an infinite series by anything which causes a carry or decimal shift)

Let's do 10n - n using the definition of n

Now, what Euler does, is subtract the 2 series,

...Except, what he failed to show everyone, or assumed was ok, was that he shifted the second series to the right by one decimal place:

How he does this is by doing to following:

I believe Colin pointed this out at one point and said this was "ok" to do. At first glance, the fractions are all equivalent... except...
...the SERIES has changes slightly:

[ Notice the starting index value. ]

This allows the subtraction to occur as :

is really equal to

So, by the Property of Addition and Subtraction of Summations:

we can see that we shouldn't shift the series, because that uses the starting index: instead of

Let's use the proposed theorem 17, to subtract the 2 series, with equal starting indices:

Unaltered series:

or, using the

Altered Series with using Theorem 17 properly:

Let's do a few terms, shall we?:

I believe you get the point now, that by changing the series, you change the results.

Therefore Euler mistakenly assumed that he could slide the series over to the right, and "cancel off" all the decimal digits, incorrectly so, in order to achieve his desired result of 9n = 9.

We are not allowed, during this proof, to use the definition of 0.(9) = 1 because the point of the proof was to prove that the 2 numbers are equal "only by doing some basic algebra".

It was a nifty trick that he used, to manipulate the series in order to "cancel off" all the decimals.

We know what results we should get at EVERY point in every equation, if we define x = a, then x always equals a, and I showed that by NOT assuming things as Euler did, we do end up with the proper result and not some illusion where (a) becomes (b) with some hocus pocus. If we started with x = 1, and ended up with x = 2, would THAT be ok too? I think not...

The conclusion is that 0.999... does not = 1 in the end. It equals 0.999... so he proved NOTHING, or, maybe the fact that extra care is due when performing "basic number manipulation" on an infinite series.

Another source of the Converging Series Identities. The difference of converging series is the same logic as summation of series:

3. ## Re: Euler's Mistake in the Digit Proof

Originally Posted by caper
Euler's "blunder" revealed in rigorous detail:

Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"

First off, since when did become a variable that we needed to solve for ?? It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".

The huge blunder here is the combination of multiplying by 10 and subtracting a series from the resulting series, namely, 10 x 0.999... - 0.999...
There are other underlying assumptions too, but I addressed those earlier in the thread (specifically, multiplying an infinite series by anything which causes a carry or decimal shift)

Let's do 10n - n using the definition of n

Now, what Euler does, is subtract the 2 series,

...Except, what he failed to show everyone, or assumed was ok, was that he shifted the second series to the right by one decimal place:

How he does this is by doing to following:

I believe Colin pointed this out at one point and said this was "ok" to do. At first glance, the fractions are all equivalent... except...
...the SERIES has changes slightly:

[ Notice the starting index value. ]

This allows the subtraction to occur as :

So, that's the "correct" way? 9.999... - .0.999 equals 8.999... ?

Weirdly enough I agree with that equation.

Probably, next post, you'll try to argue against it.

Or are you going to argue against 8.999... = 8 + 0.999...

Because if you use those two "correct" equations in your other proof, you still get

0.999... = 1

4. ## Re: Cantor's Diagonal Argument a fraud.

Originally Posted by Yuspoov
No one can. Good thing it's defined.

Ah well, if someone says it's common sense, how could anyone argue with that?
~ and if I maintain a honesty I must fess up to wanting some argument for the joy of a conversation is not wasted on me.. The contributor 'Yuspoov' has not understood the question as I do. That this is a assumption that my understanding is better, or different than his. and my reasoning is as this... Those three digits or points ... are a clear indication of intent to convey a term undefined to infinite. If I add them at the end of a numeric expression like pi as 3.142... It is a display of understanding that the term is ill defined. Yes that term can and is worked with to a very accurate end result. That the addition of the three dots just shows I understand the reality of that term. In the case of 0.999... it shows that at no point can the term be equal to another number expressed as 1. That 'Yuspoov' sees this as something else just indicates to me he does not UNDERSTAND as I do. As for the quip regarding my common sense. You can not find argument with what remains mine.

5. ## Re: Cantor's Diagonal Argument a fraud.

Originally Posted by caper
Euler's "blunder" revealed in rigorous detail:

Here is the "proof" for 0.(9) = 1, as provided by Euler, in which he claims that number manipulation can magically morph a "number" assigned as a constant, into another "number"

First off, since when did become a variable that we needed to solve for ??
"n" here represents a value we need to solve for, if we want to answer to the question which is the topic of this thread. It does not represent a variable.

It is a constant. Constants don't change their value. Try assigning a value to a constant in a computer program and see if the value ever "ends up as something else".
Just so. And Euler's argument is not intended to change the value of "n", it is to express the unchanged value of a recurring decimal in a different form — as a ratio of integers (or a single integer).

Originally Posted by grapes
So, that's the "correct" way? 9.999... - .0.999 equals 8.999... ?

Weirdly enough I agree with that equation.

Probably, next post, you'll try to argue against it.

Or are you going to argue against 8.999... = 8 + 0.999...

Because if you use those two "correct" equations in your other proof, you still get

0.999... = 1
Yes.

If it is correct that

9n = 8.999...

then

8n = 8.999... - 0.999... = 8

therefore

n = 8/8 = 1

6. ## Re: Cantor's Diagonal Argument a fraud.

So far so good. Just added equal numbers to both sides

wrong. You can't subtract n from one side and .999... from the other. n has to be subtracted from both sides. So

Then you have to convert n into numbers that can be treated properly.

not

That's math trickery.

7. ## Re: Euler's Mistake in the Digit Proof

Originally Posted by grapes
So, that's the "correct" way? 9.999... - .0.999 equals 8.999... ?

Weirdly enough I agree with that equation.

Probably, next post, you'll try to argue against it.

Or are you going to argue against 8.999... = 8 + 0.999...

Because if you use those two "correct" equations in your other proof, you still get

0.999... = 1
I'm sorry? Did you have a counter argument? Are there any mistakes ?
Didn't think so.

8. ## Re: Cantor's Diagonal Argument a fraud.

Originally Posted by Colin
"n" here represents a value we need to solve for,
NO we don't ! It is not a variable!
n = 0.999... !! Right from the start. That is really bad understanding of variables and constants.

Yes.
If it is correct that
9n = 8.999...
then
8n = 8.999... - 0.999... = 8
therefore
n = 8/8 = 1
More fail for you and grapes both. Sorry. Didn't you learn anything from the comment I posted?? Would you wager your house on that ?

Cross check our work:

As it should. The more mistakes you make, the more I will point them out. Haven't you realized yet, that this proof is WRONG? If it is mathematically invalid, there is nothing you can do get the result you want when it is done properly.

Originally Posted by grapes
Because if you use those two "correct" equations in your other proof, you still get 0.999... = 1
No. No you don't. Well, maybe you do, but it is wrong.

Any object assigned to a letter is always that object. I can repeat that if you like. It doesn't get any more basic than that.

Originally Posted by astrotech

That's math trickery.
Except that 9.999... - 0.999... = 8.999...
Therefore:
8.999... = 8.999...

9. ## Re: Cantor's Diagonal Argument a fraud.

Originally Posted by caper
9.999... - 0.999... = 8.999...
Nope. Wrong.

10. ## Re: Cantor's Diagonal Argument a fraud.

Originally Posted by astromark
The contributor 'Yuspoov' has not understood the question as I do.
We agree on that point!

Originally Posted by astrotech

So far so good. Just added equal numbers to both sides

wrong. You can't subtract n from one side and .999... from the other. n has to be subtracted from both sides.
[text]\vdots[/tex]
That's math trickery.
Oh, this sounds important. Let me make a note of this.

and . But to assert that is math trickery.

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