1 is 0.9999999999999............

[Quantum Quack]

You should, it's very simple maths and you agreed to it before.
10 apples - 1 apple = 9 apples
10 x 0.999... - 0.999... = 9 x 0.999...

You know, we can make this even simpler.
Try this:

0.999... + 0.999... = 2 x 0.999... = 1.999... = 1 + 0.999...

I am not sure this is going to make it easier because the automatic inclusion of the infinitesimal on my calculator is screwing things up.

according to my calculator 2 x 0.999.... = 2 [rounding happening]
yet if we reduce the decimal places to say 8 we end up with
1.99999998 so I really don't know what to say

0.999999999999999999999999999999999999999999999999999999
plus
0.999999999999999999999999999999999999999999999999999999
----------------------------------------------------------------
1.999999999999999999999999999999999999999999999999999998

addition long hand
so the answer can only be no
1+0.999... = 1.999...

not 1.999999999999999999999999...8

 

[Quantum Quack]

To be specific:
0.999... + 0.999... = 2 x 0.999... = 1.999...8 Does Not Equal 1 + 0.999...

 

[Pete]

QQ, the result of 2 x 0.999... is 1.999..., just like (as you yourself pointed out) 9 x 0.999... = 8.999...
There are infinite 9's after the decimal point. There is no 8 at the end, because there is no end.


But, let's look at this equation again, one that you agreed to in post 83, but since changed your mind.

10 x 0.999... - 0.999... = 9 x 0.999...

This is very simple arithmetic:
10 x 0.999... - 0.999... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... - 0.999...
= 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...
= 9 x 0.999...

 

[kwhilborn]

@ QQ,
Is this not a thread asking if 1 = 0.999... ? You can see they are the same when you get the result of zero when you subtract one from the other. Why you need to drag variables into this is not very clear.

 

[Trippy]

nah stupid mistake using a stupid example.. sorry
9x=7
therefore
1x = 7/9
1x= 0.777777778

No.

Therefore 1x= 0.777777777... or 1x = 0.7777777(7)

It's an infinitely recurring sequence of 7's. Stop relying on your calculator, get out a pen and some paper, and do the long division for yourself.

0.777777778 is not the decimal expression of $$\frac{7}{9}$$. It is an approximation o fthe decimal expression of $$\frac{7}{9}$$ that is accurate to 8 decimal places.

 

[rpenner]

kwhilborn,
As per my posts [POST=3126046]#61[/POST] and [post=3126315]#84[/post], understanding the difference between the reals and the rationals is understanding the difference between a finite number of terms and the limit of an infinite number of terms.

So the calculation $$1 - 0.999... = 1 - \lim_{n\to \infty} \sum_{k=1}^{n} \frac{9}{10^k} = 1 - \lim_{n\to \infty} ( 1 - 10^{-n} ) = \lim_{n\to \infty} 1 - ( 1 - 10^{-n} ) = \lim_{n\to \infty} 10^{-n} = 0$$ requires acceptance of formal axioms that QQ shows no sign of interest in. Therefore QQ is not talking about the real numbers.

As post [post=3126315]#84[/post] shows, you can split an infinite sum into a finite sum with an enormous number of terms and the quasi-infinitesimal remainder. But such a treatment specifically has a "last digit" of the finite sum and a non-zero difference from the original rational expression whereas the real numbers have no such last digit and no such non-zero infinitesimal thus 1 = 0.999.... .

The very concept of a last digit requires that there be only a finite number of terms under discussion, for the last digit can have no address value k when written as
$$ x =\sum_{k=1}^{\infty} \frac{a_k}{10^k}$$
If the claim K is the address of the last digit, such a claim is contradicted by the existence of a term at K+1.

// Added minus signs to exponents to 10 when 10^k is no longer a denominator as per #109

 

[Trippy]

10X - X = 9 <<<:replaces 9X with a real number... logical inconsistency (loss of context)
9X = 9
X = 1 <<: irrational number now converted to real

Therefore X = 1 = 0.9999.... <<: 1 is now an irrational number

0.99999999... or 0.9999999(9) is not an irrational number.

It is both real and rational.

From wikipedia:

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for binary, hexadecimal, or any other integer base.

Rational Numbers on Wiki

0.111(1) is both rational and real. It repeats the same finite sequence '1' over and over. It is the decimal expression of $$\frac{1}{9}$$
0.333(3) is both rational and real. It repeats the same finite sequence '3' over and over. It is the decimal expression of $$\frac{1}{3}$$.
0.0(52631578947368421) is both rational and real. It repeats the same finite sequence '52631578947368421' over and over. It is the decimal expression of $$\frac{1}{19}$$

0.999(9) repeats the same finite sequence '9' over and over. It fits the definition of a rational number. The fact that it fits the definition of a rational number implies that it is also a real number and can be located on the number line.

 

[Trippy]

rpenner - would you mind terribly writing up the LaTeX illustrating the long division of 1 by 3 to illustrate for QQ's benefit (among I would guess others) where the inifinite string of 3's comes from in the decimal expression of $$\frac{1}{3}$$. I'm short on time and not really all that familiar with LaTeX.

 

[Tach]

kwhilborn,
As per my posts [POST=3126046]#61[/POST] and [post=3126315]#84[/post], understanding the difference between the reals and the rationals is understanding the difference between a finite number of terms and the limit of an infinite number of terms.

So the calculation $$1 - 0.999... = 1 - \lim_{n\to \infty} \sum_{k=1}^{n} \frac{9}{10^k} = 1 - \lim_{n\to \infty} ( 1 - 10^n ) = \lim_{n\to \infty} 1 - ( 1 - 10^n ) = \lim_{n\to \infty} 10^n = 0$$ requires acceptance of formal axioms that QQ shows no sign of interest in. Therefore QQ is not talking about the real numbers.

You surely mean: $$1 - 0.999... = 1 - \lim_{n\to \infty} ( 1 - 10^{-n }) = \lim_{n\to \infty} 10^{-n} = 0$$

To be exact: $$1 - 0.999... = 1 - \lim_{n\to \infty} ( 1 - 10^{-(n+1) }) $$

 

[Tach]

rpenner - would you mind terribly writing up the LaTeX illustrating the long division of 1 by 3 to illustrate for QQ's benefit (among I would guess others) where the inifinite string of 3's comes from in the decimal expression of $$/frac{1}{3}$$. I'm short on time and not really all that familiar with LaTeX.

Try $$\frac{1}{3}$$

 

[Trippy]

Try $$\frac{1}{3}$$

Thanks - I had my slash in the wrong direction...

 

[rpenner]

You surely mean: $$1 - 0.999... = 1 - \lim_{n\to \infty} ( 1 - 10^{-n }) = \lim_{n\to \infty} 10^{-n} = 0$$

Fixed in original. Thanks.

To be exact: $$1 - 0.999... = 1 - \lim_{n\to \infty} ( 1 - 10^{-(n+1) }) $$

That's true because it also converges to 1, but that does not follow by the algebra I was doing since $$0.9 = 1 - 10^{-1}$$, $$0.9 + 0.09 = 1 - 10^{-2}$$, $$0.9 + 0.09 + 0.009= 1 - 10^{-3}$$, etc. As I am summing from 1 to n, there are n terms and the last term is of the form $$\frac{9}{10^n}$$.
So the exponent is now correct in original and not so good in the proposed re-write.

 

[Quantum Quack]

QQ, the result of 2 x 0.999... is 1.999..., just like (as you yourself pointed out) 9 x 0.999... = 8.999...
There are infinite 9's after the decimal point. There is no 8 at the end, because there is no end.


But, let's look at this equation again, one that you agreed to in post 83, but since changed your mind.

10 x 0.999... - 0.999... = 9 x 0.999...

This is very simple arithmetic:
10 x 0.999... - 0.999... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... - 0.999...
= 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...
= 9 x 0.999...

Pete what is you point with the above?
You can see that:
9.999.... - 0.999... = 9

and you can see that:
9 x 0.999... produces 8.999.... and not 9

so the statement that 1= 0.999... is false due to the misuse of the infinitesimal.

I am still interested in seeing a mathematical proof that derives 1= 0.999.. with out the use of algebraic symbols.

And so far all we have proven is that

(9.999... -0.999...) Does not equal (9 x 0.999...)

and that the notion that 1 = 0.999... is false
And discovered that the infinitesimal would need to be applied on one side of the equation to fudge the results so that the claim of 1=0.999... can be made.

9.999... - 0.999... = 9
9 x 0.999... + 1/infinity = 9

Therefore one can conclude that to achieve 1=0.999... as valid, we have to ignore how the infinitesimal is used. To me that is a false proposition.
[Although I am sure there is much mathematical proof out there that would show that the use of the infinitesimal in such a manner is quite justified]

For 1 to equal 0.999.... , 1/infinity must equal zero and as far as I know it doesn't and can't equal zero [re: Zeno's paradox]

 

[Quantum Quack]

@pete

QQ, the result of 2 x 0.999... is 1.999..., just like (as you yourself pointed out) 9 x 0.999... = 8.999...
There are infinite 9's after the decimal point. There is no 8 at the end, because there is no end.

if the result of 0.999... plus 0.999.... = 1.999....

then the notion of 1=0.999... is proved false.


because if 1= 0.999... the answer to 2x 0.999... = 2 not 1.999... as you have agreed to it being equal to.

In other words you have only proved the contra ....

to clarify:
Proposition:

We can use 2 x 0.999... to prove that the notion 1=0.999... is false.

0.999.. + 0.999... = 1.999.... [and not 2]

an infinitesimal must be included somewhere for 1=0.999... to hold true.


Summary response to the OP

For 1 to equal 0.999.... , 1/infinity must equal zero

 

[Quantum Quack]

0.99999999... or 0.9999999(9) is not an irrational number.

It is both real and rational.

From wikipedia:

Rational Numbers on Wiki

0.111(1) is both rational and real. It repeats the same finite sequence '1' over and over. It is the decimal expression of $$\frac{1}{9}$$
0.333(3) is both rational and real. It repeats the same finite sequence '3' over and over. It is the decimal expression of $$\frac{1}{3}$$.
0.0(52631578947368421) is both rational and real. It repeats the same finite sequence '52631578947368421' over and over. It is the decimal expression of $$\frac{1}{19}$$

0.999(9) repeats the same finite sequence '9' over and over. It fits the definition of a rational number. The fact that it fits the definition of a rational number implies that it is also a real number and can be located on the number line.

yeah Trippy, and others, I apologize for lack of correct use of these terms... learning on the fly can be so treacherous...
And from what I have read so far I am only doing what the majority of math students do when confronted with the counter intuitive nature of the notion 1= 0.999...

Painful but necessary...

 

[Quantum Quack]

There is I believe a number of thought experiments to highlight issues surrounding the use of infinity and infinitesimals.

One example comes to mind:

We have a clay house brick.
Take an infinite number of slices [thickness = 1/infinity] until the brick is only slices

to rebuild the brick it would take a finite number of slices.

This appears I believe because if 1/infinity does not equal zero then IT MUST have dimension. [ substance = slice of brick ]

The issues surrounding this question of 1=0.999... are strongly associated I believe...
"In one direction it works out and yet in the other it doesn't"
how this apparent "asymmetrical", "paradoxical" use of the infinitesimal is justified in mathematics is well beyond me.

 

[Dinosaur]

The proofs which do not relate to the limit of a geometric series are not valid.

They might seem valid & result in the same conclusion, but will not be found in any mathematics text.

 

[Undefined]

Hi guys.

If I may make this humble observation on the starting manipulation used in 'proofs' which has 10 x 0.999... = 9.999...?

I naively observe for your joint consideration that when we multiply by 10 we effectively add a "0" to the last place of a string. Yes?

BUT in the expression 0.999... there IS no 'last place' in that string, so the multiplication by 10 cannot logically add a "0" at the last place in order to 'shift' the first 9 to the left' to in front of the 'decimal point' (in that decimal notation format).

Else we would have 9.999...0. Which would be a nonsense in the same axiomatic treatment which multiplied by ten. Yes?

This illustrates what QQ has been pointing out all along. Ie, unless there is an 'infinitesimal' added on the end of the 0.999... string, we cannot go from there to here by adding a "0" via multiplication by 10.


My further humble observation is that QQ seems to be the only other person here that recognizes the importance of "contextual axioms" rather than isolated axioms which when followed blindly lead up a dark alley where trivial and conta/undefined situations occur which must require further isolated 'exceptions axioms/definitions' to paper over the axiomatic gaps which open up as QQ and others have tried to point to.

That's all I ant to observe at this juncture. I only did so at this time because I felt that what QQ was trying to illustrate was being unintentionally 'hidden' by 'proofs' from others which depended on (to me) trivial manipulations of algebraic symbols and algebraic 'rules' which make no sense when brought back to the fundamental actions. One example of which is the multiplication by 10 of the string 0.999... and assuming (without proper care or cause) that the result is 9.999... instead of 9.999...0 (which last place "0" would logically represent the 'infinitesimal' which QQ is referring to?).

In the universal reality there IS a "last infinitesimal of PHYSICAL effectiveness' which delineates the border/boundary 'condition between effective reality and ineffective reality scaleextent/strength etc of the various fundamental physical forces/entities which produce the Quantum Mechanics arena/phenomena we treat in reality via maths and logical modeling which finds its LIMITS in reality to that "last infinitesimal QUANTUM of physical effectiveness" which our quantum Mechanics already recognizes mathematically and logically in its modeling constructs. Yes?

That is why IN MY OPINION and obesrvation, contextual maths and reality physics are actually logically 'one' and consistent when treated under contextually complete 'rules' rather than partial 'isolated' axioms/postulates, because there IS a last infinitesimal of effectiveness in both any number string and in any physical modeling construct. In MATHS, that last infinitesimal represents the final 'quantum step' into a new STATE OF TRANSITION/BALANCE etc depending on what the process/states on either side of the continuum division involves in math/physical reality/properties.

Thanks for your exhaustive and interesting discussions/contributions, everyone! I have appreciated it all, every bit. Bye and good luck in all your discussions, QQ, Pete, everyone.

 

[Tach]

Hi guys.

If I may make this humble observation on the starting manipulation used in 'proofs' which has 10 x 0.999... = 9.999...?

I naively observe for your joint consideration that when we multiply by 10 we effectively add a "0" to the last place of a string. Yes?

No.

 

[Undefined]

No.

As per admin advice, that troll post from the Tach nuisance, empty of any science or math logic or argument should be ignored. He just excerpts out of context and makes useless 'noises' like "No" and leaves it at that as if he has argued something through. Suggest others ignore him likewise when encountering that same clueless and unimaginative troll.