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

[Undefined]

Right, but 7 is just a lazy way of writing 7.000(0) so even in your example all we've actuallu done is move the decimal one place to the right to make it 70.000(0).

Hi Trippy. How are you and yours going, ok I trust!

As to the point involved here, it is the 'triviality' aspect of leading/trailing zeros (as I and Pete already agree about), but now with the further 'triviality' involved in invoking a "." decimal point to "move" in any direction. It is especially evident in my example eliminating ALL trivialities (leading/trailing "0" AND unnecessary "." decimal point) that only the "0" brought to the 10 x 7 operation is a fundamental consideration and without needing to "move" ANY (in my example) non-existent 'decimal point' at all? That was the point. Triviality of leading/trailing zeros and non-necessary "." decimal point or imaginary necessity to invoke/move same at all.

Cheers, Kiwi!

 

[arfa brane]

because 0.999... + 0.999 can not equal 1.999... with out the use of an infinitesimal.

No that's wrong. Adding 0.999... to itself can be seen as the process of summing 0.9 and 0.9, then adding the sum of 0.09 and 0.09, and so on 'infinitely'. The last digit cannot be an 8, because you sequentially add 1 to it, like you do with all the other 8s in the partial sum.

 

[TruthSeeker]

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.. it magically turns from 0.9999999... to 9.000000...

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.

It is simpler than that. If you multiply 10 by 0.99999... you should get 10 times what 0.9999... is, or 9.9999.... to turn it into 10 is essentially rounding up, it is cheating.

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?

Yes and no. Planck units are technically the limits of physical matter, but in theory there could be something beyond that. Limits are just a finite way to express something which can't be defined. 0.33333.... goes on forever, but 1/3 is a finite way to describe that infinite number.

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.

Precisely.

 

[Quantum Quack]

@pete
try this set of finite string and ignore the need for ellipses

0.99999999+0.99999999=1.99999998

1.99999998+0.99999999=2.99999997

2.99999997+0.99999999=2.99999996
You will note that the trailing digit is reducing with every addition of 0.99999999

now consider the same sets in infinite terms

0.999.... + 0.999... = 1.999...8
etc

we can not ignore the trailing digit no matter how much we may want to
So
9.999... - 0.999 = 9 works clean
yet addition by way of multiplication provides an issue that at this level of discussion can't be ignored
9 x 0.999... +1/infinity = 9

 

[Quantum Quack]

No that's wrong. Adding 0.999... to itself can be seen as the process of summing 0.9 and 0.9, then adding the sum of 0.09 and 0.09, and so on 'infinitely'. The last digit cannot be an 8, because you sequentially add 1 to it, like you do with all the other 8s in the partial sum.

ok...
a different take on the process thanks....addition from left to right rather than right to left...ahhh

can you show this in some sort of long hand version please...

 

[arfa brane]

now consider the same sets in infinite terms

0.999.... + 0.999... = 1.999...8
etc

Again, this is wrong.

Add 0.9 to 0.9, you get 1.8. This is a partial sum.
Add 0.09 to 0.09, you get 0.18, add this to the partial sum, you have: 1.98

So the first decimal place has a 9 in it. By induction so does every decimal place, so there is no 8 at the "end", because there is no last decimal place, but the first, second, and nth places must be 9 (for any value of n).

 

[Trippy]

Hi Trippy. How are you and yours going, ok I trust!

As to the point involved here, it is the 'triviality' aspect of leading/trailing zeros (as I and Pete already agree about), but now with the further 'triviality' involved in invoking a "." decimal point to "move" in any direction. It is especially evident in my example eliminating ALL trivialities (leading/trailing "0" AND unnecessary "." decimal point) that only the "0" brought to the 10 x 7 operation is a fundamental consideration and without needing to "move" ANY (in my example) non-existent 'decimal point' at all? That was the point. Triviality of leading/trailing zeros and non-necessary "." decimal point or imaginary necessity to invoke/move same at all.

Cheers, Kiwi!

But they're neither un-neccessary nor trivial. That's your first mistake.

That's not what significance actually means. 1 is an approximation that could mean any number between 0.5 and 1.5 rounded to 1 significant figure. 1.0(0) is an exact number.

Remember, we're dealing with real numbers here, not integers or natural numbers.

 

[Pete]

That was also true in my example. Moreover, in my example we had the added aspect that the "." decimal point was also not relevant

Rubbish. Your example was 7 x 10. Shifting the decimal point to the right means we then have to write a zero we didn't bother to before, to get 70.

Bye.

If you don't mean it, don't say it.

 

[Pete]

You now agree with this equation:
10 x 0.999... - 0.999... = 9 x 0.999... <<disagrees due to the need for the infinitesimal...

QQ, you just agreed to that equation!

Post 172:

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...

yep...agrees

 

[Trippy]

Rubbish. Your example was 7 x 10. Shifting the decimal point to the right means we then have to write a zero we didn't bother to before, to get 70.

Which is exactly what I said.

 

[Quantum Quack]

QQ, you just agreed to that equation!

Post 172:

ok... I did... however did you read why I reject the result... of


9.999...- 0.999... = 9 = 9 x 0.999...

with out the addition of an infinitesimal

Gosh this is hard to explain... sorry...
The nine in the middle is not the same nine depending on which side of the equation you are focussed on....

try this as proof of my point.

Theoretically the proof should also work as follows
Compare:

9.999... - 0.999... = 9 [exact with out the need for any "rounding"]

9 - 0.999... = 8 ["rounding" needed]


Contention:
The value of the distance between
9.999... and 9 is less than the distance between 9 and 8 [therefore "rounding" is being applied inconsistently]

If they read as follows
10 - 0.999... = 9
and
9 - 0.999... = 8
I would have no real problem and accept that rounding is being applied consistently.
However this would not be able to be used as proof that 0.999.. = 1

Do you understand what I am suggesting with the above..?

 

[Quantum Quack]

deleted as unnecessary

 

[arfa brane]

I would have no real problem and accept that rounding is being applied consistently.
However this would not be able to be used as proof that 0.999.. = 1

There is no rounding.
0.999... is a repeating decimal, it isn't rounded up or down.

You need to abandon the idea of rounding, it doesn't apply to repeating decimals, because they repeat infintely.

 

[Pete]

ok... I did...

So, you now have an arithmetic proof that shows that
9 x 0.999... = 9
and therefore, 0.999... = 1

however did you read why I reject the result... of

9.999...- 0.999... = 9 = 9 x 0.999...

with out the addition of an infinitesimal

Yes. Have you read the explanations in the thread and on Wikipedia for why there is no infinitesimal involved?

You now have an arithmetic proof that 0.999... = 1, that conflicts with your idea that there is an infinitesimal difference between them.
One of those must therefore be wrong.

 

[leopold]

The point is that it is very clear that:
10 x 0.999... - 0.999... = 9 x 0.999...

Look:
10 x 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... = 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999... + 0.999...

Agreed?


Thus proving that 1.999... = 2


Thus proving that 8.999... = 9


If $$1/\infty$$ has any value, then that value is zero. Or do you think that Achilles really can't catch the tortoise?

if i were still in school i would test this by giving the answer 1.999 . . . to the question of what is 1+1.
i am quite positive the teacher would mark it wrong.

 

[Quantum Quack]

So, you now have an arithmetic proof that shows that
9 x 0.999... = 9
and therefore, 0.999... = 1


Yes. Have you read the explanations in the thread and on Wikipedia for why there is no infinitesimal involved?

You now have an arithmetic proof that 0.999... = 1, that conflicts with your idea that there is an infinitesimal difference between them.
One of those must therefore be wrong.

Are you prepared to deal with what I raise in the post a few up #191?
oh, btw I already accept that I am "wrong"... sheesh! 100's of years of math theorums... ha not a hope... "I may be crazy, but I am not totally stupid"

 

[Quantum Quack]

to say that
9.999... - 0.999...= 9
only says that the distance in value between 9.999.... and 9 is in fact 0.999... , thus proving that 0.999... = 0.999... and does not equal 1

 

[Quantum Quack]

Pete:
If 1/infinity [sorry no latex] has any value, then that value is zero. Or do you think that Achilles really can't catch the tortoise?

well this I can say with out any doubt what so ever if Achilles beats the tortoise to the tortoise's position in the race we as a universe are in deep shit.....
ask Heisenberg..he'll tell ya...
1/infinity can not equal zero....1/infinity must be >0 that's the whole point of infinitesimals to begin with.

 

[elte]

0 is a place holder to mark an empty digit in a number. When multiplying a number by ten it basically means to add the number together ten times.

As a digit, 0 is used as a placeholder​

http://en.wikipedia.org/wiki/0_(number)

 

[Undefined]

Hi Trippy.

But they're neither un-neccessary nor trivial. That's your first mistake.

That's not what significance actually means. 1 is an approximation that could mean any number between 0.5 and 1.5 rounded to 1 significant figure. 1.0(0) is an exact number.

Remember, we're dealing with real numbers here, not integers or natural numbers.

Yes, mate; already understood long since. As indicated in my post (#165, quoted below) to Pete mentioning that very aspect in connection with use for trivial leading/trailing zeros in certain contexts. I have highlighted the bit indicating that aspect was already understood...

Hi Pete.

Agin I beg to differ. Here is why...

The decimal NOTATION construct has PRE-EXISTING 'empty places' (STRING of "empty or zero symbol) from the units position upwards (extending towards the left from the decimal point SEPARATOR symbol which denotes TRANSITION to a PRE-EXISTING 'empty places' extending towards the right from the "tenths" position.

All 'leading zeros and trailing zeros are TRIVIAL inclusions for convention purposes to clarify 'level of accuracy' termination and not for any fundamantal 'value' information as such.


So, the decimal notation expression of 000007.00000 is trivially extended and can be reduced to fundamentals by writing it as 7. That's it.

So the multiplication operation by 10 is adding a "0" to 7, making it 7 x 10 =70 and that's it. anything more is trivial and unnecessary. So any 'proofs' depending on trivial unnecessary manipulations which hide this fundamental operation is not a 'proof' at all, but rather a trivial exercise with no point to it at all except the circuitous 'proofs' we have been seeing.

Anyhow, see where there is NO "." decimal point TO move, either direction, in the fundamental treatment?

Whereas there IS a "0" to add in some position in the "units" place position which FORCES the 10 in the target string to move to the left and so placing the 1 and 0 of the original 10 notation into the hundreds and tens place respectively while the "0" in the units place is the additional "0" effectively brought by the multiplier 10 to the string?


It is simple and straightforwardly more fundamental to think of it that way than it is to think of some decimal point being moved (especially when in this case NO "." decimal point is non-trivially invoked/involved at all in the string/operation?).

That was all I wanted to point to. No more than that aspect which makes all the trivial manipulations/format 'proofs' totally unnecessary/unconvincing in the more fundamental reality context.

As to your point about "dealing with real numbers, not integers or natural numbers etc", in my opinion/observation that is yet another abstraction away from the fundamentals (entities/operations) involved.

For example, the expression you and others have been using is 0.999... = 1 so the "1" in that statement is the most fundamental expression before any further axiomatic abstractions are piled onto it. Yes?

Hence my bringing the point back to the most fundamental action/entity/property involved. And by using that same "1" notation for the expression 10 x 7, I was being consistent to your/others' use of "1", and not the specific extension into non-fundamental expression like 1.0(0) which is trivial at the fundamental level (while it is useful for higher level expressions in the further need to express "accuracy" etc, as I already recognized as per my abovequoted post to Pete, yes?).

That is where I am coming from. In my opinion/observation, the zero (as a number) brought to the most fundamental operation/expression of Pete's example of 10 x 7 is that zero which is effectively placed after the 7 to give 70. That's it.

The mere FORMATTING conventions which would bring in more zeros as purely formatting symbols is non-fundamental in that case, and is therefore essentially a trivial manipulation like "moving a decimal point" which does not come into it fundamentally in that case, but only as a non-fundamental 'overlay' FORMATTING device in the decimal system. This trivial 'overlay' is not fundamental or needed at all in some other systems. Yes?

So "moving the decimal point" instead of just recognizing the fundamentality of the numbers/entities/operations involved is mere convention and choice and not what actually happens: ie, the 10 brings the zero which is added to the target 7 to make it 70 as the result of that multiplication. No decimal point involved fundamentally.

That was my point. That is where I was coming from: fundamental versus convention choices and trivial notation/proofs based on same. I can't say it any more directly than that, Trippy.