Simple mathematical problem (?)

[xouper]

LuxFerum: I think the problem of that notation is that it mixes the process with the limit.

You're close. I'm guessing that perhaps the problem is with those who mistakenly think the notation involves a process. The number can certainly be described in terms of a process, but the number itself is not a process, nor is the notation 0.999... a process.

Perhaps it only seems like there is a process mixed in there. We're here to disabuse anyone of that notion.

Jargon review - the notation for a number is not the number, just like the map of a city is not the city (credit BillHoyt for that analogy in another thread). The map represents the city, and the notation 0.999... represents the number 1. The proofs that have been offered in this thread are intended to demonstrate the equivalence of the notations. Does this help?

 

[LuxFerum]

xouper
yes it does.

ceptimus

Seriously Lux, when you see an ordinary number, like 0.428, do you think, "four tenths, no wait.. forty-two hundredths, hang on... four hundred and twenty eight thousandths" ?]

Most of the time I will pic 0,43 and assume a small error.

Like those cheap calculators that when you do sqrt(2)*sqrt(2) you get 1,99999999 you just assume that the calculator cant handle those number perfectly.

 

[Colloden]

Short summary anyone ?

 

[Skeptoid]

Yes, the number 0.999... is exactly equal to the number 1.

 

[Iamme]

*Simple* mathematical problem! 363 posts! LMAO

 

[Suggestologist]

That is not even a number. It's nonsense! Please try again.

Under the definition of decimal notation, 0.00000....0001 doesn't even make sense.

At best, it's equal to zero, but that's a loose interpretation that is being generous and would have you agreeing with us that 1 = 0.999... anyway!

How can you even suggest that answer when xouper has shown already that 0.0000...0001 is not even a number? And if you had actually read Cabbage's post, you would also realize that it is meaningless under decimal notation.

Traveller

Your post was the first one I've ever reported. Usless.

Adam

Wrong. That notation is correct and is used by mathematicians. 0.00000__0001 is the correct notation for iota (or "epsilon" of the surreal numbers) in the real number system. If 0.00000.....0001 is meaningless under decimal notation then so is 0.99999999.....

All of the digits of 0.9999999.... are 9 after the decimal point. What is the last real digit of the number 0.999999999.....? If it is 0, then congratulations, you've got an actual argument that it always equals 1.000000000........000. If not, then you haven't.

 

[xouper]

Suggestologist: That notation is correct and is used by mathematicians. 0.00000__0001 is the correct notation for iota (or "epsilon" of the surreal numbers) in the real number system. If 0.00000.....0001 is meaningless under decimal notation then so is 0.99999999.....

Wrong.

The notation 0.000...001 does not represent a real number.

The notation 0.999... is a real number.

Surreal numbers are not real numbers and have no relevance to the equality 0.999... = 1. The surreal number system is an extension of the reals and since 0.999... = 1 in the reals, it is also equal in the surreals.

All of the digits of 0.9999999.... are 9 after the decimal point. What is the last real digit of the number 0.999999999.....?

There is no last digit in 0.999... so your argument is specious.

If it is 0, then congratulations, you've got an actual argument that it always equals 1.000000000........000. If not, then you haven't.

Not only is that wrong, but it is wrong on a deeply fundamental level. All the arguments presented in this thread that show that 0.999... =1 are mathematically valid and do not require the "last" digit of 0.999... to be zero.

Why do you think you are right and mathematicians have all lost their marbles?

Since you propose that 0.999... is not equal to one, then you are also proposing that there exist two positive integers A and B such that A / B = 0.999... and that A does not equal B. Please provide a mathamatical proof of that assertion.

 

[Suggestologist]

Wrong.

The notation 0.000...001 does not represent a real number.

The notation 0.999... is a real number.

Where is your evidence?

Surreal numbers are not real numbers and have no relevance to the equality 0.999... = 1.

Any real number can be represented as a surreal number.

The surreal number system is an extension of the reals and since 0.999... = 1 in the reals, it is also equal in the surreals.

No, it isn't equal in the surreals. 1-epsilon != 1.

There is no last digit in 0.999... so your argument is specious.

ALL positions after the decimal point are 9. Therefore, the LAST position is also 9.

Here's the problem: If you take .999999__ to be equal to 1 by the idea that 1/1 == .9+.1/1 = .99+.01/1 = .999+.001/1 = etc. then you have a remainder after the last digit which equals epsilon. When you add epsilon back (the remainder), then you get 1. But that remainder is not consistent with all uses of .99999999999.......

Not only is that wrong, but it is wrong on a deeply fundamental level. All the arguments presented in this thread that show that 0.999... =1 are mathematically valid and do not require the "last" digit of 0.999... to be zero.

Why do you think you are right and mathematicians have all lost their marbles?

Where is your evidence that all mathematicians agree with you?

Since you propose that 0.999... is not equal to one, then you are also proposing that there exist two positive integers A and B such that A / B = 0.999... and that A does not equal B. Please provide a mathamatical proof of that assertion.

I posted that a long time ago.

The ratio would look like: 999999......./1000000......

The fact that this amounts to (infinity-1/infinity) which is undefined, goes to show that the answer depends on the usage.

 

[ceptimus]

This thread is like the Hydra.

 

[TillEulenspiegel]

Yes it is isn't it?
I think much of the confusion stems from the fact that many are thinking in terms of arithmatic equalities and the notions of sequences, limits, infinities and other basic math premises are unknown or ill understood.

 

[xouper]

Hydra indeed.

xouper: The notation 0.000...001 does not represent a real number.

Suggestologist: Where is your evidence?

I already posted it. Go look for my reply earlier in this thread with the spoiler box.

Any real number can be represented as a surreal number.

So what. Any integer can also be represented by a real number, but it still retains its integer properties. Likewise, repesenting a real number using surreal notation does not cause it to lose any of its properties as a real.

No, it isn't equal in the surreals. 1-epsilon != 1.

Which has nothing to do with 0.999... = 1.

1-epsilon does not equal 0.999... either.

ALL positions after the decimal point are 9. Therefore, the LAST position is also 9.

Are you being intentionally obtuse? There is no last digit in an infiinite decimal expansion of a real number. Please stick to traditional mathematics here instead of making up your own nonsense.

Here's the problem: If you take .999999__ to be equal to 1 by the idea that 1/1 == .9+.1/1 = .99+.01/1 = .999+.001/1 = etc. then you have a remainder after the last digit which equals epsilon.

By definition, there is no last digit in 0.999...

Where is your evidence that all mathematicians agree with you?

Duh. What planet are YOU living on? Go find any professional mathematician who says 0.999... is not equal to 1. There are none.

xouper: Since you propose that 0.999... is not equal to one, then you are also proposing that there exist two positive integers A and B such that A / B = 0.999... and that A does not equal B. Please provide a mathamatical proof of that assertion.

Suggestologist: I posted that a long time ago.
The ratio would look like: 999999......./1000000......

According to the definition of a rational number, A and B must be positive integers. Those two things you gave are not integers since neither has a finite value. Try again.

 

[xouper]

TillEulenspiegel: I think much of the confusion stems from the fact that many are thinking in terms of arithmatic equalities and the notions of sequences, limits, infinities and other basic math premises are unknown or ill understood.

I would add the clarification that mathematicians and other sufficiently math literate people seem to understand those notions rather well (or well enough) and that the confusion is on the part of those who do not sufficiently understand "the notions of sequences, limits, infinities and other basic math premises".

Edited to add - maybe that wasn't much of a clarification. The short version is, some do and some don't. The people who do understand those notions are not confused.

 

[Suggestologist]

Hydra indeed.

I already posted it. Go look for my reply earlier in this thread with the spoiler box.

So what. Any integer can also be represented by a real number, but it still retains its integer properties. Likewise, repesenting a real number using surreal notation does not cause it to lose any of its properties as a real.

Which has nothing to do with 0.999... = 1.

1-epsilon does not equal 0.999... either.

Sometimes it does.

Are you being intentionally obtuse? There is no last digit in an infiinite decimal expansion of a real number. Please stick to traditional mathematics here instead of making up your own nonsense.

By definition, there is no last digit in 0.999...

By whose definition?

Duh. What planet are YOU living on? Go find any professional mathematician who says 0.999... is not equal to 1. There are none.

There are none? Interesting.

According to the definition of a rational number, A and B must be positive integers. Those two things you gave are not integers since neither has a finite value. Try again.

They generate .9999999.... for the same reason that 1/1 can be made to generate .9999999...... They are what would be required to fit the role of the denominator and nominator of the ratio.

Begin with Nominator = 9 and Denominator = 10. Multiply by 10/10 without reducing the ratio. Add 9 to the nominator. Repeat.

 

[Skeptoid]

Nominator... is that a surreal numerator?

 

[Suggestologist]

Haven't read the entire thread so not sure if someone else have posted the proof yet.

x = 0.999... <=> 10x = 9.999... of these two equations we get:

10x - x = 9.999... - 0.999... <=>

9x = 9 <=>

x = 1

I can't see any errors in the proof here, and find it strange that such a simple proof has caused such a long thread. Maybe I'm missing something.

You have to look at the infinitessimal end of the numbers to see where the circularity is.

When you multiply .9999999.... by 10, is? the LAST digit -- the same position that the LAST 9 used to occupy -- now a 0 or now a 9? Multiplying by 10 means that it is now a 0. The LAST two digits that used to be "99" come to be occupied by "90" after the multiplication, not "99".

So:

X = .9999999......
10X = 9.99999999.......90
10X-X = 8.999999999.....91
9X = 8.99999999999.....91

So you only get 9X=X if you assume that .999.... actual equals 1 from the beginning -- which is circular. In that case, pushing the infinite zeroes over and adding another zero would produce 10.

 

[Suggestologist]

If what he says is true, that 1 does not equal 0.999..., then since both numbers are rational, and since there are an infinite number of reals between any two rational numbers, should be a piece of cake, eh? Or he could simply agree that no such real number x exists. No shame in that.

I agree that no such real number exists (at that level of granularity). As I said earlier, there is no real number closer to and less than 1 than .9999999...... . The idea that numbers exist between any two real numbers does not apply to infinite (non-zero) digits of real numbers.

The fact is that .999999......... inserts talk of infinity into the discussion. Whether by limits or series or recurrence of a digit or whatever. Trying to limit use of infinity to what YOU think is appropriate is like trying to have your cake and eat it too.

 

[sorgoth]

1/3 = .33(recurring, which means an infinite number of 3s)
3/3 = 3 * .33(recurring) = 0.9(recurring, which means an infinite number of 9s) = 1

Not quite. You see, 1/3 is an infinitely small bit larger than 0.333...

 

[Yahweh]

I've got some wood on the fireplace, its cozy and warm in my house...

I'm going to make myself some nice hot cocoa...

You kids, get on back to fighting...

 

[Cabbage]

Suggestologist:

ALL positions after the decimal point are 9. Therefore, the LAST position is also 9.

When you multiply .9999999.... by 10, is? the LAST digit -- the same position that the LAST 9 used to occupy -- now a 0 or now a 9? Multiplying by 10 means that it is now a 0. The LAST two digits that used to be "99" come to be occupied by "90" after the multiplication, not "99".

I want to thank you for these quotes, they illustrate perfectly your stupidity.

 

[Suggestologist]

Suggestologist:


I want to thank you for these quotes, they illustrate perfectly your stupidity.

Do you know how to compute a derivative using infinitessimals? Or do you only know that epsilon-delta limit crap?