Is 1 = 0.9999......

[Dari]

since this thread refuses to die, can someone here answer me this: I saw what I would consider the ultimate equation a couple of years ago. It had sine, pi, 1, i (imaginary number), and e (euler). Can someone show me a link to it? I think it was a book about euler or pi or something. But it was e=sin^(pi*i) or something of that nature. Please find it for me?

 

[Kyteland]

Originally posted by: Dari
since this thread refuses to die, can someone here answer me this: I saw what I would consider the ultimate equation a couple of years ago. It had sine, pi, 1, i (imaginary number), and e (euler). Can someone show me a link to it? I think it was a book about euler or pi or something. But it was e=sin^(pi*i) or something of that nature. Please find it for me?

Is this what you mean?

e^(pi*i) = -1

link

 

[Dari]

Originally posted by: Kyteland

Originally posted by: Dari
since this thread refuses to die, can someone here answer me this: I saw what I would consider the ultimate equation a couple of years ago. It had sine, pi, 1, i (imaginary number), and e (euler). Can someone show me a link to it? I think it was a book about euler or pi or something. But it was e=sin^(pi*i) or something of that nature. Please find it for me?

Is this what you mean?

e^(pi*i) = -1

link

yep, this is exactly what I meant. I found this when I was still in high school but completely forgot about it during college. Now it all makes sense. Ahhhhh, mathematics truly is an art in and of itself.

 

[Dufusyte]

Originally posted by: silverpig
2.000... whether you deem the zeros necessary or not, is an interminal decimal.

"2" is not interminal. It terminates.

"Terminate" and "interminate" refer to the process by which the number is derived. For example, if you derive 2 by: 10/5, then you have a nice little process that terminates, namely:

10/5=2 there is no remainder, so the process terminates. Or, if you are picky, the remainder is Zero, which also signifies that the process has terminated, since there is zero (nothing) remaining.

By contrast, if you derive 0.3333... from 1/3, you can easily see that the process of division does not terminate since there is always a remainder:

1/3=.3 with a remainder of 0.1, necessitating contining the process (interminally).

So 10/5=2 is not interminal. If you were not aware that division terminates when there is no remainder (or when the remainder = zero), then I pity you, and suggest you go back to second grade.

 

[bleeb]

PLEASE PEOPLE... 0.9999.... != 1.

 

[spidey07]

Please people! The earth is flat!!!

 

[Kyteland]

Originally posted by: spidey07
Please people! The earth is flat!!!

We know, we know. And the sun revolves around the earth and 0.9999.... != 1. bleeb has made this argument many times already.

 

[bleeb]

Its just that I dont' want this thread to die.

 

[Kyteland]

Originally posted by: bleeb

Its just that I dont' want this thread to die.

Death eventually comes to everyone, even this thread. I thought it gave up the ghost last week. I was wrong...

 

[RossGr]

Originally posted by: Kyteland

Originally posted by: RossGr

But I still don't see how you can apply rule of forever to one number and not the other.

What puzzels me is how us saying 9s forever is the same as saying 0s forever then a 1. When I say forever I am done, you say forever then a 1. How can the 1 be there if it is indeed 0s forever. Don't you see that that statement is self condraticding?

Ah, but can't you use a regular expression to construct such a number?

0.0*1

S -> 0A
A -> .B
A -> .C
B -> 0B
B -> 0C
C -> 1

Can't this number have an orbitrary number of 0's? If that's the case then their concept of 0.00....r1 would be described by this.

Too bad a Turing machine would never recognize it....

Sure you can have an arbitrary number of zeros, but if you specify some other digit following the zeros, there will be a finite number of zeros because the specified numeral must be indexed by an integer. If you cannot specify the location of a digit it is not part of a Real number. Because all digits of a real number can be specified by a integer. that is simply a fundamental definition which cannot be ignored.

Let D(i) represent the digits of a Real number where i is what we call a dummy variable, and represents some integer to be specified. A Real number consists of the infinite sum of D(i)*10^-i .

Now
We can specify all the digits of .999..... by D(n) = 9 FOR ALL n. So this number is fixed all of its digits are specified.



How do you specify all the digits of an infinite number of zeros followed by a 1

The first part says D(n) = 0 for all n. Then second part, (which says followed by a 1), implies some N such that D(N) = 1 , but doesn't the existance of such an N contradict the first part of the spec? So either you did not mean for all n but for n = 1 to N-1 D(n)=0 then D(N)=1 or N simply does not and cannot exist. It is either an infinite number of zeros or it is not you cannot have both.

 

[silverpig]

Originally posted by: Dufusyte

Originally posted by: silverpig
2.000... whether you deem the zeros necessary or not, is an interminal decimal.

"2" is not interminal. It terminates.

"Terminate" and "interminate" refer to the process by which the number is derived. For example, if you derive 2 by: 10/5, then you have a nice little process that terminates, namely:

10/5=2 there is no remainder, so the process terminates. Or, if you are picky, the remainder is Zero, which also signifies that the process has terminated, since there is zero (nothing) remaining.

By contrast, if you derive 0.3333... from 1/3, you can easily see that the process of division does not terminate since there is always a remainder:

1/3=.3 with a remainder of 0.1, necessitating contining the process (interminally).

So 10/5=2 is not interminal. If you were not aware that division terminates when there is no remainder (or when the remainder = zero), then I pity you, and suggest you go back to second grade.

But 10 = 10.000...
and 5 = 5.000...

Do the long division including all of the zeros and you will get 2.000...

I can also tell you that 10/5 (or any other division) is NOT how the number 2 is "derived".

2 = {0,1}

However, since we are dealing with a decimal number system, we define 2 to be "zero tens, 2 ones, 0 tenths, 0 hundredths" etc...

0.999... is NOT a process. It is a number.

 

[Kyteland]

Originally posted by: RossGr

Originally posted by: Kyteland

Originally posted by: RossGr

But I still don't see how you can apply rule of forever to one number and not the other.

What puzzels me is how us saying 9s forever is the same as saying 0s forever then a 1. When I say forever I am done, you say forever then a 1. How can the 1 be there if it is indeed 0s forever. Don't you see that that statement is self condraticding?

Ah, but can't you use a regular expression to construct such a number?

0.0*1

S -> 0A
A -> .B
A -> .C
B -> 0B
B -> 0C
C -> 1

Can't this number have an orbitrary number of 0's? If that's the case then their concept of 0.00....r1 would be described by this.

Too bad a Turing machine would never recognize it....

Sure you can have an arbitrary number of zeros, but if you specify some other digit following the zeros, there will be a finite number of zeros because the specified numeral must be indexed by an integer. If you cannot specify the location of a digit it is not part of a Real number. Because all digits of a real number can be specified by a integer. that is simply a fundamental definition which cannot be ignored.

Let D(i) represent the digits of a Real number where i is what we call a dummy variable, and represents some integer to be specified. A Real number consists of the infinite sum of D(i)*10^-i .

Now
We can specify all the digits of .999..... by D(n) = 9 FOR ALL n. So this number is fixed all of its digits are specified.



How do you specify all the digits of an infinite number of zeros followed by a 1

The first part says D(n) = 0 for all n. Then second part, (which says followed by a 1), implies some N such that D(N) = 1 , but doesn't the existance of such an N contradict the first part of the spec? So either you did not mean for all n but for n = 1 to N-1 D(n)=0 then D(N)=1 or N simply does not and cannot exist. It is either an infinite number of zeros or it is not you cannot have both.

Which is exactly why I said a turing machine would not recognize 0.000.....r1. It would never halt. Non-halting turing machines are bad news.

 

[Dufusyte]

To the dude who said something about the numberline being filled with holes:

The line is a continuum. All places on the continuum can be described by a fixed number. Yes, we have plenty of numbers to go around, and they can get very very accurate.

Any number that cannot be expressed as a fixed value is very fishy, and should not be allowed to play with the other numbers.

 

[bleeb]

hmm so how is it possible, if 0.9999... (an infinitely repeating, neverending decimal) be equal to a number that is non-repeating, ending, decimal number?

0.9999... != 1

 

[ElFenix]

Originally posted by: BigNeko
RossGR,
I will concede that my EXPRESSION of the concept .000...1 is technically incorrect,
however, I am asking you to overlook the expression (in our finite system) and look more to the concept itself.

If one number can go on forever, so can the other. It is so simple.

The nines are already there? Okay.
So are the zeroes, with that pesky one at the non-end.

Come to think of it, my expression is technically correct,
because my zeroes are merely place-holders for a little one that
is infinitely to the right of the decimal point. You may not be able to see it,
but its way-y-y-y down there.

Trust me.

look, you're changing the rules and going outside the defined system. like i said earlier, if you go outside the system, you can prove anything you want. in this case you're changing the concept of infinity, just the same as madrat is doing.

 

[MadRat]

Originally posted by: RossGr

Sure you can have an arbitrary number of zeros, but if you specify some other digit following the zeros, there will be a finite number of zeros because the specified numeral must be indexed by an integer. If you cannot specify the location of a digit it is not part of a Real number. Because all digits of a real number can be specified by a integer. that is simply a fundamental definition which cannot be ignored.
Let D(i) represent the digits of a Real number where i is what we call a dummy variable, and represents some integer to be specified. A Real number consists of the infinite sum of D(i)*10^-i .
Now
We can specify all the digits of .999..... by D(n) = 9 FOR ALL n. So this number is fixed all of its digits are specified.

A little while ago someone just said that .999... does not end. So if n=infinity then what is the digit? Oh, that's rightttttttt.... Now, your point earlier said that there is no infinite position. You just thought outside the box to prove it... We can add one more set of logic to provide for .000...r1 by saying that D(n) = 0 FOR ALL n < infinity, else if n=infinity then D(n) = 1.

Originally posted by: ElFenix

look, you're changing the rules and going outside the defined system. like i said earlier, if you go outside the system, you can prove anything you want. in this case you're changing the concept of infinity, just the same as madrat is doing.

Look outside the box, fella, and progress can be made. To stay inside the box one must hide from the rest of the universe. :Q

 

[RossGr]

Originally posted by: MadRat

Originally posted by: RossGr

Sure you can have an arbitrary number of zeros, but if you specify some other digit following the zeros, there will be a finite number of zeros because the specified numeral must be indexed by an integer. If you cannot specify the location of a digit it is not part of a Real number. Because all digits of a real number can be specified by a integer. that is simply a fundamental definition which cannot be ignored.
Let D(i) represent the digits of a Real number where i is what we call a dummy variable, and represents some integer to be specified. A Real number consists of the infinite sum of D(i)*10^-i .
Now
We can specify all the digits of .999..... by D(n) = 9 FOR ALL n. So this number is fixed all of its digits are specified.

A little while ago someone just said that .999... does not end. So if n=infinity then what is the digit? Oh, that's rightttttttt.... Now, your point earlier said that there is no infinite position. You just thought outside the box to prove it... We can add one more set of logic to provide for .000...r1 by saying that D(n) = 0 FOR ALL n < infinity, else if n=infinity then D(n) = 1.

Originally posted by: ElFenix

look, you're changing the rules and going outside the defined system. like i said earlier, if you go outside the system, you can prove anything you want. in this case you're changing the concept of infinity, just the same as madrat is doing.

Look outside the box, fella, and progress can be made. To stay inside the box one must hide from the rest of the universe. :Q

Madrat,
Unfortunately the box is well defined, the Real number system is not outside of the box. The ONLY way we can arrive at meaningful results is by staying in our well defined box, if you leave the box you are not talking about the real number system. I am constrained by the box to carefully define what I mean by a Real number and the methods I can use to manipulate them. Sorry if this offends you but that is Mathematics. Notice that I capitialize Real, this is the name of a very formally defined system. I do not mean lower case 'real," as in something that exists in general. I am refering to the Real number system and actually much of what I say applies to the subset of the Reals called the Rational numbers. I am not talking about Irrational numbers when I insist that every digit be defined. To use the term Real Number carries a lot of baggage, all of it must be tended to, if you ignore a single rule you are lost and any result following is meaningless.

While the box is is well defined it is defined in such away as to leave lots of freedom of thought and exploration.

 

[MadRat]

The box I spoke lay in the boundaries of one's mind.

The box you speak of is a flat line segment, {0,1}, and my argument nicely fits within it.

 

[BigNeko]

hmm so how is it possible, if 0.9999... (an infinitely repeating, neverending decimal) be equal to a number that is non-repeating, ending, decimal number?

That is a very good question!

I am sensing that the fact that the nines go on and on somehow allows people here to feel they have a grasp on it.

But, is it a number? Numbers represent values. If you cannot define the value of a number, is it still a number?

Just because it is made up of numbers does not mean it IS a number. See square root of -1 for an example.

Indeterminate is just a nice way of saying "We have no f***ing clue."

A little outside the box

 

[RossGr]

Originally posted by: BigNeko

hmm so how is it possible, if 0.9999... (an infinitely repeating, neverending decimal) be equal to a number that is non-repeating, ending, decimal number?

That is a very good question!

I am sensing that the fact that the nines go on and on somehow allows people here to feel they have a grasp on it.

But, is it a number? Numbers represent values. If you cannot define the value of a number, is it still a number?

Just because it is made up of numbers does not mean it IS a number. See square root of -1 for an example.

Indeterminate is just a nice way of saying "We have no f***ing clue."

A little outside the box

Not even slightly outside of the box, yes it is a very well defined number. Only those of you who not understand the construction of the Real Number line have troubles with it. It is to bad that you cannot understand that you are getting some execellent insight to the details of math not seen outside of upper division or graduate level courses. Seems like many have preconcieved notions which they cannot let loose of. It is the disbelievers who need to get out of the box. Matrat is , and prehaps you also, are locked deep inside of a box and seem unable to let the Math speak to you. You both need to stare at the definition of equality presented in my proof until you realize that it means exactly what it says. Equality is not defined by the way the numbers look but by how close together they are. This is not something I dreamed up just to prove this specific point but it is a definition from real analysis. I must admit that none of what I have presented is original thinking it is all straight from text books.

When it comes to .999... no ne is using the term indeterminate it is a well defined number. And you are absolutly correct, just because it is made up of numbers does not mean it IS a number, such is the state of .00...01 where the ellipsis is misused to represent an infinite number of missing zeros. It contains numbers but it is not a Real Number.

 

[BigNeko]

RossGR,
To say infinity is well defined is not part of my "box".

I have been quite reasonable during this discussion.
Have even given you credit for a good representation of what you are trying to say.
In return I get veiled insults and a condescending attitude?

This is what has brought ATOT down.

So, YOU break outside that particular box and knock that sh!t off.

Keep it adult, keep it on topic and keep at it. You may yet convince the unconvinced.

 

[ElFenix]

great, so you're arguing some definition of infinity that you just made up, where theres numbers beyond that. if you're not going to use the same constraints then arguing is completely pointless.

 

[bleeb]

Guys, there is no need to attack each other. Can't we all just get along? We need to solve this in a civilized manner. Please! For all the guys who say that 0.9999... = 1, your proof(s) obviously aren't enough to prove it beyond a reasonable doubt. There are people who still have an argument against 0.9999... = 1. So, please continue to present your reasoning and eventually the truth will be known!

BTW.... 0.9999... != 1!

 

[silverpig]

Originally posted by: bleeb
Guys, there is no need to attack each other. Can't we all just get along? We need to solve this in a civilized manner. Please! For all the guys who say that 0.9999... = 1, your proof(s) obviously aren't enough to prove it beyond a reasonable doubt. There are people who still have an argument against 0.9999... = 1. So, please continue to present your reasoning and eventually the truth will be known!

BTW.... 0.9999... != 1!

Our proofs are obviously enough for anyone with an understanding of mathematics as the ENTIRE mathematical community acknowledges, understands, and agrees with the results obtained from the proofs.

The people who have an argument against it have only that, an argument. Not once has a proof been offered, not once has any kind of formal statment been produced, and not once has any part of any of the proofs offered thus far by us been discredited with any merit.

So far all arguments have been written in english; mathematics is a separate language in itself. Any credible counterargument must be written in the vernacular of a formal mathematical setting.

This thread will continue on until one of the following occurs:

1. The people who think 0.999... != 1 provide a sound, formal mathematical proof stating so.
2. The people who think 0.999... != 1 realize that no such proof exists.

I can say right now that the first condition will NOT be satisfied, and it is therefore up to the participants of this thread to satisfy the second condition.

 

[bleeb]

Originally posted by: silverpig

Originally posted by: bleeb
Guys, there is no need to attack each other. Can't we all just get along? We need to solve this in a civilized manner. Please! For all the guys who say that 0.9999... = 1, your proof(s) obviously aren't enough to prove it beyond a reasonable doubt. There are people who still have an argument against 0.9999... = 1. So, please continue to present your reasoning and eventually the truth will be known!

BTW.... 0.9999... != 1!

Our proofs are obviously enough for anyone with an understanding of mathematics as the ENTIRE mathematical community acknowledges, understands, and agrees with the results obtained from the proofs.

The people who have an argument against it have only that, an argument. Not once has a proof been offered, not once has any kind of formal statment been produced, and not once has any part of any of the proofs offered thus far by us been discredited with any merit.

So far all arguments have been written in english; mathematics is a separate language in itself. Any credible counterargument must be written in the vernacular of a formal mathematical setting.

This thread will continue on until one of the following occurs:

1. The people who think 0.999... != 1 provide a sound, formal mathematical proof stating so.
2. The people who think 0.999... != 1 realize that no such proof exists.

I can say right now that the first condition will NOT be satisfied, and it is therefore up to the participants of this thread to satisfy the second condition.

I wanted to let you know of a few historical facts. People once thought the world was FLAT. Or that the moon was made of Cheese. Or that Mars had higher life forms. Just because the entire mathematical community agrees with something, doesn't necessarily mean that its correct. You're SO close minded that you have no openness for possibilities which might represent the truth. Remember, people who found the truth are often the ones who have gone against the "conformist" views of the masses. So I say to you, quit being so Anti-Explorative in your views and let the "true geniuses" explore the other possibilities. Where would we be if Christopher Columbus didn't sail the oceans? Or if Einstein didnt follow is own beat?