Is 1 = 0.9999......

[RossGr]

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

OK, fine..

What number system are we working with? What are your basic axioms? How, if infinty is not well defined, can you have something after infinity? Is something after infiity well defined? if it is not how do you use it? If it is, how so?

If you want to have a meaningful discussion about the Real Numbers you simply cannot arbitraly ignore the definitions of the Real Number system. I am sorry if it offends you, that indepth knowledge of the Real Number line can best be gained at the feet of professors in one of the most boring courses taught in the unversity system. I do not mean it as an insult or to be condecending, it is simply a fact.

If you wish to discuss the Real Number system then read my posts carefully and try to understand what I am attempting to tell you as I have sat of the feet of the professors and sweated blood attempting to understand the material presented. I make post after post of pure information which seem to be totally ignored, but let me imply that Madrat is an ignorant a$$ and I get all sort of replys. Perhaps if you spent more effort asking questions of the information I am attempting to provide and less worrying about your self esteem I would not get frustrated with the failure to communicate, and imply (or even say sometimes) things which clearly offend you. There I go again, I have done it again. I am sure that you will be insulted. My apologies.

edit: Spelling!

 

[RossGr]

. You're SO close minded that you have no openness for possibilities which might represent the truth.

Who is close minded? Us for understanding and appriciating the meaning of a mathematical proof or you, simply arguing from ignorance? You do not KNOW that 1 <> .999... but you certianly believe it. Just like the mediveal peasant believed the earth to be flat. It is YOU who are closed minded refusing to accept sound mathematicaly results simply because the result is not completely obvious.

Remove your blinders, make an effort to understand the wealth of information to which you are being exposed.

There is no way to think outside of the box if you do not know where the box is.

 

[BigNeko]

Perhaps if you spent more effort asking questions of the information I am attempting to provide and less worrying about your self esteem

Not insulted personally. Just don't want to see this degenerate into flames. And you making assumptions about how much time I do wonder about this question doesn't weaken my credibility (or self-esteem) in the least.
The fact that you seem to be falling back on using such tactics (as the above quote) on me, who (or is it whom) has not flamed or insulted you, indicates a possible crack in your armor.

So I shall now look for that crack (he-he).


Off to study infinity. (Be back soon )

Who said this? Sound familiar?

I realise that in this undertaking I place myself in a certain opposition to views widely held concerning the mathematical infinite and to opinions frequently defended on the nature of numbers.

 

[RossGr]

BigNeko,
I have appriciated your attitude, I see humor and evidence of true efforts to understand. I again will apologize for any apprarent personal attacks, these are more results of my inability to communicate my thought. I find these forums something less then ideal for presenting any detailed mathematical concepts. An equation editor is necessary. Unfortunatly since my last upgrade the Word Equation Editor on ALL of my home systems has ceased to function, and at work where I have a working equation editor I do not have time to type this stuff up nor acess to my web space. So I struggle along.

good luck with your explorations of infinity. If you have access to a University Liberary look for Real Analysis by Rudin, or Royden. Royden may be a bit to much, but Rudin is at a pretty good level for a first time through.

 

[BigNeko]

RossGR,
No harm, no foul. And personally I am GLAD there is no equation editor because then my mathematical ignorance would be bare to the world

Above quote was by Georg Cantor. Probably out of context, but I'll still use it.

And I am comforted by the fact that if I am wrong, I am wrong in good company.

The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited.

Thomas Aquinas.

 

[BigNeko]

As to the
.9+.1=1
.99+.01=1
.999+.001=1
and so on towards infinity
example, Blaise Pascal said this:

Even though this proposition may have an infinite number of cases, I shall give a very short proof of it assuming two lemmas. The first, which is self evident, is that the proposition is valid for the second row. The second is that if the proposition is valid for any row then it must necessarily be valid for the following row. From this it can be seen that it is necessarily valid for all rows; for it is valid for the second row by the first lemma; then by the second lemma it must be true for the third row, and hence for the fourth, and so on to infinity.

lemma is a preliminary proposition or thereom accepted for immediate use in a demonstration of some other proposition.

So according to Pascal, this type of proof does work.

 

[udonoogen]

to me ... in college food terms ... a $0.99 cent chicken sandwhich = $1 dollar chicken sandwich

therefore they are the same. haha. +1

 

[BigNeko]

This is what I was driving at earlier;

we attempt, with our finite minds, to discuss the infinite, assigning to it properties which we give to the finite and limited; but I think this is wrong, for we cannot speak of infinite quantities as being the one greater or less than or equal to another.

Galileo

 

[DanTMWTMP]

ehh...mods...lock...this...thread...please....no...matter...what......people will just argue this over and over w/ no end and stuff.....

 

[BigNeko]

Bleeb said:

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?

Galileo agrees:

... the totality of all numbers is infinite, and that the number of squares is infinite.; neither is the number of squares less than the totality of all numbers, nor the latter greater than the former; and, finally, the attributes "equal", "greater", and "less" are not applicable to the infinite, but only to finite quantities.

 

[Haircut]

As to the
.9+.1=1
.99+.01=1
.999+.001=1
and so on towards infinity

You will never reach infinity by counting though.

Imagine this:
You have a circle, you start at the top of the circle, lets call this 0 degrees.
You draw an infinitessimally thin line from the point you are at on the circumference to the centre of the circle.

Now, every sqrt(2) degrees you move around the circumference you draw another line to the centre.
I have chosen sqrt(2) as it in an irrational number, so no matter how many lines you draw you will never cross a line that is already drawn.


Now, no matter how many lines you draw you will never fill in the entire circle.
In terms of this circle 0.999... = 1 = the full, entirely coloured in circle.

 

[MadRat]

Originally posted by: RossGr
BigNeko,
I have appriciated your attitude, I see humor and evidence of true efforts to understand. I again will apologize for any apprarent personal attacks, these are more results of my inability to communicate my thought.

I'm still waiting for your apologies to me.

 

[RossGr]

Originally posted by: BigNeko
As to the
.9+.1=1
.99+.01=1
.999+.001=1
and so on towards infinity
example, Blaise Pascal said this:

Even though this proposition may have an infinite number of cases, I shall give a very short proof of it assuming two lemmas. The first, which is self evident, is that the proposition is valid for the second row. The second is that if the proposition is valid for any row then it must necessarily be valid for the following row. From this it can be seen that it is necessarily valid for all rows; for it is valid for the second row by the first lemma; then by the second lemma it must be true for the third row, and hence for the fourth, and so on to infinity.

lemma is a preliminary proposition or thereom accepted for immediate use in a demonstration of some other proposition.

So according to Pascal, this type of proof does work.

This is called proof by induction, yes it is a formal proof style taught and used in all levels of math. I perhaps could/should use an induction proof to further formalize my proof.

The difference between your set of equations and mine is that I add a small number to .999.... only to form your equations + a infinite string of 9s which sum to a number > 0 . so I claim that 10^-n + .999.... = 1 + (some more) for all real numbers. Notice that I am lumping the sum to infinity into an inequality, I do not claim that it is some specific number, I only claim that it is greater then zero.
Think about it.

 

[MadRat]

Originally posted by: silverpig

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.

You're taking liberties with that statement that are not necessarily true. The ENTIRE mathematical community does not acknowledge, understand, nor agree with the results obtained by the proofs. Infinity is understood to be something entirely different than what some of you argueing .999...=1 make it out to be. Its counterproductive to accept .999... as a Real number and then to not accept .000...1 using the same pretenses. Either both exist or neither exists by the definition given for infinity.

 

[MadRat]

Originally posted by: RossGr

The difference between your set of equations and mine is that I add a small number to .999.... only to form your equations + a infinite string of 9s which sum to a number > 0 . so I claim that 10^-n + .999.... = 1 + (some more) for all real numbers. Notice that I am lumping the sum to infinity into an inequality, I do not claim that it is some specific number, I only claim that it is greater then zero.
Think about it.

Using infinity as the base number it is possible to prove my point. With infinity as your base number you can stay well within the box (between 0 and 1) to prove any number in a formal proof without creating any magic numbers. The difference between .999... and 1 is that tiny sliver of .000...1, a "1" at the infinitely small position, that I've been telling you does exist all along.

Using infinity as the base numbering system works against your proof. That little sliver I pointed out does exist.

 

[RossGr]

Using infinity as the base number

What? I am not using infinity as a base number! Are you trying to say that .999.... = infinity? I thought you were claiming that it is less then 1, how did it get infinitly large?

I clearly show that .999.... + (any real number) = 1 + (a infinite long tail of 9s).

Are you trying to say that this number is less then zero? Or what are you trying to say?

. Its counterproductive to accept .999... as a Real number and then to not accept .000...1 using the same pretenses. Either both exist or neither exists by the definition given for infinity.

I say that .999.... is a real number because I can tell you what digit is at EVERY position in the sum. Every 9 is associated with some integer, therefore it is a Real number.

What integer is assigned to your 1? If you say none, then you have proven that you are not talking about a Real number. If you specifiy an N, then it is not at infinity, again you have contradicted yourself.

.00...001 can only be a Real number if the ellipsis represents a finite number of zeros.

I am being very consistent with this requirement, you continue to ignore the basic structure of a Real Number. Are we or are we not talking about the Real Number system?

 

[GoingUp]

*poop*

 

[Kyteland]

Originally posted by: bleeb

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 didn?t follow is own beat?

Actually if you look back in history the people who thought that the moon was made of cheese were the ignorant masses. Granted there were some intelligent people who thought that it was true, but for the most part it was the educated elite who finally came around to the fact that there was no cheese on the moon. In spite of that, it took them a long time to convince the aforementioned masses about the moon's lack of cheese. I mean, come on, look at it. Common sense would tell you that it is obviously made of cheese, right?

The same holds true for the world being flat. When you look at it, it looks flat. Therefore it is flat. It took the educated quite some time to convince the masses otherwise.

As for life on Mars, well there's life on Earth so there must be life on Mars, right? Common sense. We are now certain that there is no life on Mars (intelligent anyways).

Which brings us back to the 0.99...=1 debate. If you take a look at the poll it quite clearly reflects the history presented above. 278 people (50.82%) claim that they are not equal. I will label these people "Exhibit A" aka "The Ignorant Masses". 224 people (40.95%) claim that they are equal. I will label these people "Exhibit B" aka "The Educated Elite". I think we all know which category you fall in to bleeb.

Therefore, 0.99...=1 QED

 

[silverpig]

Originally posted by: bleeb


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?

People originally THOUGHT the world was flat until someone PROVED it to not be so. Try and apply that to this situation. Upon first glance, I too thought that 0.999... must be less than 1, but very soon after seeing a proof I realized it was not so.

 

[silverpig]

Originally posted by: BigNeko
As to the
.9+.1=1
.99+.01=1
.999+.001=1
and so on towards infinity
example, Blaise Pascal said this:

Even though this proposition may have an infinite number of cases, I shall give a very short proof of it assuming two lemmas. The first, which is self evident, is that the proposition is valid for the second row. The second is that if the proposition is valid for any row then it must necessarily be valid for the following row. From this it can be seen that it is necessarily valid for all rows; for it is valid for the second row by the first lemma; then by the second lemma it must be true for the third row, and hence for the fourth, and so on to infinity.

lemma is a preliminary proposition or thereom accepted for immediate use in a demonstration of some other proposition.

So according to Pascal, this type of proof does work.

That's a proof by induction:

Prove for n = 1, show for n = 2, assume true for n = k, show for n = k+1.

If you can do that, then you can prove the statement true for all finite k.

 

[MadRat]

Originally posted by: RossGr

What? I am not using infinity as a base number!

RossGr,
You are the king of misquotes. Clip that in a few other places and I'm sure you can make it look like I said the moon is made of cheese...

You also have this incessant need to use general rules that apply to Real numbers for infinity, in the case of finding the difference from 1 and .999... numbers. This task eludes you because of your inability to understand argument. The task is easily possible by defining both numbers (1 and .999...) in a base numbering system using infinity as the base. Using infinity as the base its simple to deduce the difference is simply "1". Since you cannot fathom the argument I'll just save my breath repeating the equation again.

Originally posted by: RossGr
Are you trying to say that .999.... = infinity? I thought you were claiming that it is less then 1, how did it get infinitly large?
I clearly show that .999.... + (any real number) = 1 + (a infinite long tail of 9s).
Are you trying to say that this number is less then zero? Or what are you trying to say?
I say that .999.... is a real number because I can tell you what digit is at EVERY position in the sum. Every 9 is associated with some integer, therefore it is a Real number.

What is the positional value of the lowest position in your .999...? If you dare say its "-infinity" then you just confirmed the existense of an infinite position. C'mon, you can do it. Its pathetic that you cannot simply admit that you are using infinity as the positional limit of your 9's. To say that there is no end to the nines only confirms that you are using infinity.

Originally posted by: RossGr

What integer is assigned to your 1?

The integer is a "1", as I've repeatedly said. Lets see, your question asked something it answered...

Originally posted by: RossGr
If you say none, then you have proven that you are not talking about a Real number. If you specifiy an N, then it is not at infinity, again you have contradicted yourself.
.00...001 can only be a Real number if the ellipsis represents a finite number of zeros.
I am being very consistent with this requirement, you continue to ignore the basic structure of a Real Number. Are we or are we not talking about the Real Number system?

I do say it is at N with N being equal to -infinity, just as you'd need to define the infinitely placed last "9" in the .999...! LOL. You just don't get it that you're either going to prove we're both right or neither theory is right. You cannot have a truth that fails this simple litmus test of validity.

 

[bleeb]

Originally posted by: Kyteland

Originally posted by: bleeb

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 didn?t follow is own beat?

Actually if you look back in history the people who thought that the moon was made of cheese were the ignorant masses. Granted there were some intelligent people who thought that it was true, but for the most part it was the educated elite who finally came around to the fact that there was no cheese on the moon. In spite of that, it took them a long time to convince the aforementioned masses about the moon's lack of cheese. I mean, come on, look at it. Common sense would tell you that it is obviously made of cheese, right?

The same holds true for the world being flat. When you look at it, it looks flat. Therefore it is flat. It took the educated quite some time to convince the masses otherwise.

As for life on Mars, well there's life on Earth so there must be life on Mars, right? Common sense. We are now certain that there is no life on Mars (intelligent anyways).

Which brings us back to the 0.99...=1 debate. If you take a look at the poll it quite clearly reflects the history presented above. 278 people (50.82%) claim that they are not equal. I will label these people "Exhibit A" aka "The Ignorant Masses". 224 people (40.95%) claim that they are equal. I will label these people "Exhibit B" aka "The Educated Elite". I think we all know which category you fall in to bleeb.

Therefore, 0.99...=1 QED

By stating the fact that the people who say that 0.9999 != 1 are "ignorant masses" no doubt labels you as a "close-minded bastard". Please try to keep this reasonable please. What the "masses" believed at the time was dependent on the available technology and relative state of knowledge. The world was flat to people because they were immersed in it. They weren't able to determine it otherwise...

And the statement regarding the cheese. I'm not sure if people actually believed the moon was made of cheese, but there was a greater point that I was trying to make. And that point was to show that the people believed in one thing because they were limited by the current state of knowledge. This might be the case now with you.

0.9999.... != 1

Q.E.D.

 

[Evadman]

.9999999... = 1. Just because I said it. I now command this thread to die.

 

[bleeb]

I'd like to see this thread NEFFED to at least a 1000 posts. Then I'll give in.

 

[spidey07]

neffity-bumpity-bump-bump.