Simple mathematical problem (?)

[LW]

If someone doesn't get it by now, I don't hold much expectation they ever will.

Somehow this thread reminds me of a quote from The Godfather (the book, perhaps also in movie). Unfortunately, I couldn't find the original one so I'll have to retranslate it from Finnish:

"It is impossible to reason with this man".

 

[LuxFerum]

Your observation is correct, but this is not how the number 0.999.... is defined. So your objection is moot.

Good point, what is the definition of 0,999....?
It is not the limit of the serie (9/(10^k)),k=1...infinity, because that is 1.
That notation does not exist.1 is one and 0,99.. is just a bunch of nines.
I never see that notation in any math books.

Now we are getting somewhere. You recognize the flaw in Zeno's Paradox. In the same way that the runner does in fact catch the turtle, 0.999.... is exactly equal to 1.

Perhaps the reason you do not see this is because you are trying to use a flawed method of analysis (using Zeno's process).

Yes, there is a flaw, and that flaw is the repetition of nines, that goes nowhere like the zenos paradox.



Here is a definition of limit by the merriam-webster, bold mine.
source

6 a : a number whose numerical difference from a mathematical function is arbitrarily small for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers or that are sufficiently large positively or negatively

infinity is not a number, he is nowhere in the euclidean space, he does not make parallel lines meet, he does not make and endless line of 9's became 0 and 0 became 1.


And this is my last post in this thread.
This is a waste of mensurable time in unmensurable things.

 

[69dodge]

I have the third edition (third printing, June 1962) of the book by Thomas that Zep mentioned (excellent book, by the way). Perhaps this quote will be helpful. It's from section 16--1, p. 769, which in my edition is two pages before the theorem that Zep quoted.

Remark. It is worth noting that the sum<blockquote>u₁ + u₂ + . . . + u_n</blockquote>is always defined (if n is a positive integer and u₁, u₂, . . . , u_n are finite numbers). When the series <blockquote>u₁ + u₂ + . . . + u_n + . . .</blockquote>has a "sum," in the sense that its sequence of partial sums has a limit, the symbol "+" has acquired new meaning, as has the word "sum." We can now "add" infinitely many numbers, in certain cases, not by actual addition, but rather by the process of finding a limit. This is a typical example of the way in which mathematics generalizes familiar concepts to increase their usefulness.

 

[DrMatt]

Let me try another angle.

I know of no mathematician who thinks 0.999... is not exactly equal to one. Why do you suppose that is? Have they all lost their marbles?

I know it is not valid to use an argumentum ad populum, but I offer it only as a hint that just maybe those of you who think 0.999.... is not equal to one, might wanna take a look at where you may be going wrong.

When most everyone else says the world is round and offers sound evidence of it, and you are still saying it's flat, perhaps it's time to rethink your position.

It's much easier to go with and grind through the definition.

Definition: 0.999... is SUM[n=1, inf] (9 * 10^(-n))

The series of approximations of 0.999... is
0, 0.9, 0.99, 0.999, ...
The radius is 0.1 < 1, therefore the series converges to a limit,
therefore the sum is equal to the limit of the series.



1) Suppose 0.999... < 1.
2) From line (1), it follows that 1 - 0.999... > 0
3) From line (2), there exists a positive number E such
that 1 - 0.999... = E
4) Since in line (3) E is positive, there exists a natural
number N such that E >= 10^(-N)
5) By line (4) , 1 - E < 0.999...99900000000000... where
the first (N-1) digits to the right of the decimal place are 9
and the rest are all 0.
6) Combining lines (5) with line (3) we get that
0.999... < 0.999...99900000000000...
7) From line (6) we get that 9 < 0. This is a contradiction.
Therefore the supposition in line (1)is not logically
consistent with arithmetic.

 

[xouper]

LuxFerum: And this is my last post in this thread. This is a waste of mensurable time in unmensurable things.

Yes, please stop wasting people's time in this thread with your flawed mathematics. Since you aren't coming back, I won't bother answering your rhetorical question or refuting the mistaken notions in your latest post.

 

[LuxFerum]

It's much easier to go with and grind through the definition.

Definition: 0.999... is SUM[n=1, inf] (9 * 10^(-n))

SUM[n=1, inf] (9 * 10^(-n))
That is 1 and not 0,999...

1) Suppose 0.999... < 1.
2) From line (1), it follows that 1 - 0.999... > 0
3) From line (2), there exists a positive number E such
that 1 - 0.999... = E
4) Since in line (3) E is positive, there exists a natural
number N such that E >= 10^(-N)
5) By line (4) , 1 - E < 0.999...99900000000000... where
the first (N-1) digits to the right of the decimal place are 9
and the rest are all 0.
6) Combining lines (5) with line (3) we get that
0.999... < 0.999...99900000000000...
7) From line (6) we get that 9 < 0. This is a contradiction.
Therefore the supposition in line (1)is not logically
consistent with arithmetic.

I change 0,999.. for a to follow your analise#
1)a<1
2)1-a>0
3)1-a=e
4)e>=10^(-N)
5)1-e<0,999...9990000 I disagree, 1-e=a and 1-e>0,999....99900




Why do you guys think they call it Limit?
Why do you say find the limit?
Why they don't call it the value of a function in infinity?


as posted by 69dodge

We can now "add" infinitely many numbers, in certain cases, not by actual addition, but rather by the process of finding a limit. This is a typical example of the way in which mathematics generalizes familiar concepts to increase their usefulness.

yes I know that I said that I wont post anymore, but I just couldn't.

 

[xouper]

hal bidlack: This post has been reported by several people. I responded to Traveller at length elsewhere. Simply put, please reduce your sig block as a matter of courtesy.

This post, as posted, is not a violation of the formal rules. While a tad silly, it can remain.

One might wonder about how the jerk rule might apply here, but as you well know, I do not require that posts like that be censored. His comments cannot hurt me. I am not worried that other people will think less of me based on the blatherings of a drive-by nimnul.

I have traveller on ignore so I don't even see his nonsense unless someone else feels the need to comment on it, and then I am again reminded why I have him on ignore. For the record, I didn't report it. My recommendation for people who dislkie his posts is to to put him on ignore. Chances are very high with an attention whore like traveller, that once he realizes no one is paying any attention to him any more, he will lose interest.

 

[xouper]

LuxFerum: SUM[n=1, inf] (9 * 10^(-n))
That is 1 and not 0,999...

You are wrong. You are so fundamentally wrong, I don't even know how far back to the basics we need to go to find common ground of agreement.

I change 0,999.. for a to follow your analise#
1)a<1
2)1-a>0
3)1-a=e
4)e>=10^(-N)
5)1-e<0,999...9990000 I disagree, 1-e=a and 1-e>0,999....99900

Sheesh.

Why do you guys think they call it Limit?
Why do you say find the limit?
Why they don't call it the value of a function in infinity?

Please pay attention to what I have already posted.

The infinite sequence of partial sums converges on a limit. The infinite series, however, is equal to that limit.

yes I know that I said that I wont post anymore, but I just couldn't.

If you want to stay in the thread, then I challenge you to answer one of the two challenges previously issued by others in this thread:

• Find a real number between 1 and 0.999... [edited to add: and express your example number in decimal notation.]
  
  or
  
• Find two positive integers A and B such that A / B = 0.999.... and A is not equal to B.

If you cannot do either of those two things, then your position is simply untenable. You can disagree all you want, but without a proper mathematical proof to back up your disagrement, you are wasting your time (and ours).

 

[DrMatt]

SUM[n=1, inf] (9 * 10^(-n))
That is 1 and not 0,999...

By definition, SUM[n=1, inf](9 * 10^(-n)) = 0.9 + 0.09 + 0.009 +.... which, BY DEFINITION, = 0.999... so you have just
agreed that 0.999... = 1. Thank you.

I change 0,999.. for a to follow your analise#
1)a<1
2)1-a>0
3)1-a=e
4)e>=10^(-N)
5)1-e<0,999...9990000 I disagree, 1-e=a and 1-e>0,999....99900

Re-read my step 5 again. You are off by a power of 10 in your step 5.

Why do you guys think they call it Limit?
Why do you say find the limit?
Why they don't call it the value of a function in infinity?

Because we are dealing here with the distinction between numerals and numbers. 1.000... and 0.999... are different numerals for the same number. They represent the same number
in the same way as the phrase "apples" and the phrase
"the fruits of apple trees" represent the same set. Limits are
a way of evaluating syntax (numerals) to find the semantics
(numbers).

as posted by 69dodge



yes I know that I said that I wont post anymore, but I just couldn't.

Nobody ELSE controls you.

I have more for you to ponder.
What do we mean when we say that two sets have the same number of elements? The number 3 is an abstraction which means that if I have 3 apples and you have 3 oranges, then if you and I are clever, we can make bags each containing exactly one of my apples and one of your oranges and have no apples or oranges left over without a partner to join in a bag. We can pair each element of one set with an element of the other uniquely--make an invertable function between all the elements of the two sets.
Suppose I have as many apples as there are counting numbers 1, 2, 3, and so-on. And suppose you have only as many oranges as there are EVEN numbers 2, 4, 6, and so on. Well, guess what. If you and I are clever enough, we'll put my apple number 1 and your orange number 2 in a bag, my apple number 2 and your orange number 4 in a bag, and in general my apple number N and your orange number 2N in a bag. Lo, an invertable function between all the elements of the two sets. What does that mean? Even though you may think you're missing half the elements of my set, your set has exactly as many oranges as mine has apples! Perhaps this result is counterintuitive, but it's the natural consequence of the way we compare the sizes of sets with each other.
Homework: use this kind of reasoning to show us why, if I have as many apples as there are counting numbers, and you have as many oranges as there are REAL numbers between 0 and 1, then no matter how clever you and I are in bagging our fruit you'll always end up with at least one orange unpaired. You may search the web for help in constructing this proof. Some helpful search terms are "Cantor" and "Diagonalization".

Enjoy!

 

[DrMatt]

When I was about three years old, I enjoyed counting. I got up to a hundred, and I had the pattern--every nine numbers, the prefix changed.

Then my grandfather pointed out to me that the number after one hundred nine is one hundred ten, not two hundred. Why, I asked? Every adult I asked said the same thing: Because one hundred is such a large number. This didn't make any sense to me, so I remained skeptical that the adults knew what they were talking about. My grandfather, though, was more persistent and thoughtful than most folks. Next time I saw him, when I was about 4, he had a book to show me which explained about ones places, tens places, hundreds places, and so forth. Ding! Now HERE was an explanation I could follow--it showed me what grownups were trying to accomplish by counting one hundred nine, one hundred ten, and so forth. The book even had pictures of arrays of blocks, ten by ten, and ten by ten by ten.

LuxFerum, I'm not sure how to show you graphically what mathematicians are doing when we compute limits. All I can tell you is that 0.9 is an approximation of a number, 0.99 is a better approximation of the number, 0.999 is a better approximation of it, 0.9999 is a better approximation of it, 0.99999 is yet a better approximation of it... what is that number? That number is 1.

 

[Walter Wayne]

Here is a definition of limit by the merriam-webster, bold mine.
source

6 a : a number whose numerical difference from a mathematical function is arbitrarily small for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers or that are sufficiently large positively or negatively

Lux, I don't think you are reading the definition correcly. Notice it says "for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers." So if we use mathmatical notation we replace "mathmatical function" with f(x) and independant variable with x. The definition (with a single indepent) then becomes:

a number whose numerical difference from f(x) is arbitrarily small for all values of x that are sufficiently close to but not equal to given prescribed number or that are sufficiently large positively or negatively

For a simple function, say f(x)=x+1 we can say:

The limit of f(x) as x approaches 0 is 1, because we can make f(x) arbitrarily close to 1, by making x sufficiently close to but not equak to zero.

But look at the above f(0) = 1!!! Here is a case where a function at a given value is in fact equal to limit as that value is approached. Here the "limit of f(x) as x -> 0" = f(0).

The definition you quote does not support your case.

 

[xouper]

Walter Wayne: For a simple function, say f(x)=x+1 we can say:

The limit of f(x) as x approaches 0 is 1, because we can make f(x) arbitrarily close to 1, by making x sufficiently close to but not equak to zero.

But look at the above f(0) = 1!!! Here is a case where a function at a given value is in fact equal to limit as that value is approached. Here the "limit of f(x) as x -> 0" = f(0).

Excellent example, Walt.

I only have one question.

What is the value of 1!!!






(Yes, I know 1!!! = 1, but it looks comical to see that in a discussion that's heavy in math jargon. I always did like your sense of humor.)

 

[Nasarius]

You people are crazy.

Look, if you agree that 1/9 = .11111...
then it should be blatantly obvious that .999... = 1.

 

[Walter Wayne]

Excellent example, Walt.

I only have one question.

What is the value of 1!!!






(Yes, I know 1!!! = 1, but it looks comical to see that in a discussion that's heavy in math jargon. I always did like your sense of humor.)

((1 factorial) factorial) factorial

I'm lucky that didn't change the meaning.

 

[roger]

[BI'm lucky that didn't change the meaning. [/B]

Oh, I'm sure you'll get arguments to the contrary!

 

[T'ai Chi]

OK, good, you agree that 1 = 0.999...

Yeah, no kidding, I've been saying that all along.

We'll overlook that you tried to use 0.000...0001 which is not a real number.

It certianly is in the context of the discussion of 'does .999... = 1'. In fact, .000... = 0.

 

[DrMatt]

Oh, I'm sure you'll get arguments to the contrary!

Yeah, well, look at these two things:

1+1+1+1

4

Just by LOOKING at them you can see they're not the same.

Why, one of them is 1 byte long and the other is 7 bytes long!

So, by logic heard earlier on this thread:
"The computation of one comes out to the other, but they are not equal!"

 

[xouper]

T'ai Chi: Yeah, no kidding, I've been saying that all along.

OK, yes, you did say that in the beginning of this thread. Why then did you post all that extraneous nonsense about a tiny e and e>0? Do you now retract it?

xouper: We'll overlook that you tried to use 0.000...0001 which is not a real number.

T'ai Chi: It certianly is in the context of the discussion of 'does .999... = 1'.

No it's not.

In fact, .000... = 0.

Of course, but that's not the number you used. I was referring to your use of the notation 0.000...0001, which is not a real number in any context. If you think otherwise, then what is the exponent of the power of ten in the divisor of the last digit (the 1 digit)?

 

[patoco12]

I had a dream about Monty Hall last night! I thought HIS problem was the most controversial.

 

[T'ai Chi]

Why then did you post all that extraneous nonsense about a tiny e and e>0? Do you now retract it?

Be specific. What "nonsense"?

No it's not.

What number system do you think e>0 is referring to?

I was referring to your use of the notation 0.000...0001, which is not a real number in any context.

This is an informal discussion, not a formal mathematics paper.

If you think otherwise, then what is the exponent of the power of ten in the divisor of the last digit (the 1 digit)?

One can talk about MANY numbers in the real number system without needing to specifically specify all places of the digits in that number.