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January 12th, 2016, 05:21 PM   #11
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 Originally Posted by Karma Peny You are simply defining the summation to be equal to the limit. You claim there is no mention of infinity but you appear to stand by the definition which is a summation of infinitely many terms.
Absolutely not. As Azzajazz says, you should read up on this stuff before you claim to know what is going on. I am defining nothing. I am telling you what is the formal definition of the notation that you are abusing with your talk about infinities that you neither believe in nor understand.

Again, the definition of the notation $\sum \limits_{n=0}^\infty a_n$ is that it means $\lim \limits_{m \to \infty} \sum \limits_{n=0}^m a_n$. This limit in turn is defined to be the number $L$, if such a number exists, such that for every $\epsilon \gt 0$ there exists an integer $M$ such that $\left|L - \sum \limits_{n=0}^m a_n\right| < \epsilon$ for all $m \gt M$.

There. Not an "infinity" in sight. No "infinite" sums either. All perfectly sweet for an "extreme finitist" to agree with.

There is certainly an argument to be had over whether an "infinite sum" actually exists and, if it does, whether the "answer" to that "infinite sum" is the same number as represented by the definitions above. But that is not what this thread is about. This thread starts from the false premise that, given that a real number has an infinite representation, that we do not know what type of infinity we are talking about for the length of that representation.

I'd be grateful if you took down your blog until its content is checked by somebody who actually understands the subjects you cover to avoid confusing people who are looking for mathematical theory as opposed to uninformed guesswork.

Last edited by v8archie; January 12th, 2016 at 05:58 PM.

January 13th, 2016, 03:42 AM   #12
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Quote:
 Originally Posted by v8archie I'd be grateful if you took down your blog until its content is checked by somebody who actually understands the subjects you cover to avoid confusing people who are looking for mathematical theory as opposed to uninformed guesswork.
My blog makes no claim to represent the prevailing mathematical theory and it makes it perfectly clear that the views expressed are extreme.

I take it you object to anyone expressing any views that do not conform to prevailing philosophies. You are objecting to free speech. Your position is indefensible.

January 13th, 2016, 04:23 AM   #13
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Quote:
 Originally Posted by v8archie This thread starts from the false premise that, given that a real number has an infinite representation, that we do not know what type of infinity we are talking about for the length of that representation.
The OP did not explicitly refer to 'real numberâ€™; it talks about the decimal representation of irrational numbers.

The OP issue is with the troublesome concept of infinitely many digits.

If we follow an algorithm that tries to convert the square root of 2 into a decimal, then the fact that the algorithm cannot complete is problematic.

I do not need to see all the digits, but I do want a clear explanation of how the algorithm can complete and become equal to the square root of 2. I have exactly the same issue with 1/3 and 0.333...

The series is supposedly endless, meaning you cannot reach a point after which there are no more terms, but there are supposedly no more terms after 'infinitely many'. How can it be endless and yet have no more terms after 'infinitely many'? You will deny it, but to me this is a clear contradiction.

Either a decimal with infinitely many digits is a valid concept or it is not. Just defining the symbolic representation to be equal to the limit of the series is to completely avoid the issue.

January 13th, 2016, 04:42 AM   #14
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 Originally Posted by Karma Peny I take it you object to anyone expressing any views that do not conform to prevailing philosophies. You are objecting to free speech. Your position is indefensible.
No, I'm suggesting that content you post in an educational forum should be verified as correct before it is presented in a manner intended to convince students of its veracity.

If you wish to claim knowledge of some truth, you have a responsibility to ensure that it is accurate.

January 13th, 2016, 04:58 AM   #15
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Quote:
 Originally Posted by Karma Peny The OP did not explicitly refer to 'real numberâ€™; it talks about the decimal representation of irrational numbers. The OP issue is with the troublesome concept of infinitely many digits. If we follow an algorithm that tries to convert the square root of 2 into a decimal, then the fact that the algorithm cannot complete is problematic. I do not need to see all the digits, but I do want a clear explanation of how the algorithm can complete and become equal to the square root of 2. I have exactly the same issue with 1/3 and 0.333...
First of all, the irrationals are reals, so there is no problem there.
Secondly, the OP's issue is with the size of the infinite decimal representation, not the fact that it is infinite.

The decimal representation of a number is just that: a representation. You can use $\sqrt2$ if you prefer, but most people (in the real world, not mathematicians) find the decimal representation more useful, or rather an approximation to that decimal representation.

This is not a computation. The value $\sqrt2$ exists, and the decimal expansion exists to whatever degree of accuracy you require, in that it is always the same no matter how many times you determine it. It's accuracy is unlimited.

To insist that you must have proof that "the algorithm can complete" is to insist on some level of approximation because by definition no algorithm can produce an infinite output and then terminate. How can something that never ends terminate?

Last edited by v8archie; January 13th, 2016 at 05:04 AM.

January 13th, 2016, 05:11 AM   #16
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Quote:
 Originally Posted by v8archie The value $\sqrt2$ exists, and the decimal expansion exists to whatever degree of accuracy you require, in that it is always the same no matter how many times you determine it. It's accuracy is unlimited.
Yes, but there is a difference in saying the accuracy is unlimited and that a decimal expansion can exist with infinitely many digits.

I completely agree that a decimal expansion can only be an approximation. This is my whole argument. It sounds like we might be in agreement on this point.

Can you give me a simple yes or no on if a decimal expansion can exist with infinitely many digits, after which there are no more terms? Many mathematicians claim it can exist.

Quote:
 Originally Posted by v8archie To insist that you must have proof that "the algorithm can complete" is to insist on some level of approximation because by definition no algorithm can produce an infinite output and then terminate. How can something that never ends terminate?
Again, it sounds like you are agreeing with me... there is no such thing as a decimal representation with infinitely many digits.

So the answer to the OP is that there are no infinities involved at all.

Last edited by Karma Peny; January 13th, 2016 at 06:01 AM.

January 13th, 2016, 06:24 AM   #17
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Quote:
 Originally Posted by v8archie No, I'm suggesting that content you post in an educational forum should be verified as correct before it is presented in a manner intended to convince students of its veracity. If you wish to claim knowledge of some truth, you have a responsibility to ensure that it is accurate.
Earlier in this thread I published a link to my article in response to one of your posts to inform you of my views (rather than copying the whole text of my article into a post in this thread).

I have posted a link to my article as an OP in this maths forum to get critical reviews from mathematicians. Both in the OP and on my website it is very clear that these are my personal views and that these views are 'extreme'. But you claim I can't publish a link to my article in a maths forum until it has been reviewed by mathematicians.

Last edited by Karma Peny; January 13th, 2016 at 06:44 AM.

 January 13th, 2016, 06:42 AM #18 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra Of course the expansion exists. I said so in my post. You seem to think that a decimal expansion is a process, it's not. It's a representation of a number.
January 13th, 2016, 06:58 AM   #19
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 Originally Posted by v8archie Of course the expansion exists. I said so in my post. You seem to think that a decimal expansion is a process, it's not. It's a representation of a number.
You cannot have a definition that includes the summation from 0 to infinity and then claim it is not a process. The notion of going from 0 to infinity describes a process.

It is then claimed that the decimal representation is merely the representation of a number. This suggests it is merely an alternative symbolic representation for the number. You can't have it both ways.

The fraction 1/3 can be expressed in entirety as a fraction, or in some bases, like base 12, but not in base 10. It cannot exist in base 10 because it cannot be actually constructed or even shown how it could possibly be constructed in base 10.

Last edited by Karma Peny; January 13th, 2016 at 07:01 AM.

January 13th, 2016, 07:02 AM   #20
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Quote:
 Originally Posted by Karma Peny Earlier in this thread I published a link to my article in response to one of your posts to inform you of my views (rather than copying the whole text of my article into a post in this thread). I have posted a link to my article as an OP in this maths forum to get critical reviews from mathematicians. Both in the OP and on my website it is very clear that these are my personal views and that these views are 'extreme'. But you claim I can't publish a link to my article in a maths forum until it has been reviewed by mathematicians. Have you read Catch 22?
I've never obliged you to do anything. All this pathetic talk of about my wish to curtail your right to free speech is some persecution complex of yours.

Catch 22 is pretty boring for me.

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