1=0.999... infinities and box of chocolates..Phliosophy of Math...

[Quantum Quack]

In another thread we have the ongoing discussion of a number system [Reals] considering :

0.999... = 1 [which is currently held as a valid equation - of which the Systemic Validity is NOT in dispute]

This appears to be primarily true for this system due to the fact that it considers the following concerning the limitations in the use of infinity.

"As the number of digits used increases without bound, the difference between 1 and the repeating decimal expansion 0.999... shrinks to be smaller in magnitude than any finite number, thus the magnitude of the difference in the limit of an infinite number of steps must be zero."

On the surface this appears to be fine.. In fact it makes a lot of sense that infinite numbers of decimal places expanding to smaller and smaller values would ultimately resolve to being smaller in magnitude to "ANY" finite number.

1] Is "being of less magnitude to any finite number" actually equal to zero?
2] Is it an "effective" or pseudo zero, borne of mathematical or systemic convenience?
3] At what point would "finite" become non-finite and how could this be determined?

Also possibly worth discussing is that:
0.999.... is actually referencing the whole number 1 as a primary source of value, which in turn is referencing zero as a way of gaining value.
after all is not 0.999.... stating a condition of 1?

How does the fact that 0.999... as being a condition of the number 1 effect the reality of 0.999... = 1?

Example: 1 - 0.999... = 1

Needless to say there are many ways of looking at this issue and I am sure you guys/gals can think of a few more...

I am of the opinion that this a more a discussion about the philosophy that underpins mathematical formulations and forms than mathematics directly.

Care to discuss?

 

[Pete]

"As the number of digits used increases without bound, the difference between 1 and the repeating decimal expansion 0.999... shrinks to be smaller in magnitude than any finite number, thus the magnitude of the difference in the limit of an infinite number of steps must be zero."

What is the source of this quote? It seems to be missing an "except zero" clause.

 

[Quantum Quack]

What is the source of this quote? It seems to be missing an "except zero" clause.

see rpenners post:
http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3127165&viewfull=1#post3127165
as to where he got it from I have no real idea.. but I assume that Rpenner is a person/poster of integrity and would only quote from a reliable source.

 

[Quantum Quack]

In another thread we have the ongoing discussion of a number system [Reals] considering :

0.999... = 1 [which is currently held as a valid equation - of which the Systemic Validity is NOT in dispute]

This appears to be primarily true for this system due to the fact that it considers the following concerning the limitations in the use of infinity.



On the surface this appears to be fine.. In fact it makes a lot of sense that infinite numbers of decimal places expanding to smaller and smaller values would ultimately resolve to being smaller in magnitude to "ANY" finite number.

1] Is "being of less magnitude to any finite number" actually equal to zero?
2] Is it an "effective" or pseudo zero, borne of mathematical or systemic convenience?
3] At what point would "finite" become non-finite and how could this be determined?

Also possibly worth discussing is that:
0.999.... is actually referencing the whole number 1 as a primary source of value, which in turn is referencing zero as a way of gaining value.
after all is not 0.999.... stating a condition of 1?

How does the fact that 0.999... as being a condition of the number 1 effect the reality of 0.999... = 1?

Example: 1 - 0.999... = 0 <<<:edited due to error in OP

Needless to say there are many ways of looking at this issue and I am sure you guys/gals can think of a few more...

I am of the opinion that this a more a discussion about the philosophy that underpins mathematical formulations and forms than mathematics directly.

Care to discuss?

Just found an error in the op please see correction above.

 

[Pete]

see rpenners post:
http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3127165&viewfull=1#post3127165
as to where he got it from I have no real idea.. but I assume that Rpenner is a person/poster of integrity and would only quote from a reliable source.

I've learned that 'finite' can mean "not infinite and not zero".

That's post by rpenner is very good. You should read it properly, start to finish.

"As the number of digits used increases without bound, the difference between 1 and the repeating decimal expansion 0.999... shrinks to be smaller in magnitude than any finite number, thus the magnitude of the difference in the limit of an infinite number of steps must be zero."

Note that this quote is a wordy translation of a much more precise mathematical expression:
$$\lim_{n\to\infty} 1 - (1 - 10^{-n}) = \lim_{n\to\infty} 10^{-n} = 0$$

Given the nature of English prose, it is unsurprising that there is some ambiguity in the translation that does not exist in the original. I'll try to clarify, and hope I don't make things worse.

1] Is "being of less magnitude to any finite number" actually equal to zero?

Not in this context. What it means is that for any finite non-zero number you choose, you can find a value of n such that $$10^{-n}$$ is smaller than your chosen number.

2] Is it an "effective" or pseudo zero, borne of mathematical or systemic convenience?

Maybe. I don't think it's a useful way to think of it... it's more useful to understand the rigorous meaning of a limit. In other words, it would be better for you to ignore the English translation and instead learn what the mathematical expression means.

3] At what point would "finite" become non-finite and how could this be determined?

If you're stepping through the process, never.
You can't determine the result of an infinitely long algorithm by stepping through the algorithm.
You can determine the result of an certain infinitely long algorithms by calculating their limit.

 

[Tach]

after all is not 0.999.... stating a condition of 1?

How does the fact that 0.999... as being a condition of the number 1 effect the reality of 0.999... = 1?

Example: 1 - 0.999... = 1

Basic arithmetic says that you laid an egg. While pretending to be discussing high level mathematics.

 

[Quantum Quack]

Basic arithmetic says that you laid an egg. While pretending to be discussing high level mathematics.

see post#4

or are you just here to do what?

What do you hope to achieve Tach, as a member of sciforums?

 

[someguy1]

0.999.... is actually referencing the whole number 1 as a primary source of value, which in turn is referencing zero as a way of gaining value.
after all is not 0.999.... stating a condition of 1?

I'll answer that if you'll answer me this one first:

Does 2 + 2 state a condition of 4?

If by "state a condition of ..." you really mean "is another way of representing ..." then I agree. Because .999... is just another way of representing the same number that's represented by 1.

But if you mean some kind of mystical "becoming" or something, then you are misunderstanding the nature of mathematical notation. Notations are just names for numbers. A given number can have lots of names. .999... and 1 are just two different names for the same number. The only reason you think 1 is special is because you learned it first.

I will grant you a subtle mathematical point, though. The expressions 1 and .999... each represent the same real number. But when you first learned about numbers, they taught you about the whole number 1. And when we say .999... = 1 we are asking you to think about 1 in its role as a member of the real numbers. Infinitary processes are a crucial aspect of the real numbers. By infinitary processes I mean taking limits of infinite sequences, sums of infinite series, etc. The very notion of real number implies infinitary reasoning.

In other words when we say that .999... = 1 we are asking you to forget about your intuitions about whole numbers; and implicitly grant all the rules about real numbers ... including infinitary operations.

Now in terms of how whole numbers and real numbers are defined in terms of set theory, the whole number 1 is not actually equal to the real number 1. They are distinct sets. But the whole numbers are "identified," as we say, with their counterparts in the real numbers; since the real numbers contain a perfect copy of the whole numbers.

So in that sense not only is the real number .999... not equal to the whole number 1; but the real number 1 is not even equal to the whole number 1.

 

[Tach]

see post#4

or are you just here to do what?

What do you hope to achieve Tach, as a member of sciforums?

Point out the fact that you are a pretender who trolls the threads in the hope of running your post count.

 

[Quantum Quack]

I've learned that 'finite' can mean "not infinite and not zero".

That's post by rpenner is very good. You should read it properly, start to finish.

Note that this quote is a wordy translation of a much more precise mathematical expression:
$$\lim_{n\to\infty} 1 - (1 - 10^{-n}) = \lim_{n\to\infty} 10^{-n} = 0$$

Given the nature of English prose, it is unsurprising that there is some ambiguity in the translation that does not exist in the original. I'll try to clarify, and hope I don't make things worse.


Not in this context. What it means is that for any finite non-zero number you choose, you can find a value of n such that $$10^{-n}$$ is smaller than your chosen number.

Maybe. I don't think it's a useful way to think of it... it's more useful to understand the rigorous meaning of a limit. In other words, it would be better for you to ignore the English translation and instead learn what the mathematical expression means.

If you're stepping through the process, never.
You can't determine the result of an infinitely long algorithm by stepping through the algorithm.
You can determine the result of an certain infinitely long algorithms by calculating their limit.

Firstly thanks, Pete for your post. Much appreciated.
yes Rpenner's post was rather terrific I must agree, even If I can only surmise it's purport.

I would be very interested to know of the original authors name for the translation.
As this I feel it is critical, in not so much in understanding the accepted validity of 0.999...=1 but more about the nature of the zero in
1- 0999... = 0

It is the nature of this resultant zero that is my personal focus.

Trying to get into the original concept, obviously, is not that easy.

However Billy T, I believe [and Billy T can correct me If I am wrong], has mentioned that infinite limits cannot truly be calculated and that evaluation may be required.
I do hope that he will post a clarification on the distinction so that I and others can benefit from such a seemingly "on the surface trivial" yet fundamentally important detail. ["lazy" reference to Godel]

Post#401 http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3128192&viewfull=1#post3128192
Can you give an outline at least as to how 0.5^n is ACTUALLY CALCULATED as n approaches infinity? I think (but may be wrong) that all you can do is say its value is less than any value one wants to chose, but must admit it is still non- zero for any CALCULATION you can actually make.

Until you do, I stick with the POV that the limit is defined to be zero, not actually calculated. Certainly those with some math experience have come to belief the limit can be CALCULATED, but in truth, I think it is not, so it is DEFINED.

There are other examples of definitions being adopted. For example 0 / 0 is often defined as 1 by extension of the rule if numerator and denominator are identical, then their ratio is unity. I.e. it is convenient to define it this way rather than add a "footnote" to that rule. Thus A / A = 1 without any need of footnote stating: "Except when A = 0."

and

post#406 http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3128212&viewfull=1#post3128212
Do you recall from your first calculus course, the class when the concept of a limit was being introduced / defined? Well ALL limiting processes are fundamentally like than and what I said (In quote below.) Why I said the limits are DEFINED by showing something can be made arbitrarily small, NOT BY CALCULATION.

and most importantly

post#408 http://www.sciforums.com/showthread.php?136842-1-is-0-9999999999999&p=3128227&viewfull=1#post3128227

Tach I think we use the term "calculation" differently. I agree one can "evaluate" limits based on the procedure we agree is needed (or have been used so often that we know the results without doing those limiting procedures in detail again).- I.e. by showing something finite can be calculated to be as close to the limit as arbitrarily required, but I reserve "calculation" for evaluations of finite expressions, not their infinite sequence limits. That I call an evaluation. Perhaps you don't distinguish between these two concepts?

For example every thing done by a digital computer is finite evaluation so is a calculation. I.e. for me, calculations are a sub-set of evaluations. All limits are evaluations, not calculations.

It may be a little pedantic to make this distinction. For example the evaluation of the geometric series can be, with a little algebra, reduced to a quick calculation.

By edit after reading Tach's post 409:
Perhaps others well versed in mathematic will comment on weather or not the distinction I make between "calculation" and "evaluation" is desirable. It does permit more precise speech.

Unfortunately Tachs posting behavior makes sharing such details very difficult...

 

[Quantum Quack]

Point out the fact that you are a pretender who trolls the threads in the hope of running your post count.

So your sole purpose here at sciforums is to stalk and harass other members? by your own admission...

What do you think is my purpose in engaging you?

 

[Quantum Quack]

I'll answer that if you'll answer me this one first:

Does 2 + 2 state a condition of 4?

If by "state a condition of ..." you really mean "is another way of representing ..." then I agree. Because .999... is just another way of representing the same number that's represented by 1.

But if you mean some kind of mystical "becoming" or something, then you are misunderstanding the nature of mathematical notation. Notations are just names for numbers. A given number can have lots of names. .999... and 1 are just two different names for the same number. The only reason you think 1 is special is because you learned it first.

I will grant you a subtle mathematical point, though. The expressions 1 and .999... each represent the same real number. But when you first learned about numbers, they taught you about the whole number 1. And when we say .999... = 1 we are asking you to think about 1 in its role as a member of the real numbers. Infinitary processes are a crucial aspect of the real numbers. By infinitary processes I mean taking limits of infinite sequences, sums of infinite series, etc. The very notion of real number implies infinitary reasoning.

In other words when we say that .999... = 1 we are asking you to forget about your intuitions about whole numbers; and implicitly grant all the rules about real numbers ... including infinitary operations.

Now in terms of how whole numbers and real numbers are defined in terms of set theory, the whole number 1 is not actually equal to the real number 1. They are distinct sets. But the whole numbers are "identified," as we say, with their counterparts in the real numbers; since the real numbers contain a perfect copy of the whole numbers.

So in that sense not only is the real number .999... not equal to the whole number 1; but the real number 1 is not even equal to the whole number 1.

and no doubt all the above is for very good reasoning.
Thank you for taking the time to post... much appreciated.
regards 2+2 = 4
not exactly...

4 = 4/2 + 4/2
or
4= (4-2)+(4-2)
then yes

however see below:

In regards to my comment about 0.999... being a condition of 1, I was using a mathematically vague philosophical definition of the word condition.
What I meant by it's use was that
the mere fact that for example, 0.08 exists, is because it is in fact 8*100(ths) of 1 so the starting condition to me is 1 of which we have 8*100(th) of.
How that "gells" with your post I am unsure...which is I guess why I mentioned it as possible point for discussion regarding 0.999.. = 1 in the OP.

that 0.999... is .(9) of 1 which is in an intuitive discord with the validity of 0.999... equaling one.
so
0.999... x 1 = (1 of 1) ~quantity
0.999... x 0.999... = (1 = 1) ~quality
and so on
the distinction is subtle and probably trivial.

 

[Pete]

I would be very interested to know of the original authors name for the translation.
As this I feel it is critical, in not so much in understanding the accepted validity of 0.999...=1 but more about the nature of the zero in
1- 0999... = 0

It is the nature of this resultant zero that is my personal focus.

It's zero. It's just a number. There's no hidden meaning.

However Billy T, I believe [and Billy T can correct me If I am wrong], has mentioned that infinite limits cannot truly be calculated and that evaluation may be required.

Billy is using his own personal intuitive notion of what 'calculation' means. There's no meaningful distinction between 'calculating' a limit and 'evaluating' a limit.

To determine the value of a limit, we use a well defined algorithm, just as a calculator uses a well defined algorithm to determine the value of 2x3.

 

[Quantum Quack]

It's zero. It's just a number. There's no hidden meaning.


Billy is using his own personal intuitive notion of what 'calculation' means. There's no meaningful distinction between 'calculating' a limit and 'evaluating' a limit.

To determine the value of a limit, we used a well defined algorithm, just as a calculator uses a well defined algorithm to determine the value of 2x3.

Perhaps Billy T will clarify? I will PM him and see if he wishes to or not...

 

[Billy T]

... Billy is using his own personal intuitive notion of what 'calculation' means. There's no meaningful distinction between 'calculating' a limit and 'evaluating' a limit. ...

Yes, it may not be a widely used, but I think it is useful for greater precision of speech (less redundancy of words) to make a distinction between "calculate" and "evaluate" as I do.

For me calculate is a mathematical sub-set of evaluations, which often have no math involved, like evaluation of an actor's performance. For me, calculations ONLY use the four basic operations and their direct extensions,* and are always done in a finite number of steps, not by a limiting processes, which always boils down to showing that something which can be calculated in a finite set of operations will approach some value as closely as you like, if the number terms calculated is made large enough.

There are many functions common in math that cannot be calculated (in a finite set of steps using the basic four operations). For example all the trig functions, logs of various bases, powers of irrational numbers, limits of most infinite series (but some, like the convergent geometric series, can be operated on by the basic four functions to transform them into calculable values), etc. Often these "not calculable" function are published to some specified accuracy (long ago in books or in modern times their APPROXIMATIONS are calculated by algorithms in computers).

I claim that the limit of a convergent infinite series that cannot be evaluated by finite number of the basic four operations of calculations, is not calculated, but defined to have the value of its least upper bound. The limit of the convergent infinite geometric series, is however, calculable:
I.e. S = 1+a+a^2 +a^3 ... then by multiplication: aS = a +a^2 + a^3 ... Then by subtraction: S - aS = S(1-a) = 1 then by division: S = 1 / (1-a) but for the convergent series, 1 > a is required.

Others may not like to make the distinction in mathematics that I do between "calculate" and "evaluate" and in truth, it serves little purpose but when distinctions can be defined, IMHO, they should be. (If two words have identical meanings, then one should cease to be used, except perhaps by poets).

* For example one can ask what number multiplied by itself is equal to some other number and then we say the first smaller number is the "square root" of the second number. In some cases, the square root can be calculated and in others it can only be approximated. For example 4^(0.5) = 2 but 3^(0.5) and only be approximated.

 

[Tach]

1. I claim that the limit of a convergent infinite series than cannot be evaluated by finite number of the basic four operations of calculations is not calculated, but defined to have the value of its least upper bound.
2. The limit of the convergent infinite geometric series, is however, calculable:

You realize that you are:

1. Contradicting yourself (in the space of only two successive sentences )
2. Using non-mainstream terminology, invented by you
3. Creating a false dichotomy (non present in any mainstream literature) between "calculated" and "evaluated" (or "defined").

I trace your errors to a misconception that you espoused earlier, you seem to think that there is a difference between $$\lim_{x \to a}f(x)$$ and $$\lim_{x \to \infty}f(x)$$. You seem to think that the former is "calculated" while the latter is "defined". There isn't such dichotomy in standard calculus, both are calculated.

The limit of the convergent infinite geometric series, is however, calculable:
I.e. S = 1+a+a^2 +a^3 ... then by multiplication: aS = a +a^2 + a^3 ... Then by subtraction: S - aS = S(1-a) = 1 then by division: S = 1 / (1-a) but for the convergent series, 1 > a is required.

You can do it the way you did it or you can do it by :

$$S_n=1+a+...+a^n$$
$$S=\lim_{n \to \infty}S_n=\lim_{n \to \infty} \frac{1-a^{n+1}}{1-a}=\frac{1}{1-a}$$ because
$$\lim_{n \to \infty}a^{n+1}=0$$ if $$a<1$$.

 

[Billy T]

You realize that you are:

1. Contradicting yourself
2. Using non-mainstream terminology, invented by you
3. Creating a false dichotomy (non present in any mainstream literature) between "calculated" and "evaluated" (or "defined").

I trace your errors to a misconception that you espoused earlier, you seem to think that there is a difference between $$\lim_{x \to a}f(x)$$ and $$\lim_{x \to \infty}f(x)$$. You seem to think that the former is "calculated" while the latter is "defined". There isn't such dichotomy in standard calculus, both are calculated.

I'm a little dyslexic. In your 1. the word after the red part is that not than, but probably you read "that" from context when looking at "than" so this is just for the record.

I see a clear difference between your $$\lim_{x \to a}f(x)$$ and $$\lim_{x \to \infty}f(x)$$. For example (not special) if a =10 & f(x) = 1/x then then the value of the first at x = 10 is calculated to be 1/10 = 0.1 but the value of the second at x = $$infiy$$ can only be defined, (based on a limiting process as x becomes very large.) as the least upper bound, which is zero.

If you don't agree, Please show me how you calculate (instead of define) that zero (without some resort to a limiting process as x becomes large).

I'm not "evangelic" - trying to get your to adopt my POV. If you wish to have two redundant (identical) terms in mathematics, -fine.

Sorry my guess at tex version of infinity was not quite right. Perhaps this will be: x = $$\infiy$$ or $$\infiy\$$ Oh well you know what I mean to write.

 

[Tach]

I'm a little dyslexic. In your 1. the word after the red part is that not than, but probably you read "that" from context when looking at "than" so this is just for the record.

Even after replacement, your sentences still contradict each other.

I seem clear difference in your $$\lim_{x \to a}f(x)$$ and $$\lim_{x \to \infty}f(x)$$. For example (not special) if a =10 & f(x) = 1/x then then the value of the first at = 10 is calculated to be 1/10 = 0.1 but the value of the second at x = $$infiy$$ can only be defined, (based on a limiting process as x becomes very large.)

Yes, therein (in redline) lies your error, you made it earlier and you are repeating it above. The limits to $$\infty$$ are calculated, not defined. I gave earlier an example to the "Quack" how it is done.

Sorry my guess at tex version of infinity was not quite right. Perhaps this will be: x = $$\infiy$$

$$\infty$$

 

[Quantum Quack]

Yes, it may not be a widely used, but I think it is useful for greater precision of speech (less redundancy of words) to make a distinction between "calculate" and "evaluate" as I do.

For me calculate is a mathematical sub-set of evaluations, which often have no math involved, like evaluation of an actor's performance. For me, calculations ONLY use the four basic operations and their direct extensions,* and are always done in a finite number of steps, not by a limiting processes, which always boils down to showing that something which can be calculated in a finite set of operations will approach some value as closely as you like, if the number terms calculated is made large enough.

There are many functions common in math that cannot be calculated (in a finite set of steps using the basic four operations). For example all the trig functions, logs of various bases, powers of irrational numbers, limits of most infinite series (but some, like the convergent geometric series, can be operated on by the basic four functions to transform them into calculable values), etc. Often these "not calculable" function are published to some specified accuracy (long ago in books or in modern times their APPROXIMATIONS are calculated by algorithms in computers).

I claim that the limit of a convergent infinite series that cannot be evaluated by finite number of the basic four operations of calculations, is not calculated, but defined to have the value of its least upper bound. The limit of the convergent infinite geometric series, is however, calculable:
I.e. S = 1+a+a^2 +a^3 ... then by multiplication: aS = a +a^2 + a^3 ... Then by subtraction: S - aS = S(1-a) = 1 then by division: S = 1 / (1-a) but for the convergent series, 1 > a is required.

Others may not like to make the distinction in mathematics that I do between "calculate" and "evaluate" and in truth, it serves little purpose but when distinctions can be defined, IMHO, they should be. (If two words have identical meanings, then one should cease to be used, except perhaps by poets).

* For example one can ask what number multiplied by itself is equal to some other number and then we say the first smaller number is the "square root" of the second number. In some cases, the square root can be calculated and in others it can only be approximated. For example 4^(0.5) = 2 but 3^(0.5) and only be approximated.

Thanks BillyT for your post. Most enlightening.
Just for the record: As translation maybe an issue here and just to clarify:
you posted:

I claim that the limit of a convergent infinite series than cannot be evaluated by finite number of the basic four operations of calculations is not calculated, but defined to have the value of its least upper bound.

You use the word "than". Should that word read "then" or "therefore" instead?

I claim that the limit of a convergent infinite series, THEREFORE cannot be evaluated by a finite number of the basic four operations of calculations, is not calculated, but defined to have the value of its least upper bound.

 

[Billy T]

... The limits to $$\infty$$ are calculated, not defined. I gave earlier an example to the "Quack" how it is done.

what thread and post number. I'll read it. Thanks for tex way to make:
$$\infty$$