Obscure & Useless

[Resha Caner]

The problem with the above is that you cannot start at the right of an infinite number.

Exactly. So how does one multiply a number with an infinite number of digits as Blayz did? My point was that one can't. You must make an approximation. It's the approximation coupled with proper choice of the multiplier that makes the trick work. So, 1 is a good approximation of 0.9..., but it is not an exact representation.

 

[pgp_protector]

Because with 0.99999 recurring, the 9s never end.

x = 0.99999999 recurring
10x = 9.99999999 recurring

In 10x, the number of 9s after the decimal place is the same as in x, i.e. infinity. So when you subtracting x from 10x, everything after the decimal place cancels out, leaving 9.

Um wouldn't 10x-x = 9.00000..... with a 9 at the end* given you've shifted x by a decimal point?

*even though you can't reach the "end" of an infinite regression.

 

[GrowingSmaller]

1) The "hard problem of consciousness"....

David Chalmers on the "hard problem" of consciousness - The Consciousness Chronicles Vol. 1 - YouTube


Also 2) the issue of mental causation and the evolution of qualia/phenomenology.

These are probably not useless if the science actually develops, but I think they are probably quite obscure in terms of mainstream psychology and evolutionary psychology.

 

[Tinker Grey]

Exactly. So how does one multiply a number with an infinite number of digits as Blayz did? My point was that one can't. You must make an approximation. It's the approximation coupled with proper choice of the multiplier that makes the trick work. So, 1 is a good approximation of 0.9..., but it is not an exact representation.

How would you respond to the comment that repeating decimals are an artifact resulting from trying to represent a rational number ill-suited to the base.

E.g., 1/9 = .11111.... in base 10. However, in base 9 it is simply 0.1

.9999.... can result from multiplying .111111.... by 9. That is, .9999... ==1 because 9 x (1/9) = 1.

 

[pgp_protector]

How would you respond to the comment that repeating decimals are an artifact resulting from trying to represent a rational number ill-suited to the base.

E.g., 1/9 = .11111.... in base 10. However, in base 9 it is simply 0.1

.9999.... can result from multiplying .111111.... by 9. That is, .9999... ==1 because 9 x (1/9) = 1.

Same problem though, when multiplying, you start on the Right side of the number. so even in your example, you approximate.

their is no end to point to start the multiplication on with an repeating number

 

[GrowingSmaller]

What about this? 9.999999.... is either a number or not. If it is a number then it is not representative is an infinite series, because an infinite series is not a point on a number line. Or, if it is not a number then it does not belong in an algebraic equation, where variables like "x" etc represent numbers. In both cases the proof being discussed is invalid. That is the dilemma I present.

 

[Resha Caner]

1) The "hard problem of consciousness"....

Yeah, it's fun to watch scientists try to deal with topics like consciousness. But I guess I qualify as part of the peanut gallery on that topic. Since I'm convinced science will never get it right, I tend to criticize without offering anything constructive. Even though it's fun, it's not a very defensible position to take.

 

[Resha Caner]

How would you respond to the comment that repeating decimals are an artifact resulting from trying to represent a rational number ill-suited to the base.

E.g., 1/9 = .11111.... in base 10. However, in base 9 it is simply 0.1

.9999.... can result from multiplying .111111.... by 9. That is, .9999... ==1 because 9 x (1/9) = 1.

It's part of the fascination. It means we can never symbolically represent everything we know - a bow to Godel. IMO the implications of that are HUGE.

Digression: My dad is a retired math teacher. When I was a kid he used to give me problems to solve, and I was geeky enough to do them. Number bases was one of the things he gave me to work on. When my teachers at school found out about it, they started giving me extra problems as well ... and I did those too.

When one digs into it, "number" is one of the most difficult concepts people have ever wrestled with. I love it!

 

[Tinker Grey]

It's part of the fascination. It means we can never symbolically represent everything we know - a bow to Godel. IMO the implications of that are HUGE.

It may be true that we can never represent everything we know, but I don't think my example applies. We can represent 1/9th exactly (as I just did 2 step before this parenthetical) and we can do it decimal-like as 0.1 in base 9 or 0.01 in base 3.

That we can't adequately represent 1/9 in base 10 (and I'm sure a PhD in math would disagree that we can't; that is, 0.11111... with the indication of infinite repetition is accurate), has nothing to do with Godel.

 

[Gracchus]

Um wouldn't 10x-x = 9.00000..... with a 9 at the end* given you've shifted x by a decimal point?

*even though you can't reach the "end" of an infinite regression.

No, because .999 ... = 0 + .9 +.09 + .009 ... Thus when you multiply 0.999 ... by 10 you multiply all the terms by ten. (In mathematics the elipsis "..." means repeat an infinite number of times.)

The Dedikind Cut has nothing to do with this.

a: x = 0.999 ... (Premise, by definition)
b: 10x = 9.999 ... (If equals are multiplied by equals the products are equal.)
c: 10x - x = 9x = 9 (If equals are subtracted from equals the differences are equal.)
thus d: x = 1

What we have shown is that there is no place between 0.999 ... and 1 to make a "cut". You can make a cut between any two distinct numbers, but you can't make a "cut" between a number and the same number.

1 = 0.999 ... is not an approximation any more than 1 + 1 = 2 is an approximation.

1 + 1 does not approximate 2 just because it is written with different symbols, nor is 2/2 different from 1 except in form.

 

[Resha Caner]

It may be true that we can never represent everything we know, but I don't think my example applies. We can represent 1/9th exactly (as I just did 2 step before this parenthetical) and we can do it decimal-like as 0.1 in base 9 or 0.01 in base 3.

That we can't adequately represent 1/9 in base 10 (and I'm sure a PhD in math would disagree that we can't; that is, 0.11111... with the indication of infinite repetition is accurate), has nothing to do with Godel.

I was not precise enough in my statement. Though you can represent 1/9 exactly with other bases, you cannot operate on 1/9 exactly in base 10. Therefore, something that can be represented exactly in base 10 cannot operate exactly on 1/9. So, one is forced to jump back and forth between systems to do the operation. Or, if one found a system where they could both be operated on exactly, I am quite sure (in the spirit of Godel) that the new system would leave some other problem inoperable.

 

[Resha Caner]

The Dedikind Cut has nothing to do with this.

I'll send you the paper if you like. If Richman made an error, I'd like to know about it.

I'd also be curious to here your response to post #19 where I used 9 as the multiplier rather than 10. Why doesn't that work? And, to flesh out the example more, I meant it to be a repeating excercise. I used 0.999. If you tried 0.9999, and 0.99999, and then 0.999999, one would keep getting the same answer. If that is true, at what point does that series of problems suddenly change so that the answer is 1?

[edit] And, I'll introduce yet a third argument. An infinite series converges on a number. That does not mean it equates to that number. One must use these terms carefully. For example, with my dad (whose expertise is geometry) I once tried to argue that "equal" and "congruent" were equivalent terms. He was adament that they were not. Maybe somewhat synonomous, but not equivalent.

 

[Gracchus]

I was not precise enough in my statement. Though you can represent 1/9 exactly with other bases, you cannot operate on 1/9 exactly in base 10. Therefore, something that can be represented exactly in base 10 cannot operate exactly on 1/9. So, one is forced to jump back and forth between systems to do the operation. Or, if one found a system where they could both be operated on exactly, I am quite sure (in the spirit of Godel) that the new system would leave some other problem inoperable.

No. A change of basis does not affect the result. In a different basis (say base 9) you can represent 1/9 as a finite power. (So in base 9, 1/9 = 0.1)

All Goedel proved was that there are some statements in any algebra obeying the rules of arithmetic that cannot be proven. He didn't prove none of them can. I don't know that anyone has even found one.

Does anyone know of one?

 

[Resha Caner]

No. A change of basis does not affect the result. In a different basis (say base 9) you can represent 1/9 as a finite power. (So in base 9, 1/9 = 0.1)

All Goedel proved was that there are some statements in any algebra obeying the rules of arithmetic that cannot be proven. He didn't prove none of them can. I don't know that anyone has even found one.

Does anyone know of one?

First of all, I'm more interested in a response to #32.

Second, I'm confused by your answer. It almost seems like you're repeating what I said and then disagreeing with me. I don't get it.

 

[Gracchus]

Hmm. The Dedekind cut is one of those things that makes my head hurt. I have to think about just the definition for an hour before I can even start solving a problem. So, I don't think I would be able to be brief. My best explanation was the one I gave in my previous post - simply that because the two numbers are symbolically different, their value is different.

It doesn't matter that you can't find a rational number between them. A Dedekind cut focuses on the "gap" between two numbers. It defines a set "A" that is less than "B", and yet does not specify the greatest value for "A." Therefore, "A" is unbounded and yet less than "B" (I'm not sure I said that quite right). Anyway, the two sets (I think they're called "rings") "A" and "B" have no intersection. Therefore, it is impossible for any of their members to be equal. Richman then shows that a Dedekind cut of the number line puts 0.9... and 1.0... in two different sets. So, they are not equal.

But if the sets are disjoint, then the intersection is empty. So, there is nowhere to make a cut. The sets A and B are disjoint in that they have no elements in common, and there is no point "between them" and so their conjunction is continuous.

If it helps, think of this. Blayz arbitrarily multiplied x by 10. You don't have to do that. What if you multiplied x by 9. The trick should still work, but it doesn't. How do you do the multiplication? Maybe someone would argue that you get:

-x = -0.9...
9x = 8.9...
----------
8x = 8

x = 1

But that's not right. Think of a truncated version:

-x = -0.999
9x = 8.991
-----------
8x = 7.992

x = 0.999

What you have done is truncate an infinite series, and a truncated series is not the same as the infinite series.

Since multiplication requires starting at the right end of the number, you're actually approximating the infinite series in that algebra trick in order to bootstrap the process. You're also depending on an artifact of a 10-based system to make the trick work. If we worked in a hexadecimal system (like computers do), the trick wouldn't work because you would be using the wrong approximation. In other words, you would have to use the hexadecimal equivalent to make the trick work, and you would end up proving a different result (even though they are very similar).

Did that actually help?

Multiplication does not require that you start at the right end of the number. That is how you were taught to do it for the teacher's convenience.

Consider 4 X 124. I can start in the middle and work out: 4 x 20 = 80, 4 X 100 = 400, 4 X 4 = 16 and 80 + 400 + 16 = 496, which is what you would get if you started at the right or the left.

What about this? 9.999999.... is either a number or not. If it is a number then it is not representative is an infinite series, because an infinite series is not a point on a number line. Or, if it is not a number then it does not belong in an algebraic equation, where variables like "x" etc represent numbers. In both cases the proof being discussed is invalid. That is the dilemma I present.

Every number is a point on the number line. Pi is on the number line and it is a non-repeating infinite series. A repeating infinite series is a rational number, but all irrational numbers are also on the number line.

I'll send you the paper if you like. If Richman made an error, I'd like to know about it.

Possibly Richman made an error. Possibly you did. If you don't understand the Dedekind Cut, I think that the second possibility is more likely. If you arbitrarily cut the number line then you divide it into two sets. One set is bounded at at least one end, where you established the cut. But the other set is not bounded at the cut. That is a Dedekind Cut. Consider all the set of all real numbers between "a" and "b" That is the set of all R such that a < R < b.
If I make a cut at any point "c" between a and b then we have produced the sets A and B such that a < x <= c where x is an element of A and c < y < B where y is an element of B. Note that c is an element of A but it is not an element of B. But any number greater than c is an element of B. In a Dedekind cut one of the sets always includes the cut and the other always excludes it.

I'd also be curious to here your response to post #19 where I used 9 as the multiplier rather than 10. Why doesn't that work? And, to flesh out the example more, I meant it to be a repeating excercise. I used 0.999. If you tried 0.9999, and 0.99999, and then 0.999999, one would keep getting the same answer. If that is true, at what point does that series of problems suddenly change so that the answer is 1?

It changes when the number of terms becomes infinite. A convergent series provides the resolution of Zeno's Paradox.

[edit] And, I'll introduce yet a third argument. An infinite series converges on a number. That does not mean it equates to that number. One must use these terms carefully. For example, with my dad (whose expertise is geometry) I once tried to argue that "equal" and "congruent" were equivalent terms. He was adament that they were not. Maybe somewhat synonomous, but not equivalent.

The infinite series 0.999 ... converges to one as the number of decimal places approaches infinity. That is the same thing as saying that the sum of the infinite series 0.9 + 0.09 + 0.009 + ... = 1

I hope your dad convinced you that you were wrong. If he didn't you are way over your head here.

 

[Gracchus]

Post #32:

I'll send you the paper if you like. If Richman made an error, I'd like to know about it.

I'd also be curious to here your response to post #19 where I used 9 as the multiplier rather than 10. Why doesn't that work? And, to flesh out the example more, I meant it to be a repeating excercise. I used 0.999. If you tried 0.9999, and 0.99999, and then 0.999999, one would keep getting the same answer. If that is true, at what point does that series of problems suddenly change so that the answer is 1?

[edit] And, I'll introduce yet a third argument. An infinite series converges on a number. That does not mean it equates to that number. One must use these terms carefully. For example, with my dad (whose expertise is geometry) I once tried to argue that "equal" and "congruent" were equivalent terms. He was adament that they were not. Maybe somewhat synonomous, but not equivalent.

By all means, post a link to Richman's paper. Either he is mistaken, or you are mistaken about what he said, I'll warrant.

First of all, I'm more interested in a response to #32.

I dealt with that in post #35.

Second, I'm confused by your answer.

I gathered that.

It almost seems like you're repeating what I said and then disagreeing with me. I don't get it.

Your example is correct, but your conclusions are not.

If an infinite series converges to some number, x, then the sum of the series is x. That can serve as a definition of "convergence". The truncation of that series is an approximation, even if it is truncated at the trillionth term.

 

[Resha Caner]

Multiplication does not require that you start at the right end of the number. That is how you were taught to do it for the teacher's convenience.

Consider 4 X 124. I can start in the middle and work out: 4 x 20 = 80, 4 X 100 = 400, 4 X 4 = 16 and 80 + 400 + 16 = 496, which is what you would get if you started at the right or the left.

Yes, you are correct. My mistake. So, we can start at some arbitrary place in the middle. I still don't see how my example (9 * 0.9...) can be multiplied out to give the intended result (i.e. that 0.9... = 1) without truncating somewhere. And truncating would negate the validity of it all.

It changes when the number of terms becomes infinite. A convergent series provides the resolution of Zeno's Paradox.

I don't think one needs a convergent series to resolve Zeno's paradox (though I suppose it might provide a nice way to formalize the answer). But I think your statement about infinity is incorrect - or at least a somewhat wooden way of making your point. Infinity is not a number. Therefore, there is nothing one can point to as a "change" from finite to infinite. Dealing with finite things and infinite things requires two very different approaches.

By all means, post a link to Richman's paper. Either he is mistaken, or you are mistaken about what he said, I'll warrant.

My copy is in a PDF and I don't remember where I got it. Here is a link from Richman's homepage, but I'm not sure it is the complete paper. It's just a text version, and I didn't read it line for line to make sure it's all there. If not, I hope it's enough. I suspect I'm about to learn something.

Is 0.999... = 1?

If an infinite series converges to some number, x, then the sum of the series is x. That can serve as a definition of "convergence". The truncation of that series is an approximation, even if it is truncated at the trillionth term.

Hmm. That doesn't seem right. At least that's not how I would interpret the definition of convergence. Does that mean you disagree with the wiki definition?

Convergent series - Wikipedia, the free encyclopedia

 

[Gracchus]

Yes, you are correct. My mistake. So, we can start at some arbitrary place in the middle. I still don't see how my example (9 * 0.9...) can be multiplied out to give the intended result (i.e. that 0.9... = 1) without truncating somewhere. And truncating would negate the validity of it all.



I don't think one needs a convergent series to resolve Zeno's paradox (though I suppose it might provide a nice way to formalize the answer). But I think your statement about infinity is incorrect - or at least a somewhat wooden way of making your point. Infinity is not a number. Therefore, there is nothing one can point to as a "change" from finite to infinite. Dealing with finite things and infinite things requires two very different approaches.



My copy is in a PDF and I don't remember where I got it. Here is a link from Richman's homepage, but I'm not sure it is the complete paper. It's just a text version, and I didn't read it line for line to make sure it's all there. If not, I hope it's enough. I suspect I'm about to learn something.

Is 0.999... = 1?



Hmm. That doesn't seem right. At least that's not how I would interpret the definition of convergence. Does that mean you disagree with the wiki definition?

Convergent series - Wikipedia, the free encyclopedia

Which see:

"The reciprocals of the positive integers produce a divergent series (harmonic series):

Alternating the signs of the reciprocals of positive odd integers produces a convergent series (the Leibniz formula for pi):

The reciprocals of prime numbers produce a divergent series (so the set of primes is "large"):

The reciprocals of triangular numbers produce a convergent series:

The reciprocals of factorials produce a convergent series (see e):

The reciprocals of square numbers produce a convergent series (the Basel problem):

The reciprocals of powers of 2 produce a convergent series (so the set of powers of 2 is "small"):

Alternating the signs of reciprocals of powers of 2 also produce a convergent series:

The reciprocals of Fibonacci numbers produce a convergent series (see &#968;):

"
Notice that all of these series have an infinite number of terms, but the convergent series sum (an infinite number of terms) to real numbers, while the divergent series sum to infinity.

I know it is counter intuitive, but the mathematician Georg Cantor showed that some infinities are greater than others. Aleph 0 < Aleph 1 < Aleph 2.

People do get paid for teaching this stuff, so I am not going to rob a rice bowl by giving you a complete course, but I will correct your homework if you like. (For a reasonable fee!)

Give me a chance to explore that link to Richman's paper. I'll see what I can make of it.

 

[Resha Caner]

I'm familiar with these types of series, so I know people use this notation. But that doesn't prove your point, because I think that notation conflicts with the definition:

Given a sequence

, the nth partial sum Sn is the sum of the first n terms of the sequence, that is,

A series is convergent if the sequence of its partial sums

converges. In more formal language, a series converges if there exists a limit

such that for any arbitrarily small positive number

, there is a large integer N such that for all

,

So, I'm asking if you think one of these is wrong, or if my interpretation of it is wrong.

Give me a chance to explore that link to Richman's paper. I'll see what I can make of it.

Of course. I realize that reading it is no light matter.

People do get paid for teaching this stuff, so I am not going to rob a rice bowl by giving you a complete course, but I will correct your homework if you like. (For a reasonable fee!)

Lovely. Does that mean you won't answer my example? If I am asking too much from an Internet discussion, OK. I didn't think I was, since you chose to engage in the discussion, but I realize not everyone has the time or interest to delve the depths. I'm not asking you to be my tutor, but to explain why you differ with what I presented.

Still, if I'm pushing too hard ... well, I don't want to be unreasonable.

 

[Gracchus]

I'm familiar with these types of series, so I know people use this notation. But that doesn't prove your point, because I think that notation conflicts with the definition:

Given a sequence

, the nth partial sum Sn is the sum of the first n terms of the sequence, that is,

A series is convergent if the sequence of its partial sums

converges. In more formal language, a series converges if there exists a limit

such that for any arbitrarily small positive number

, there is a large integer N such that for all

,

So, I'm asking if you think one of these is wrong, or if my interpretation of it is wrong.

If N is an arbitrarily large number less than infinity then sum is a partial sum. But you can sum an infinite series. It is done all the time in the integral calculus.

If the partial sums of an infinite series converge to a limit then the sum of the infinite series is the limit. That is the whole point.

Look at it this way. f(x) = e^x (the number e raised to the power of x) is defined as x^0/0! + x^1/ 1! + x^2/2!+ ... + x^n/n! +... That is a function that has the interesting property that it is its own derivative, i.e. d e^x/dx = e^x Now what is fascinating about this is that e^(ix) (where i X i =-1)= cos(x) = i X sin(x), so as the x approaches 2 X (pi) then the sum of the series approaches 1.

This is not one of Richman's "intuitions". (Richman says that the sceptic must attack the assumption that x = 0.999... But that is not an assumption. x is assigned the value of 0.999... as a matter of definition.) This is not something that admits of scepticism. This is a matter of formal proof. In mathematics a formal proof trumps intuition and scepticism every time.

We have proven, using the formal rules of arithmetic that 0.999... = 1. Thus you cannot make a cut on the number line between 0.999... and 1. They represent the same point. The set of all x such that -(Inf) < x < +(Inf) where x is a real number includes all real numbers. Any "cut" will divide the line into two sets of points. The conjunction of those sets returns the entire set of reals. The intersection of those sets returns the empty set.

Lovely. Does that mean you won't answer my example?

Restate your example please. I am not sure which you are referring to.