Simple mathematical problem (?)

[Zep]

That is not correct, you will have no string after an infinite number of cuts.

I think the problem you're having is that you're thinking of 0.9999... as a process. It's not a process. 0.999... is a representation for an infinite number of 9's already being there. It is complete, so to speak.

Adam

(1) Incorrect. After the "infinite-th" cut you will still have 10% of the previous length of the string left over. And so it will be for ANY number of iterations of that process.

(2) Yes, the number is indeed a "process". The very definition of it requires some calculation to even conceive of it. Mathematically that may be as shown by a previous poster: 9/10^1 + 9/10^2... Conceptually it may be the "one iota down from one" idea.

Both these ideas are really an attempt to represent a surreal number as accurately as possible.

 

[Earthborn]

Well, when kids ask questions, that indicates an interest -- a hunger to learn that should not be fed with the same old mush, but should be rewarded with something interesting.

But sometimes they ask such questions not because of curiousity, but because they misunderstand the rules of mathematics they are supposed to learn. Teaching them something that is allowed in some other form of mathematics but will be considered wrong in a test, is only going to confuse them.

They should be encouraged to take EVERYTHING they learn, including something that seems to be as objective as mathematics, with a grain of salt in the eye.

I think that is going to hurt them more that it does them any good.

Xouper:

I'm not sure I follow your question. The first step I used is 1 divided into 10 gives 9 with a remainder of 1. That is a valid step.

So you say. Now can I also use 8? Or 7 or 6?

 

[Suggestologist]

A good mathematical notation is unambiguous.

Then there are no good mathematical notations.

 

[Suggestologist]

But sometimes they ask such questions not because of curiousity, but because they misunderstand the rules of mathematics they are supposed to learn.

But this is an opportunity to show them that their perspective is not WRONG or BAD, but has its place.

Teaching them something that is allowed in some other form of mathematics but will be considered wrong in a test, is only going to confuse them.I think that is going to hurt them more that it does them any good.

Ah, the typical pedagogical mind. Always knows what knowledge is best not learned, for their own goods.

 

[xouper]

Suggestologist: Oh, it does matter what one's perspective is. .99999.... may be the result of changing 1/3 to decimal form and then multiplying by three. Or, it may be the result of trying to translate a surreal into decimal form. If one forgets where and how one got a number, one is liable to mistreat it.

This is total nonsense.

The proofs are all tautologies.

Wrong. Tell us what is tautalogical about the proof I gave in that other thread, the one that uses the geometric series theorem. That is in fact one of the standard proofs.

There is no point in pointing out errors IN the proof, only the error of thinking you are proving anything at all.

Nonsense.

I demonstrated that the request has no meaning.

You did no such thing.

Um, 1 divided into 10 is 10 with a remainder of 0.

Yes, that's true. So is the step I used. One divided into ten gives nine with a remainder of one. There is nothing false about that statement.

 

[xouper]

Zep: (2) Yes, the number is indeed a "process".

Wrong. The notation 0.999... is not a process. It represents a specific and unambiguous number.

Both these ideas are really an attempt to represent a surreal number as accurately as possible.

Nonsense.

 

[Walter Wayne]

That's a heuristic. Which seems to fail when you use 9, since it does not produce a repeating pattern of 9s after the decimal point, does it?

let x be any single digit number.

a/9 is what in decimal. If we want decimal we desire a denominator which is power of 10. b/10 is very easy to write in decimal form. So we try to solve:

a/9 = b/10
b = (10/9)a

so if multipl a by 10/9 we can get the fraction into a form that is easy to change to decimal notation.

10/9 = 1.111.... (sorry, my equation editor is acting funny, try the long division yourself if your not convinced)

b = a x 1.111 ...
b = a.aaa.... (assuming a is a single digit)

Therefore
a/9 = b/10 = a.aaa/10 = 0.aaa...

a/9 = 0.aaa... for any a which is a single digit

Walt

 

[69dodge]

Originally posted by Suggestologist
Then there are no good mathematical notations.

Well, surely, if we have two distinct concepts in mind, namely, 1 and 1 - epsilon, using the same notation ("0.999...") for both is an ambiguity that is easily avoided.

 

[xouper]

69dodge: A good mathematical notation is unambiguous.

Suggestologist: Then there are no good mathematical notations.

I've seen some pretty ridiculous comments in this thread, but this one from Suggestologist takes the cake.

 

[Earthborn]

Ah, the typical pedagogical mind. Always knows what knowledge is best not learned, for their own goods.

Pete has 3 apples. He gives 5 apples to Mary. How many apples does Pete have left?

All I am suggesting is that people are taught mathematics with simple rules first, and that they should well understand those, before they are bothered with more complex mathematical theories. They are not going to understand the complex rules, if they don't understand the simpler underlying rules first.

 

[Walter Wayne]

(1) Incorrect. After the "infinite-th" cut you will still have 10% of the previous length of the string left over. And so it will be for ANY number of iterations of that process.

"After the infinite-th cut", how can you be after the infinite-th anything. The problem is trying to extrapolate from the finite to the infinite. We can see the end of a line-segment, what is at the end of a line?

You can get some other interesting results extrapolating from finite to infinite.

First:
1-0.9=0.1
1-0.99=0.01
1-0.999=0.001
1-0.9... = 0.0...1
0r an infinity of zeros followed by a one, how can an infinite series of zeroes have a beginning (the decimal point) and an end (the trailing 1)

Second:
1-0.9=1/10
1-0.99=1/10²
1-0.9...=1/10^infinity

And a fun one for playing with infinities:
0=(1-1)+(1-1)+(1-1)+ ...
=1 + (-1+1) + (-1+1) + ...
=1

 

[patoco12]

Pete has 3 apples. He gives 5 apples to Mary. How many apples does Pete have left?

All I am suggesting is that people are taught mathematics with simple rules first, and that they should well understand those, before they are bothered with more complex mathematical theories. They are not going to understand the complex rules, if they don't understand the simpler underlying rules first.

How do we know Pete has 3 apples? He could have 2 apples or 10 oranges. It really depends on your perspective. There could be any number of fruit depending on the season, location, etc.

You're also making a GIANT leap in assuming that Pete can actually transfer apples to Mary. I'm not ready to accept that.

What universe are you working in? In a surreal universe, which is obvisouly superior to the naive real universe, Pete and Mary can both have as many apples as they want, without consequense. As long as Mary doesn't take a bite...

 

[xouper]

Earthborn: Xouper:So you say. Now can I also use 8? Or 7 or 6?

You could, but since the subsequent remainders diverge, what would your result demonstrate?

 

[Walter Wayne]

Pete has 3 apples. He gives 5 apples to Mary. How many apples does Pete have left?

Well if he had 3, but he gave 5 away he then he must have appropriated someone else's apples to impress the girl. Is it that hard to believe a guy would do that?

 

[Earthborn]

You could, but since the subsequent remainders diverge, what would your result demonstrate?

Maybe that if you can wholly divide the first number you find with the number you are dividing it with, it doesn't make any sense to divide it by any less?

Is this:

1/1\1
  1
  -
  0

equally valid than this:

1/1\0.87...
  8
  -
  20
   7
  -
  130
   ...

Cause it doesn't make much sense to me to do it like that.

 

[Zep]

Wrong. The notation 0.999... is not a process. It represents a specific and unambiguous number.

OK, imagine that number in your head. How do you "see" it? Is it "a number just less than one but so really REALLY close that it is almost one but not quite"? Well, that is the same "process" as "one iota down from one." You have to start at some point and go to another in order to describe the number.

Consider any decimal number: e.g. 12345. The "process" of placing this number is to understand that the "1" refers to units of 10^4, the "2" refers to units of 10^3, etc. And altogether we can "read" it as that specific value. Simple baby notation stuff, yes?

But what if I showed you a number like "E56A79B"? Ah ha! A hexidecimal base 16 number, you might say (because it has "hex chars" in it). But are you sure? It could also be a base 15 number, and as such a different numerical value could be represented. The situation is now that you are faced with two different "processes" for defining a numeric value from the notation given. And herein lies the issue with the "process" of representing surreal numbers like 0.9~.

Here's a mind exercise. Consider the hex number 0.FFFF~. This, too, is a surreal number that approaches unity. Question: which one is "closer" to unity, 0.9~ base 10 or 0.F~ base 16? I have my own thoughts on this, but I'm always prepared to listen to others!

 

[xouper]

Earthborn: Maybe that if you can wholly divide the first number you find with the number you are dividing it with, it doesn't make any sense to divide it by any less?

Normally, it wouldn't, but since the question was asked, I was trying to show that it is valid to divide 1 by 1 and get 0.999... using the standard division algorithm.

 

[Zep]

And a fun one for playing with infinities:
0=(1-1)+(1-1)+(1-1)+ ...
=1 + (-1+1) + (-1+1) + ...
=1

=1-1+1-1+1-1+1-1+...
=1-(1+1)-(1+1)-(1+1)...
=-infinity

 

[xouper]

Zep: OK, imagine that number in your head. How do you "see" it?

I "see" it as exactly equal to one. I "see" all the nines already there simultaneously. All infinity of them.

Is it "a number just less than one but so really REALLY close that it is almost one but not quite"?

No.

Well, that is the same "process" as "one iota down from one." You have to start at some point and go to another in order to describe the number.

Wrong. The definition of the notation is that all the nines are already there simultaneously. There is no process in that definition. I do not accept your attempt to redefine the meaning of that standard notation. It may be convenient in some contexts to think of it in terms of a process, but that is not how the notation is defined.

Here's a mind exercise. Consider the hex number 0.FFFF~. This, too, is a surreal number that approaches unity.

No. It is exactly equal to one.

 

[patoco12]

=1-1+1-1+1-1+1-1+...
=1-(1+1)-(1+1)-(1+1)...
=-infinity

I know this was meant as a joke, but you goofed up your order of operations. You can't go from 1-1+1 to 1-(1+1).

How strong are your Math skills, Zep?