Simple mathematical problem (?)

[69dodge]

Originally posted by Suggestologist
Yes. When you convert between rational numbers and decimal numbers, you sometimes lose information. Just like when you translate a word between languages, the translated word often means something close to the original word, but it doesn't quite mean exactly the same thing. It may have slightly (or very) different connotations, or associations.

Interesting.

I think you might be attaching more meaning to rational numbers or decimals than mathematicians generally do. For example, consider "1/5" and "0.2". Both of these represent the same number. What information do you think one contains that the other lacks?

 

[Earthborn]

OK, let's work it this way with a simple mind experiment.

Consider we have a perfect string that is exactly 1 metre long. An operation on this string that we do is to cut off and discard 90% of the current length each time. Perform the operation once - we are left with 10cm of string. Do it again, 1cm. Again, 1mm. Repeat this operation ad infinitum. Question: At what point do we have "no string"? Answer: Never. There is always "some" string left over if we remove only 90% of the remainder at each operation.

This is the quivalent of refuting the mathematical notion that:

Suppose 0.9~ = 1, then 1 - 0.9~ = 0
But since we can demonstrate that 1 - 0.9~ <> 0 then it follows that 0.9~ <> 1.

Again, think Cantor Sets!

You are describing a process. But usually when people speak about 0.9(recurring), it is simply a given that there are an infinite number of 9s. It is not a process of adding 9s, they are already there from the start.
The shortening of the string, if you will, has all ready occured instantly. And how much string is left? None at all!

 

[xouper]

After reading through this thread, I can only conclude that Suggestologist and Zep have absolutely no clue what the f*ck they are talking about. Since they are not convinced by the proofs given (especially in that other thread I posted in), then I can only think of one remaining course of action:



There are way too many errors in their posts to bother addressing them all at once, but I will comment on a question that was raised here but hadn't been discussed in the previous thread.

If you divide 1 by 1 using the standard long division algorithm we all learned in grade school, you can indeed get 0.999...

Example:

 

[Suggestologist]

There are way too many errors in their posts to bother addressing them all at once, but I will comment on a question that was raised here but hadn't been discussed in the previous thread.

If you divide 1 by 1 using the standard long division algorithm we all learned in grade school, you can indeed get 0.999...

Example:

You're missing a step in your algorithm there.

 

[Suggestologist]

Interesting.

I think you might be attaching more meaning to rational numbers or decimals than mathematicians generally do. For example, consider "1/5" and "0.2". Both of these represent the same number. What information do you think one contains that the other lacks?

Sometimes words can be translated "exactly".

What I'm saying is that if you get .3333333..... by converting it from 1/3 to decimal and multiply it by 3, then the answer of .99999999... does equal 1. However, if you get .999999999..... by converting from a surreal number (up_down_up_hat, I think) into decimal form, then it definately does not equal 1.

 

[xouper]

Suggestologist: You're missing a step in your algorithm there.

No I'm not. I used an alternate (but equally valid) first step.

 

[Suggestologist]

Or, differently formulated: allowing kids to think and explore things within a set of rules they might actually understand, or confusing them even more than they are with things that make them think (even more than they already do) 'When am I ever going to need this?'

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.

And they should come away with the fact that even though an idea they now have about numbers may not be accepted as an answer on a test, that like Conway, they may be able to develop different mathematical concepts that have uses as "calculating tricks". 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.

 

[Earthborn]

Xouper, do you mean that using any number as a first step is equally valid? That's new to me.

 

[Donut]

I'd just like to point out that, even in the surreal number system;
0.999... = 1--exactly and unequivocably.

It's a byproduct of our decimal system, nothing more. The supposed problem disappears if we use another base (say, for instance, base nine where 1/3 = 0.3 and 0.3 + 0.3 + 0.3 = 1).

I can only assume that Zep and Suggestologist have never actually dealt with a surreal number system (and I place no blame here--few mathematicians bother) because, in such a system 1 - 0.999... = 0. When you want to talk about a number infinitesimally close to 1 you refer to it as "1 - epsilon" or "1 + epsilon" (or iota or whatever). This allows you to talk about surreal numbers like "1 + 2*epsilon" or whatever. Thus, we can have surreal numbers infintely close to 1 which are not equal to each other (how, using the Suggestologist sytem, would one represent a surreal number infinitely close to but greater than 1? 1.00...1?)

Anyway, unless otherwise stated, I would always assume that we are working in the reals. It is, after all, the number system most of us have been taught all our lives.

 

[Walter Wayne]

A suggested ‘proof’ is
X=0.999(recurring)
10X=9.9999
10X-X = 9X
9X=9X
X=1

I don't see what the line italized has to do with the line above, I think the way you meant to write it was.

X=0.999(recurring)
10X=9.9999
10X-X = 9.999... - 0.999...
9X=9
X=1

Walt

Ignore this entire post, somehow I thought this was a new thread and didn't see the two pages after the first.

 

[Suggestologist]

No I'm not. I used an alternate (but equally valid) first step.

I thought you wrote that you were using "the standard long division algorithm we all learned in grade school". Altering the first step, alters the algorithm. You're only helping to prove my point that mathematics is a process, with different possible meanings and results. Thanks.

 

[patoco12]

You're only helping to prove my point that mathematics is a process, with different possible meanings and results. Thanks.

Now you are being ridiculous. We have proven numerous times and with different approaches that .9~ = 1. It doesn't matter what the "perspective" is. You continue to ignore the proofs and state nonsense. Point out the flaws in all the proofs given, then maybe you'll get some credit. But I don't think that will happen.

And you never did come up with a fractional representation of .9~ that is not equal to 1. This should be rather trivial in the realm of rational numbers, right?

 

[Suggestologist]

how, using the Suggestologist sytem, would one represent a surreal number infinitely close to but greater than 1? 1.00...1?

Sometimes words are untranslatable.

up_up_down_hat would be 1.00...1

The surreals work on top of the real numbers. At least they do initially from what I see. As far as I can tell, they're not supposed to intrude on the Hyperreal system too much -- though they seem like similar ideas.

Anyway, unless otherwise stated, I would always assume that we are working in the reals. It is, after all, the number system most of us have been taught all our lives.

I see. You were never taught the imaginaries?

 

[xouper]

Suggestologist: I thought you wrote that you were using "the standard long division algorithm we all learned in grade school".

It is the standard algorithm.

Altering the first step, alters the algorithm.

No it doesn't. The algorithm doesn't change just because I choose a different number.

You're only helping to prove my point that mathematics is a process, with different possible meanings and results. Thanks.

Not the way you mean it I haven't. I haven't changed any meanings or results.

Earthborn: Xouper, do you mean that using any number as a first step is equally valid? That's new to me.

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.

 

[Walter Wayne]

Any repetitive decimal can be written as a fraction.

For example
0.323232...= 32/99
0.314314314 = 314/999

Sof for those who are wondering about the rationality of 0.99999... in can be written as a fraction (is rational). Take whatever the repetitive sequence is and write it over a bunch of 9's (number of 9's equal the length).

Walt

 

[Suggestologist]

Now you are being ridiculous. We have proven numerous times and with different approaches that .9~ = 1. It doesn't matter what the "perspective" is.

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.

You continue to ignore the proofs and state nonsense. Point out the flaws in all the proofs given, then maybe you'll get some credit. But I don't think that will happen.

The proofs are all tautologies. There is no point in pointing out errors IN the proof, only the error of thinking you are proving anything at all. You are only demonstrating how you think about the numbers, not necessarily how they should be thought about for any particular purpose.

And you never did come up with a fractional representation of .9~ that is not equal to 1. This should be rather trivial in the realm of rational numbers, right?

I demonstrated that the request has no meaning.

 

[patoco12]

Any repetitive decimal can be written as a fraction.

For example
0.323232...= 32/99
0.314314314 = 314/999

Sof for those who are wondering about the rationality of 0.99999... in can be written as a fraction (is rational). Take whatever the repetitive sequence is and write it over a bunch of 9's (number of 9's equal the length).

Walt

OK, I'll try it:
The pattern is '9'. The length is one so I'll divide it by '9'.

This gives: 9/9; Oh no! You've shown that .9~ = 1!

 

[Suggestologist]

It is the standard algorithm.

No it doesn't. The algorithm doesn't change just because I choose a different number.

Not the way you mean it I haven't. I haven't changed any meanings or results.

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.

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

 

[Suggestologist]

Any repetitive decimal can be written as a fraction.

For example
0.323232...= 32/99
0.314314314 = 314/999

Sof for those who are wondering about the rationality of 0.99999... in can be written as a fraction (is rational). Take whatever the repetitive sequence is and write it over a bunch of 9's (number of 9's equal the length).

Walt

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?

 

[69dodge]

Originally posted by Suggestologist
What I'm saying is that if you get .3333333..... by converting it from 1/3 to decimal and multiply it by 3, then the answer of .99999999... does equal 1. However, if you get .999999999..... by converting from a surreal number (up_down_up_hat, I think) into decimal form, then it definately does not equal 1.

A good mathematical notation is unambiguous. (So I am not surprised to learn from Donut that 0.999... = 1 for surreal numbers too.) Striving for clarity is much better than merely warning everyone about the ambiguity of everything.