Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

Topic: Re: It is a very bad idea and nothing less than stupid to define 1/3
= 0.333...

Replies: 42   Last Post: Oct 9, 2017 11:53 AM

 Messages: [ Previous | Next ]
 FromTheRafters Posts: 248 Registered: 12/20/15
Re: It is a very bad idea and nothing less than stupid to define 1/3 = 0.333...
Posted: Oct 3, 2017 3:55 PM

netzweltler laid this down on his screen :
> Am Dienstag, 3. Oktober 2017 14:20:11 UTC+2 schrieb FromTheRafters:
>> netzweltler brought next idea :
>>> Am Dienstag, 3. Oktober 2017 03:22:11 UTC+2 schrieb Jim Burns:
>>>> On 10/2/2017 2:47 PM, netzweltler wrote:
>>>>> Am Montag, 2. Oktober 2017 20:35:56 UTC+2
>>>>> schrieb Jim Burns:

>>>>>> On 10/2/2017 1:58 PM, netzweltler wrote:
>>>>>>> Am Montag, 2. Oktober 2017 17:59:21 UTC+2
>>>>>>> schrieb Jim Burns:

>>>>>>>> On 10/1/2017 3:22 AM, netzweltler wrote:
>>>>>>>>> Do you agree that 0.999... means infinitely many commands
>>>>>>>>> ...?

>>>>>>>>
>>>>>>>> 0.999... does not mean infinitely many commands.

>>>>>>>
>>>>>>> But that's exactly what it means.

>>>>>>
>>>>>> That's not the standard meaning.

>>>>>
>>>>> So, you disagree that
>>>>> 0.999... = 0.9 + 0.09 + 0.009 + ... ?

>>>>
>>>> Your '...' is not usable. If we say what we _really_ mean,
>>>> in a manner clear enough to reason about, then the '...'
>>>> disappears. Also, what we are left with are finitely many
>>>> statements of finite length. You will not find infinitely
>>>> many commands in those finitely-many, finite-length
>>>> statements.
>>>>
>>>> We sometimes write the set of natural numbers as
>>>> { 0, 1, 2, 3, ... }
>>>> The '...' is informal. We do not use '...' in our reasoning,
>>>> we use a correct description of what the '...' stands for.
>>>>
>>>> Do you see '...' anywhere in the following?
>>>>
>>>> The set N contains 0, and for every element x in N, its
>>>> successor Sx is in N.
>>>>
>>>> This is true of N but not true of any _proper_ subset of N.
>>>>
>>>> _Therefore_ , if we can prove that B is a subset of N
>>>> which contains 0 and which, for element x of B, contains Sx,
>>>> then B is not a _proper_ subset of N.
>>>>
>>>> B nonetheless is a subset of N, we just said so. The only subset
>>>> of N which B can be is N. Therefore, B = N.
>>>>
>>>> This is finite reasoning about the infinitely many elements
>>>> in N. Note that there is no '...' in it.
>>>>
>>>> I could continue and derive 0.999... = 1 from our definitions,
>>>> and nowhere in that derivation will be '...'. There will not be
>>>> infinitely many commands in it either.
>>>>

>>>>>> You give it some other meaning, and then you find a problem
>>>>>> with the meaning you gave it. Supposing I wanted to sort out
>>>>>> what that other meaning was, and how to make sense of it, my
>>>>>> attention to your meaning would not affect the standard meaning.
>>>>>>
>>>>>> I am not a math historian, but the impression I have
>>>>>> is that great care was taken in choosing the standard meaning
>>>>>> in order to avoid problems like the ones you are finding.
>>>>>>
>>>>>> You have the ability to create and then wallow in whatever
>>>>>> problems you choose. No one is able to take that power away
>>>>>> from you. But you can't "choose" by an act of your will to
>>>>>> make your created problem relevant to what everyone else
>>>>>> is doing. You are not the boss of us.
>>>>>>

>>>>>>> Infinitely many commands. Infinitely many additions.
>>>>>>> Infinitely many steps trying to reach a point on the number line.
>>>>>>>

>>>>>>>> There is a set of results of certain finite sums, a set of
>>>>>>>> numbers. We can informally write that set as
>>>>>>>> { 0.9, 0.99, 0.999, ... }
>>>>>>>> That is an infinite set, but we can give it a finite description.
>>>>>>>>
>>>>>>>> (Our finite description won't use '...'. The meaning of
>>>>>>>> '...' depends upon it being obvious. If we are discussing
>>>>>>>> what '...' means, it must not be obvious, so we ought to
>>>>>>>> avoid using '...')
>>>>>>>>
>>>>>>>> There is number which is the unique least upper bound of that set.
>>>>>>>> The least upper bound is a finite description of that number.
>>>>>>>>
>>>>>>>> 0.999... means "the least upper bound of the set
>>>>>>>> { 0.9, 0.99, 0.999, ... }".
>>>>>>>> That number can be show to be 1, by reasoning in a finite manner
>>>>>>>> from these finite descriptions of what we mean.
>>>>>>>>
>>>>>>>> If you give 0.999... some meaning other than what we mean,
>>>>>>>> and then it turns out there are problems of some sort with

>>>>>
>>>
>>> Sorry, no. The meaning of "..." is absolutely clear in this context and we
>>> both know that there is a decimal place for each n ? N in 0.999...

>>
>> But 0.999 repeating is a rational number, no need for repeating
>> decimals at all in the naturals. Repeating zeros is okay I guess, but
>> why use them in the naturals. In the rationals and reals, repeating
>> zeros are called 'terminating' decimal expansions and the trailing
>> zeros are elided.

>
> I'm not sure if you got what I meant. Let me rephrase it:
>
> The meaning of "..." is absolutely clear in this context and we both know
> that there is a nth decimal place for each n ? N in 0.999...

Okay, so the natural number is only an index for positions in the
endless (ad infinitum) string of nines in the decimal expansion.

We're still in the reals, good. :) Sorry for my confusion.

Date Subject Author
10/2/17 Guest
10/2/17 netzweltler
10/2/17 Jim Burns
10/3/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 bursejan@gmail.com
10/4/17 netzweltler
10/3/17 FromTheRafters
10/3/17 Jim Burns
10/3/17 FromTheRafters
10/3/17 netzweltler
10/3/17 Jim Burns
10/4/17 netzweltler
10/4/17 Jim Burns
10/4/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 netzweltler
10/5/17 Jim Burns
10/5/17 FromTheRafters
10/6/17 netzweltler
10/6/17 Jim Burns
10/7/17 FromTheRafters
10/8/17 FromTheRafters
10/8/17 netzweltler
10/8/17 Jim Burns
10/8/17 netzweltler
10/8/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/9/17 netzweltler
10/9/17 Jim Burns
10/7/17 Jim Burns