My Math Forum An Anomaly of Decimal Representation

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 July 6th, 2018, 03:35 PM #51 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,822 Thanks: 643 Math Focus: Yet to find out. Why decimal representation exactly? The heart of your argument seems to claim that we can't, with any level of confidence, assert things about processes involving infinity. Do you take issue with other infinite sums that have a convergent value? Anyway, why doesn't someone start a Google doc to write out all the arguments for and against in concise form with in depth explanations? There's loads of repeated information here that could be made better use of.
July 6th, 2018, 03:49 PM   #52
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Quote:
 Originally Posted by zylo I start walking. If every time I take a step I take another step, do I stop at some point or do I go on indefinitely. It's one or the other. And if I keep on going indefinitely, is there some place I will never reach, beyond the universe? That's mysticism. And if there is such a place, is there something beyond that? n steps is finite. n+1 steps if I take n steps isn't.
Confusing math with physics again. Nobody is claiming there's actual infinity instantiated in the real world. Not according to contemporary physics.

Quote:
 Originally Posted by zylo If it makes you happy, every time you divide the line 10^n times, divide it 10^(n+1) times.
Not following your point at all.

Quote:
 Originally Posted by zylo And of course the question, what is your definition of a real number?
The standard mathematical reals as defined axiomatically and instantiated in set theory through Dedekind cuts or some other construction. Are you proposing a different definition?

https://en.wikipedia.org/wiki/Real_number

July 6th, 2018, 07:05 PM   #53
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 Originally Posted by Micrm@ss Cool, so you believe the number line is something like the hyperreals or surreals. Or you believe that the number line is a purely geometric entity that can not be put into one-to-one correspondence with the reals or any field. I'm fine with that, those are all respectable opinions in my book. But please don't refer to the number line as being the real numbers then.
I've heard a very little about a philosopher named Charles Sanders Peirce, with that spelling. He worked in the late 19th century. He said that whatever a continuum is, it certainly must have the property that every part is identical to the whole.

The real numbers manifestly fail this standard. We can write $\displaystyle \mathbb R = \bigcup_{x \in \mathbb R} \{x\}$. If we can break up the real numbers into a union of singletons, none of which are anything like the full set of real numbers ... then the mathematical real numbers are not the right model of the continuum.

I may be paraphrasing Peirce badly here, my understanding is second or third hand. I find this a compelling line of argument regardless.

The entire Cantorian and set-theoretic paradigm cannot lead us to the continuum.

Perhaps this is something to to with the rise of Category theory and Homotopy type theory as potential alternative foundations. Peirce as I understand it was a pretty deep thinker, not as famous as some, but who perhaps anticipated the modern developments in foundations.

https://en.wikipedia.org/wiki/Charles_Sanders_Peirce

Last edited by Maschke; July 6th, 2018 at 07:14 PM.

July 6th, 2018, 10:31 PM   #54
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Quote:
 Originally Posted by zylo a non-repeating decimal
Is a non-repeating decimal a type of decimal? If not, how do you define it?

Quote:
 Originally Posted by zylo . . . how to count binary representations of the natural numbers
How can you count them?

Quote:
 Originally Posted by zylo .33..... is never 1/3, but the difference approaches 0.
You are assuming that each ".33...." isn't endless. That allows 1/3 to be represented by the endless .3333333........, doesn't it?

How would counting be done without a number system? (I asked this before, but you didn't tell me.)

July 7th, 2018, 05:26 AM   #55
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The axiomatic definition of a real number doesn't let you calculate $\displaystyle \oint_{0}^{\frac{1}{3}}f(x)dx$

What bothers me is that neither does the decimal number system because 1/3 doesn't exist there, and though [0,1/3) exists, [0,1/3] doesn't. You can waffle around it, but you really don't have an unambiquous number system at your disposal.

The rational numbers plus infinite decimals seem to cover all points on a line to the extent that they define it in an unequivocal manner, but you are really dealing with two number systems with duplications.

But maybe two number systems combined (rational and infinite decimal) isn't so bad, since it means you can always specify a point on a line without equivocation. And with that, 3(1/3)=1 $\displaystyle \neq$ .999....... And, brace yourself, since rational and reals are countable, so is their combination, even with duplication. Induction is infinity.

As for something like pi, whose only numerical definition is it's decimal expansion, there is no ambiguity.

Quote:
 Originally Posted by skipjack How would counting be done without a number system? (I asked this before, but you didn't tell me.)
One, two, three, four, ........, ie, with the natural numbers

Last edited by zylo; July 7th, 2018 at 05:31 AM.

July 7th, 2018, 08:31 AM   #56
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Quote:
 Originally Posted by zylo You are still confusing limits. There are no sums involved here, just limits of subdivisions of a line, corresponding to limits of decimal or binary or any radix places.
Nope, I'm not confusing anything. I'm working from definitions and representations of numbers.

You don't want to use the definitions and you are claiming erroneously that decimal strings are numbers. Decimal strings can be used to represent numbers. There's a huge difference.

Moreover, you refuse to learn anything about infinite objects. You insist on relying on your intuition about the infinite and your intuition is wrong. As is everybody's. As a result nothing you say about infinite objects is correct.

July 7th, 2018, 08:53 AM   #57
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Quote:
 Originally Posted by zylo One, two, three, four, ........, ie, with the natural numbers
How would counting be done without a number system?

If you have a method, please describe how it would be used to count 1, 2, 3, 4, ... (continued without end).

July 7th, 2018, 10:56 AM   #58
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Quote:
 Originally Posted by skipjack Isn't "the natural numbers" a number system? Please answer my question. How would counting be done without a number system? If you have a method, please describe how it would be used to count 1, 2, 3, 4, ... (continued without end).
Sorry, but I don't see what you are getting at. Sticks, stones, scratches on a rock?

Edit: Trying to guess what you are getting at

0 one
1 two

00 one
01 two
10 three
11 four

000 one
001 two
010 three
011 four
100 five
101 six
110 seven
111 eight
.
.
9999999........

They are defined for all n, ie, they are defined for n+1 when they are defined for n, ie, in the limit as n -> infinity, when they represent infinite divisions of a line or, with a period in front of them the real numbers
(without some rationals like 1/3)* in [0,1), or as another notation for the natural numbers (all of them as n -> infinity). In any event, countable.

* As explained in post #55

Last edited by zylo; July 7th, 2018 at 11:28 AM.

 July 7th, 2018, 12:01 PM #59 Global Moderator   Joined: Dec 2006 Posts: 20,747 Thanks: 2133 Perhaps you hadn't noticed the word "without". My question remains: how would counting be done without a number system? I'm not asking how you would use a number system to do counting. I didn't quite understand your example anyway... which column showed what you were counting, given that neither column seems to contain 1, 2, 3, 4, ... (continued without end)?
 July 7th, 2018, 12:19 PM #60 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 Sticks, stones, scratches on a rock, etc I was counting n-place binary representations, countable for all n through n approaches infinity, as proved by induction.

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