My Math Forum An Anomaly of Decimal Representation

 Real Analysis Real Analysis Math Forum

July 6th, 2018, 06:47 AM   #41
Math Team

Joined: Dec 2013
From: Colombia

Posts: 7,622
Thanks: 2611

Math Focus: Mainly analysis and algebra
Quote:
 Originally Posted by zylo The answer is that in the limit of infinite decimal places there is no number between them though they remain distinct.
No they don't. The real number literally is the limit. You can't do mathematics if you ignore definitions.

July 6th, 2018, 07:09 AM   #42
Global Moderator

Joined: Dec 2006

Posts: 20,375
Thanks: 2010

Quote:
 Originally Posted by zylo 2) Induction is the best nature gives you. Sure, you can imagine the moon is made of green cheese, or beyond the beyond.
Induction isn't "given by nature". You are avoiding responding to the assertion that you misused induction, and thereby used flawed reasoning.

Quote:
 Originally Posted by zylo Maschke, Sorry, to me trans=finite is double=talk.
If you don't like that terminology, just read on a bit to find that Maschke was referring to oridnals or cardinals that aren't finite.

Quote:
 Originally Posted by zylo There is no trans-finite
It's unreasonable to object to "trans-finite" as double-talk and then use the term. Induction isn't a limiting process, yet you readily state "in the limit" in a context where all you have is that something holds for each positive integer n. The wording "for all n" isn't "a little flaky", it's just an alternative wording. If you're unaware of a formal definition of "trans-finite" or "transfinite", that doesn't imply that such a definition isn't possible.

Quote:
 Originally Posted by zylo . . . in the limit of infinite decimal places there is no number between them though they remain distinct.
Wouldn't that allow them to be treated as distinct representations of the same number, making it trivial that there isn't a number strictly between them?

Quote:
 Originally Posted by zylo . . . one (you) might conceive that there are more points on a line than you can count with a number system.
What do you mean by "more points than you (or anyone) can count with a number system"? How would counting be done without a number system?

July 6th, 2018, 08:57 AM   #43
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 124

Quote:
 Originally Posted by skipjack Induction isn't "given by nature". You are avoiding responding to the assertion that you misused induction, and thereby used flawed reasoning.
It's a natural, logical, way to refer to everything you can associate with a natural number.

Quote:
 Originally Posted by skipjack If you don't like that terminology, just read on a bit to find that Maschke was referring to oridnals or cardinals that aren't finite.
There is nothing beyond induction. That's the limit of a precise statement of "infinity."

Quote:
 Originally Posted by skipjack It's unreasonable to object to "trans-finite" as double-talk and then use the term. Induction isn't a limiting process, yet you readily state "in the limit" in a context where all you have is that something holds for each positive integer n. The wording "for all n" isn't "a little flaky", it's just an alternative wording. If you're unaware of a formal definition of "trans-finite" or "transfinite", that doesn't imply that such a definition isn't possible.
I agree, you can define anything, like trans-trans-finite, or beyond the beyond. Trans-finite only arose because Cantor didn't know how to count binary representations of the natural numbers.

Trans-finite is an imaginary concept which serves no purpose other than to throw a wrench into the gears while pretending to be logical, mathematical, and unassailable. None of the above. But it impresses the hell out of freshmen. I intentionally didn't say freshwomen- they're not that easily impressed.

Quote:
 Originally Posted by skipjack Wouldn't that allow them to be treated as distinct representations of the same number, making it trivial that there isn't a number strictly between them?
.33..... is never 1/3, but the difference approaches 0.

Quote:
 Originally Posted by skipjack What do you mean by "more points than you (or anyone) can count with a number system"? How would counting be done without a number system?
"more points may exist on a line than you (or anyone) can count with a number system"

Quote:
 Originally Posted by v8archie No they don't. The real number literally is the limit. You can't do mathematics if you ignore definitis.
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. Think of limit of a nest of closed intervals.

Decimal or any radix definition works fine for anything I can think of in analysis or topology (completeness, neighborhood, nearness of points, etc).

July 6th, 2018, 09:32 AM   #44
Senior Member

Joined: Oct 2009

Posts: 753
Thanks: 261

Quote:
 Originally Posted by zylo .33..... is never 1/3, but the difference approaches 0.
In the same way the difference between 2 and 3 approaches 1?

July 6th, 2018, 09:35 AM   #45
Senior Member

Joined: Oct 2009

Posts: 753
Thanks: 261

Quote:
 Originally Posted by zylo "more points may exist on a line than you (or anyone) can count with a number system"
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.

 July 6th, 2018, 10:46 AM #46 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 Seems we haven't defined a line. A line is all n-place decimals, or radixes, for all n. It can be pictured physically as the points on a light ray between two points divided into 1/10^n segments, for all n. It follows that m(1/m) = 1 for all m, m=10^n, and $\displaystyle \infty 0$=1. Edit. If you like, as the number of sub-divisions becomes larger, the numbers (points on the line) become “real.” That’s about the best you can do without yoga. Last edited by zylo; July 6th, 2018 at 11:36 AM.
July 6th, 2018, 11:59 AM   #47
Senior Member

Joined: Aug 2012

Posts: 2,209
Thanks: 651

Quote:
 Originally Posted by zylo A line is all n-place decimals, or radixes, for all n.
Your "line" consists only of terminating decimals. Not much a line. It's full of holes and doesn't even include all the rationals.

July 6th, 2018, 12:43 PM   #48
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 124

Quote:
 Originally Posted by Maschke Your "line" consists only of terminating decimals. Not much a line. It's full of holes and doesn't even include all the rationals.
You missed “for all n.” Granted, the interval [0,1/3] doesn’t exist (in decimal or binary notation) unless you interpret 1/3, or pi say, as endless decimals. It’s “for all n” that gives you the reals (the holes) and lets you use the shortcuts 1/3 or pi.

I know, I know, no matter what n is it’s a rational number. But that’s just semantics. As time goes by, your division construction fills up all the holes.

And where is the line you use as a yardstick for your numbers? It doesn’f exist except as a line of molecules, so you have to define it.

July 6th, 2018, 12:57 PM   #49
Senior Member

Joined: Aug 2012

Posts: 2,209
Thanks: 651

Quote:
 Originally Posted by zylo You missed “for all n.” Granted, the interval [0,1/3] doesn’t exist (in decimal or binary notation) unless you interpret 1/3, or pi say, as endless decimals. It’s “for all n” that gives you the reals (the holes) and lets you use the shortcuts 1/3 or pi.
By "for all n" I assume you mean for all natural numbers n. In that case I reiterate my remark. Your line consists only of the terminating rationals, which are few and far between in the scheme of things.

If you mean something else by "for all n" please say what you mean.

Quote:
 Originally Posted by zylo I know, I know, no matter what n is it’s a rational number. But that’s just semantics. As time goes by, your division construction fills up all the holes.
You are contradicting yourself. Just semantics? Your definition of a line consists of only terminating rationals.

Quote:
 Originally Posted by zylo And where is the line you use as a yardstick for your numbers? It doesn’f exist except as a line of molecules, so you have to define it.
Confusing math with physics.

The "real line" is defined as the set of real numbers.

As Micrm@ss noted, there are other philosophical conceptions of line. Are you referring to one of those alternate ideas?

Last edited by Maschke; July 6th, 2018 at 01:10 PM.

July 6th, 2018, 01:59 PM   #50
Banned Camp

Joined: Mar 2015
From: New Jersey

Posts: 1,720
Thanks: 124

Quote:
 Originally Posted by Maschke By "for all n" I assume you mean for all natural numbers n. In that case I reiterate my remark. Your line consists only of the terminating rationals, which are few and far between in the scheme of things. You are contradicting yourself. Just semantics? Your definition of a line consists of only terminating rationals. The "real line" is defined as the set of real numbers. As Micrm@ss noted, there are other philosophical conceptions of line. Are you referring to one of those alternate ideas?
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.

If it makes you happy, every time you divide the line 10^n times, divide it 10^(n+1) times.

And of course the question, what is your definition of a real number?

Some people see behind the beyond. Most people don't even see beyond the behind. Couldn't resist that old joke.

 Tags anomaly, decimal, representation

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post zylo Real Analysis 14 October 22nd, 2016 11:19 AM zylo Real Analysis 87 October 19th, 2016 08:55 PM mhhojati Linear Algebra 0 November 1st, 2015 11:37 PM John Creighton Number Theory 2 March 14th, 2011 10:03 AM demipaul Linear Algebra 2 November 19th, 2009 05:42 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top