January 21st, 2019, 09:48 AM  #31  
Senior Member Joined: Jun 2014 From: USA Posts: 505 Thanks: 39  This is the funny part : Quote:
I tell you what zylo... I'll agree with your definition of the reals in $(0,1)$ based on their decimal expansions if you clean up your error involving reals of the form $\frac{n}{10^m}$, where $n,m \in \mathbb{N}$.  
January 21st, 2019, 10:50 AM  #32  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
Definition of real numbers in [0,1): List the natural numbers, put a period after them, and read them in reverse. 1. > .1 2. > .2 . 10. > .01 . 95141. > .14159 And I don't have the slightest idea what you mean by "cleanup" What is your definition of the real numbers, or what was the point of bringing up Dedekind cuts? Edit: Actually, I did answer the OP, specifically and succinctly: I now realize it is a very good question. Thanks. Last edited by zylo; January 21st, 2019 at 11:34 AM.  
January 21st, 2019, 12:26 PM  #33  
Senior Member Joined: Jun 2014 From: USA Posts: 505 Thanks: 39  Quote:
If you instead define the set $A = \{0,1,2,3,4,5,6,7,8,9\}$ and then define each real number in $(0,1]$ as an infinite string of the form: $$0.x_1x_2x_3\dots, \text{ where each } x_i \in A \text{ such that no string contains a } j \in \mathbb{N} \text{ where all } x_{p \geq j} = 0$$ then we'll be much closer. It's sloppy still and no real has a finite expansion, but hey. Maybe you could clean up the above a little bit. Last edited by AplanisTophet; January 21st, 2019 at 12:37 PM.  
January 21st, 2019, 12:49 PM  #34 
Global Moderator Joined: Dec 2006 Posts: 20,467 Thanks: 2038  You've consistently been unable to provide that definition without being unable to account for 1/9, for example. When I've asked about that, you've never done anything better than repeat the socalled definition (sometimes with subtle unexplained changes) without answering my question. Often, you don't reply to my specific questions at all.

January 22nd, 2019, 08:12 AM  #35  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125 
Once again, the decimal representation of 1/9 is .11... where the sequence of 1's is endless. It is a number in the sequence: Quote:
EDIT Back to OP A cut for an irrational (endless) decimal is an infinite sequence of cuts: Define a cut for $\displaystyle S_{n}=\sum_{i=1}^{n}a_{i}, \quad a_{i}=0,1,...,9$ for all n. Of course each step in the process is a rational number but the result is cut of an irrational number. A cut is really a meaningless figment of the imagination, but as an isolated concept interesting. It assumes you can divide all the rational numbers (endless) into two sets A and B st every member of A is less than every member of B and A has no largest member. You can of course assume it can be done. It only becomes insidious when it intrudes on analysis and casts an impenetrable fog of obfuscation and cripples the ability to think and understand. A rational version of decimals and the real numbers is given here: Decimals and the Continuum Last edited by zylo; January 22nd, 2019 at 08:48 AM.  
January 22nd, 2019, 09:21 AM  #36 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,634 Thanks: 2620 Math Focus: Mainly analysis and algebra  
January 22nd, 2019, 09:42 AM  #37 
Senior Member Joined: Jun 2014 From: USA Posts: 505 Thanks: 39  
January 22nd, 2019, 09:46 AM  #38 
Global Moderator Joined: Dec 2006 Posts: 20,467 Thanks: 2038  
January 22nd, 2019, 12:37 PM  #39  
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 125  Quote:
1/9 is the infinite (unending) decimal .111......., by standard (High School) division. This is an example of how one arrives at decimal representation. What is the infinite (unending) decimal .111.....? An infinite decimal whose sum evaluated to n places gets arbitrarily close to 1/9 as n increases indefinitely. The comparison can be made because .1111 .. . evaluated to n places is a rational number. What is 1/9 in radix 9 notation? .1: Divide a unit line into nine intervals. 1/9 is end of first interval. That's the way a ruler works: Divide the standard inch in half: 1/2 Divide each interval in half: 1/4 Divide each interval in half again: 1/8 ................................................ A carpenter doesn't deal in decimals. His ruler (a tape) is divided in 1/32's. He works to 1/32's or maybe 1/16's. A cabinet maker would work to 1/64th's. His ruler is probably a yard stick graduated to 1/64th's. A machinist deals in thousandths. His ruler is a micrometer or vernier caliper which reads to three (.001) or four (.0001) decimal places. There is constant conversion in practice: 1/16 = .0625 for example To convert Pi to a decimal representation: Take a circle of diameter 1. Then unwind the circle onto a line with same unit 1. The decimal representation of the point where the line ends can be found as in: Decimals and the Continuum  
January 22nd, 2019, 12:41 PM  #40 
Senior Member Joined: Aug 2012 Posts: 2,255 Thanks: 681  Aren't you claiming that every decimal expansion is derived from reversing the digits of a positive integer? How does .111... get generated in your system?


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