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January 22nd, 2019, 09:34 AM  #1 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Decimals and the Continuum
To define the real numbers in [0,1) list the natural numbers, put a period after them, and read them in reverse. Call them "decimals" 1. > .1 2. > .2 . 10. > .01 . The decimals can be associated uniquely with points on the line (the continuum) as follows: Given a line segment, label its end points 0 and 1. Choose a point. Divide the line into ten closed intervals. If the point is, say, in third interval, S$\displaystyle _{1}$ =.2. Divide third interval into ten closed intervals. If point is in seventh interval, S$\displaystyle _{2}$=.26. Continue the process until the point is on an interval, which ends the sequence, or you have an infitite decimal (nested sequence} which uniquely defines the point. To find the point corresponding to an infinite decimal. Divide the line into 10 closed intervals. If S$\displaystyle _{1}$=6, the point is in the fifth closed interval. Continue in this manner to define an infinite nest which uniquely defines a point on the line. If S$\displaystyle _{n}$ is finite, the point is at the end of an interval. The above can be repeated for any radix to associate uniquely with every point on the line a radix fraction. If radix =2, Then any fraction in [0,1) is of the form .100110..., where each place is determined by a nest of intervals, divide the line in half. Then divide the interval in which the point resides in half again, and repeat till you get a finite or unending binary fraction. EDIT: The fact that decimals are a field follows from the fact that they are a field for n decimal places for all n. Last edited by skipjack; January 22nd, 2019 at 02:10 PM. 
January 22nd, 2019, 09:45 AM  #2 
Senior Member Joined: Oct 2009 Posts: 906 Thanks: 354 
Why the need to make a new thread on it. Doesn't this fit perfectly in your other thread?

January 22nd, 2019, 10:09 AM  #3 
Banned Camp Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 126  Foundations of Analysis. 
January 22nd, 2019, 10:23 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,697 Thanks: 2681 Math Focus: Mainly analysis and algebra 
This has nothing to do with the foundations of analysis. It has nothing much to do with mathematics.

January 22nd, 2019, 10:38 AM  #5 
Senior Member Joined: Oct 2009 Posts: 906 Thanks: 354  
January 22nd, 2019, 11:17 AM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,975 Thanks: 1157 Math Focus: Elementary mathematics and beyond 
At some point, you're going to have to admit the existence of natural numbers with an infinite amount of digits. Your ideas keep "getting buried" because noone is agreeing with what you are saying. Is this just a remake of your assertion that the reals form a countable set? In the interest of keeping this a discussion (as opposed to a dissertation) this thread is closed. Please continue the discussion in your previous thread. Last edited by greg1313; January 22nd, 2019 at 11:45 AM. 
January 22nd, 2019, 02:57 PM  #7 
Global Moderator Joined: Dec 2006 Posts: 21,110 Thanks: 2326  

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