June 28th, 2018, 04:49 AM  #21  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,647 Thanks: 119  Quote:
The theorem on term by term multiplication only applies if you think of "infinite decimals" as limits rather than sums. A sum does not have a limit, it depends on the number of terms. A sum is literal. Limit is a defined $\displaystyle \epsilon$ $\displaystyle \delta$ operation on a sum. .3333..... to n terms is a sum which depends on n. You can satisfy all the axioms of an ordered field with the operations of elementary algebra without ever thinking about limits. The axiom on lub forces you to think about limits, as pointed out in OP. EDIT In this context note my last post: Last edited by skipjack; June 28th, 2018 at 06:28 AM.  
June 28th, 2018, 05:32 AM  #22 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,647 Thanks: 119 
Everything works out simply and transparently if you list all the decimals in [0,1) to nplaces as: .00.......0 .00.......1 .00.......2 ............. .9999999 for all n. Add, multiply. and divide any two members of the list by the rules of elementary algebra and you get another member of the list. If you like, you can label repeating decimals in the list as unique rational numbers. Last edited by zylo; June 28th, 2018 at 05:46 AM. Reason: Add .00.......0 
June 28th, 2018, 06:12 AM  #23  
Senior Member Joined: Oct 2009 Posts: 713 Thanks: 238  Quote:
Sure, I don't necessarily disagree with you. But this would be a decimal representation of the number of length (the class) of all ordinals. Or something of length $\omega+1$. This is not necessarily a wrong or bad view, but it is nonstandard. What people mean with the standard decimal expansion is one of length $\omega$. So you might be right if you mean other kind of expansions (I haven't checked, but it seems plausible), but that's just not what people usually mean when they say decimal expansion.  
June 28th, 2018, 09:10 AM  #24 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,647 Thanks: 119 
The list is archetypal. It gives every real number, as defined axiomatically, a unique symbol (representation). It is quite clean except for the mystical "for all n." You can interpret members as limits of sums, such as repeating decimals, and you can note that any member that ends in 0's can be summed to a rational number, or call .49999.... .5, or note 1/2 is in the list but 1/3 isn't, but none of that changes the list, nor does it create a system of real "numbers." The real numbers are the list. 
June 28th, 2018, 09:49 AM  #25 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895 
What do you mean by "The list is archetypal. It gives every real number, as defined axiomatically, a unique symbol (representation)." when you then say "1/2 is in the list but 1/3 isn't"? If 1/3 is not in the list then how does this list give "every real number" a unique symbol? 
June 28th, 2018, 10:10 AM  #26  
Senior Member Joined: Oct 2009 Posts: 713 Thanks: 238  Quote:
 
June 28th, 2018, 10:28 AM  #27 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,647 Thanks: 119 
You can assign rational numbers to the list: Sum .5000000..... to 1/2 Assign 1/3 to .3333........ You can limit sum .49999.......= $\displaystyle .4\bar{9}$ to .5, just as you can millions of numbers before it, it makes no sense. You can assign $\displaystyle .5\bar{0}$ to $\displaystyle .4\bar{9}$. Why bother, $\displaystyle .5\bar{0}$ is already on the list. The list is a primal realization of the real numbers in [0,1). 
June 28th, 2018, 11:14 AM  #28  
Senior Member Joined: Oct 2009 Posts: 713 Thanks: 238  Quote:
Last edited by skipjack; June 28th, 2018 at 02:18 PM.  
June 28th, 2018, 12:41 PM  #29  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,647 Thanks: 119  Quote:
Divide out any rational number to get its unique decimal expansion. Use its decimal expansion to find it's decimal representation on the list, which contains all decimal representations once. 1/3=.3333.......... 1/2=.5000.......... I prefer to call the members of the list decimal representations to reflect how the list was constructed, which wasn't by decimal expansion. I said 1/3 wasn't on the list in the sense that there was no term on the list whose decimal representation summed to 1/3. Sum of .33333333...... isn't 1/3. I was being careful about the difference between sum and limit. There are lots of members on the list whose limit is 1/3. There is only one member of the list whose representation sums to 1/2. Last edited by skipjack; June 28th, 2018 at 02:17 PM.  
June 28th, 2018, 02:20 PM  #30 
Senior Member Joined: Aug 2012 Posts: 2,139 Thanks: 623  Zylo, What do you mean by this? Surely 3/10 + 3/100 + 3/1000 + ... is a geometric series that sums to 1/3, isn't it?


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