May 28th, 2016, 06:49 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,399 Thanks: 100  Infinite Decimal is a Natural Number
Definitions: Infinite=countably infinite Decimal=whole decimal An infinite decimal is one defined for any number n of decimal places. Ex: fractional part of pi considered as a decimal: 1415926......... A finite decimal is only defined for n decimal places: Ex: 1845 It follows that any decimal, finite or infinite, is a natural number. The above is true for any radix, such as binary (replace "decimal" with "binary"). EDIT: It follows that any real number is countable because either side of the period of its complete decimal representation is 1:1 with a natural number. Last edited by skipjack; May 31st, 2016 at 06:49 AM. 
May 28th, 2016, 07:21 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,342 Thanks: 2465 Math Focus: Mainly analysis and algebra 
That's just wrong, as you were shown yesterday. Are you not able to read? With what did you disagree? And why are you not able to converse about such points in that thread? Do you imagine that starting a new thread somehow makes these arguments less stupid?
Last edited by skipjack; May 31st, 2016 at 06:51 AM. 
May 28th, 2016, 07:23 AM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 
First, you don't appear to know what the word "definition" means! You cannot define a word in terms of itself so you cannot define "infinite" as "countably infinite" nor "decimal" as "whole decimal"! I expect that you mean that you are using the word "infinite" in the limited sense of "countably infinite". I don't know what a whole decimal could be. Do you mean just a "whole number"? "Decimal" refers to writing a number in a particular way, it is not a type of number. Also you say "an infinite decimal is one defined for any number n of decimal places" and "a finite decimal is only defined for n decimal places". Those are exactly the same! I assume you meant "an infinite decimal is one with no limit on the number of decimal places". But a very simple finite decimal such as 0.5 is NOT a "natural number" by the usual definition of "natural number". You appear to be consistently using words whose specific definitions you don't know. You may have meant "a natural number divided by some power of 10". That would be true if it were correct that "an infinite decimal is one defined for any number n of decimal places" which, grammatically means a specific number, n, of decimal places. However, that is not correct. The rational number, 1/3, which in decimal form is 0.333333... is not any whole number divided by a power of 10 (and certainly is not a "whole number" itself). If this is not all some kind of joke (and I am still wondering about that) then you would do well to look up specific definitions of such things as "natural number", "whole number", etc. 
May 28th, 2016, 07:43 AM  #4 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,399 Thanks: 100 
infinity = countably infinite is a semantic definition to save repetition, and also as a reminder that there is no other definition of infinity. decimal= whole decimal to save repetition. A decimal number consists of a whole portion and a fractional portion. I would think that decimal in the context of my OP would be clear, but I didn't want to take a chance. In general, "decimal" is ambiguous. "any n" and "n" are not the same thing. It is a critical distinction. See Peano Axioms for definition of natural number. I assert there is a 1:1 correspondence between fractional portion of any decimal and the natural numbers established by assigning to the fractional portion the whole number defined by its digits. .141593.... > 141593.... Last edited by zylo; May 28th, 2016 at 07:58 AM. Reason: add Peano Axioms, 1:1 crrespondence 
May 28th, 2016, 10:03 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,342 Thanks: 2465 Math Focus: Mainly analysis and algebra 
If you understood anything about the subject you would know that "Infinite=countably infinite" is utterly redundant in the context. All infinite decimals have a countably infinite number of digits. It's still not clear what you mean by "whole decimal". You explanation talks about whole numbers, but your op talks about the fractional parts of decimals. If you don't want "any $n$" and "$n$", you should define what they do mean. Your statement that "an infinite decimal is defined for any number $n$ of decimal places" is false. An infinite decimal does not have $n$ decimal places for any natural number $n$, it has an infinite quantity of decimal places. It certainly can't have every possible number of decimal places. A decimal can't have both 2 and 3 decimal places. $0.14 \ne 0.141$ they are two different numbers. We can approximate an infinite decimal to any finite number of decimal places, but we can do that for finite decimals too (although most such approximations are completely accurate). Your assertion is utter nonsense that simply repeats the same claim you have made many times before, that there exists a natural number with an infinite quantity of digits. Not only have other people shown you proofs that this is incorrect, but you youself have accepted the proofs and stated that all natural numbers are finite. If you don't understand that your assertion contradicts this, ask why that is the case or put a counter argument in the thread where the point is being discussed. DON'T SIMPLY POST THE SAME STUPID CLAIM YET AGAIN as if nobody had said given any counterargument to it. Repeating it won't make it any more true. 
May 31st, 2016, 04:21 AM  #6 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,399 Thanks: 100 
If ri is a decimal digit, r1r2.....rn is a natural number for all (any) n. .r1r2.....rn is a decimal fraction for all (any) n. 1:1 correspondence between decimal fractions and natural numbers: 0$\displaystyle \leq$x<.1 add .1 and remove decimal point: .00001...>.10001>10001 .1$\displaystyle \leq$x<1 remove decimal point: .14159....>14159..... .333333.....>3333.... Remove trailing 0's. 
May 31st, 2016, 07:06 AM  #7  
Global Moderator Joined: Dec 2006 Posts: 19,289 Thanks: 1681  Quote:
The axiom of induction (which is one of the Peano axioms) effectively means that the list 1, 2, 3, 4, . . . (continued without end) contains all the positive natural numbers. As, for example, 333333... (where "..." indicates nonterminating), isn't in that list, it's not a natural number. Hence your attempt to regard 333333... as a natural number is in conflict with the Peano axioms. You haven't defined what "all (any)" means, and you haven't explained what "n" is. As you are attempting to define a natural number, you can't require n to have a value that is a natural number, as that would make your definition "circular".  
May 31st, 2016, 08:23 AM  #8  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,342 Thanks: 2465 Math Focus: Mainly analysis and algebra  Quote:
Not to mention that, yet again, you have repeated the same false claim that 333.... is a natural number. As usual, it's a claim without any proof or justification and you have completely ignored all representations that have been made to the contrary. This means that we can't really hope to explain properly why you are wrong, because you won't say which part of the proofs you have been given you believe to be wrong or don't understand. Despite $\color{#039}{\bf{skipjack}}$'s entirely merited comments on your definition, I believe that I can see what you are attempting to do in your statements. Quote:
In a similar vein, $0.r_1r_2r_3 \ldots r_n$ is a number between zero and 1 for all $n$. But this does not define all such fractions, because there are fractions such as $\frac13$ and $(\pi  3)$. The fractions that your statement talks about are those consisting of finite (and thus terminating) strings of decimal digits. Whereas $\frac13$ and $(\pi  3)$ have infinite (and thus nonterminating) strings of decimal digits. Last edited by skipjack; June 4th, 2016 at 10:25 PM.  
June 3rd, 2016, 07:27 AM  #9 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,399 Thanks: 100 
.333... is defined for n digits, ALL n. The association .333.. to the natural number 333... is valid for n digits, ALL n. Last edited by zylo; June 3rd, 2016 at 07:35 AM. Reason: removed sum analogy 
June 3rd, 2016, 08:03 AM  #10  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,342 Thanks: 2465 Math Focus: Mainly analysis and algebra  Quote:
How can a single number have lots of different quantities of digits? You are saying that this number has 3 digits, 27 digits, 934 digits and 27 million digits. Which is it? Or does it just have an infinite quantity of digits? A number with 3 digits terminates after those three digits: .125 A number with 27 digits terminates after those twentyseven digits: .125673491825764986152379451 But a number with an infinite quantity of digits does not terminate. You apparently wish to associate $\frac13$ with $\{3, 33, 333, 3333, 33333, \ldots\}$. Why?  

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