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May 27th, 2016, 08:32 AM   #21
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Originally Posted by v8archie View Post
Yes. The irrationals are uncountable.
That was actually in a final exam past paper at my school
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May 27th, 2016, 09:03 AM   #22
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Associate with each number to the right of the decimal point a number to the left of the decimal point. Now forget about the decimal point. These are natural numbers: 1,2,3,4,....... which include all repeating and non-repeating sequences of digits.

So the question is, can you associate any number in the sequence of natural numbers with its decimal number (not fraction) representation? Yes
Can you associate any decimal number (not fraction) with the natural number it represents? Yes.

V8archie and skipjack point of view: Pi is not a sequence of natural numbers, it is the limit of a sequence of natural numbers. I understand that.

But are there enough natural numbers to uniquely define the sequence, ie, make the limit a countable number?
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May 27th, 2016, 09:07 AM   #23
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Originally Posted by 123qwerty View Post
That was actually in a final exam past paper at my school
I hope you didn't use Cantor.
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May 27th, 2016, 09:29 AM   #24
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Originally Posted by zylo View Post
Associate with each number to the right of the decimal point a number to the left of the decimal point. Now forget about the decimal point. These are natural numbers: 1,2,3,4,....... which include all repeating and non-repeating sequences of digits.
What is this supposed to mean? What is this mapping? It's not a bijection because a real number has a finite integer part and a (possibly) infinite number of digits after the decimal point.

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Originally Posted by zylo View Post
So the question is, can you associate any number in the sequence of natural numbers with its decimal number (not fraction) representation?
Again, what is this association?

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Originally Posted by zylo View Post
Can you associate any decimal number (not fraction) with the natural number it represents?
And here? What natural number does a decimal number represent? How is a decimal number not a fraction? Try using the proper terminology and instead of just saying "yes", provide a proper proof of your claims.

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Originally Posted by zylo View Post
V8archie and skipjack point of view: Pi is not a sequence of natural numbers, it is the limit of a sequence of natural numbers. I understand that.
That's nothing like what I claim. You misunderstand. The digits of the decimal sequence of $\pi$ are not a limit. They are an infinite (non-terminating) sequence. Most sequences of natural numbers have no limit, the digits of $\pi$ certainly don't.

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Originally Posted by zylo View Post
But are there enough natural numbers to uniquely define the sequence, ie, make the limit a countable number?
Again, there is no limit here.

Last edited by skipjack; May 27th, 2016 at 02:32 PM.
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May 27th, 2016, 10:10 AM   #25
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Quote:
Originally Posted by v8archie View Post
Quote:
Originally Posted by zylo View Post
V8archie and skipjack point of view: Pi is not a sequence of natural numbers, it is the limit of a sequence of natural numbers. I understand that.
That's nothing like what I claim. You misunderstand. The digits of the decimal sequence of $\pi$ are not a limit. They are an infinite (non-terminating) sequence. Most sequences of natural numbers have no limit, the digits of $\pi$ certainly don't.
Thank goodness. That had me worried.

As for the rest:

We are talking about decimal representation of numbers. By decimal fraction I mean .xxxx, by decimal number xxxx. I was trying to save on verbiage.

.1845 <-> 1845

Given any decimal representation of a natural number, say .14159... to n places, does it occur in the sequence of natural numbers for any n? I note the natural numbers have no limit.

Last edited by skipjack; May 27th, 2016 at 02:44 PM.
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May 27th, 2016, 10:25 AM   #26
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Every finite sequence of decimal digits appears in at least one natural number (and there is an obvious bijection of finite sequences of decimal digits with the natural numbers).

But there is no infinite sequence of decimal digits in any natural number. $\pi$ itself is the limit of a sequence of rational numbers, but the digits of $\pi$ do not form a convergent sequence, even though the sequence is infinite.

It is incorrect to talk of limits in this context. Limits are expressions of potential infinities. We are dealing with actual infinities.

Last edited by skipjack; May 27th, 2016 at 02:45 PM.
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May 27th, 2016, 10:50 AM   #27
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Given pi = 3.1415926... , I am claiming that the natural number whose decimal representation is 1415926... to n places exists for every n.

Last edited by skipjack; May 27th, 2016 at 02:45 PM.
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May 27th, 2016, 11:16 AM   #28
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Yes. Every finite $n$. But that's not the whole sequence because the sequence is infinite.
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May 27th, 2016, 11:38 AM   #29
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f(n) =14159.. to n places.

https://en.wikipedia.org/wiki/Sequence
"Formally, a sequence can be defined as a function whose domain is either the set of the natural numbers (for infinite sequences) or the set of the first $n$ natural numbers (for a sequence of finite length $n$)."

Last edited by skipjack; May 27th, 2016 at 02:54 PM.
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May 27th, 2016, 11:54 AM   #30
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I know what a sequence is. Every term in your sequence is finite. It has $n$ digits where $n$ is a natural number, and therefore finite. So which term of this sequence of terms of finite length do you claim is equal to the infinite sequence of decimal numerals that represents the fractional part of $\pi$?

Which natural number $n$ is infinite? There isn't one, from the definitions and proofs provided before now.
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