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April 8th, 2016, 12:49 PM   #1
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Real Number as Binary Sequence

Binary sequence: r =.011.... = p1×(1/2)^1 + p2×(1/2)^2 + p3×(1/2)^3 + ......

Let r be in L= [0,1)

1) divide L by 2
p1=0 if r in L1=[0,1/2], r1 = .0

2) divide L1 by 2
p2 =1 if r in [1/4,1/2], r2 = .01

3) divide L2 by 2
p3=1 if r in [3/8,1/2], r3 = .011
......

r=.011......... = lim rn

The digits in r are countable (countably infinite).

Last edited by skipjack; April 8th, 2016 at 07:10 PM.
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April 8th, 2016, 12:55 PM   #2
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Of course the digits in a real number are countably infinite! They are indexed by the natural numbers in the form of the indices on the place values $2^{-1},2^{-2},2^{-3},\ldots$.

At least you've got something right for once. The only problem is that you think it's some sort of groundbreaking discovery, which probably means there's something you are misunderstanding and that this will be followed by something that doesn't make any sense at all.

I'm not sure that your proof makes an awful lot of sense though.

Last edited by skipjack; April 8th, 2016 at 06:47 PM.
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April 8th, 2016, 01:39 PM   #3
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Quote:
Originally Posted by v8archie View Post
Of course the digits in a real number are countably infinite! They are indexed by the natural numbers in the form of the indices on the place values $2^{-1},2^{-2},2^{-3},\ldots$.

At least you've got something right for once. The only problem is that you think it's some sort of groundbreaking discovery, which probably means there's something you are misunderstanding and that this will be followed by something that doesn't make any sense at all.

I'm not sure that your proof makes an awful lot of sense though.
How do you know:
1) A real number can be expressed as a countably infinite series of binary digits?
2) How to find the digits?

See OP for the answers.

I note that the set of all countably infinite binary sequences enumerates into a countably infinite list, not a finite one.

EDIT:
The process in OP also shows that every countably infinite binary sequence is a real number.

Last edited by skipjack; April 8th, 2016 at 07:06 PM.
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April 8th, 2016, 04:49 PM   #4
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Quote:
I note that the set of all countably infinite binary sequences enumerates into a countably infinite list, not a finite one.
Wrong - it's an uncountably infinite set - you are back to where you erred in the first place.

Last edited by skipjack; April 8th, 2016 at 07:07 PM.
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April 8th, 2016, 05:56 PM   #5
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So you can do division. Well done.

You are wrong to say that the process shows that every infinite binary sequence can be identified with a real number because your process doesn't produce every infinite binary sequence. In particular, all those ending with an infinite string of 1s are missing. Even more specifically 1111… is missing because 0.1111…= 1 which is specifically omitted from the interval from which L is taken.

This thread is a mixture of the blindingly obvious and the wrong (mostly where the obvious turns out to be false).
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April 8th, 2016, 07:38 PM   #6
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Quote:
Originally Posted by zylo View Post
I note that the set of all countably infinite binary sequences enumerates into a countably infinite list, not a finite one.
You merely assert that now, without having given any such enumeration.

Quote:
Originally Posted by v8archie View Post
You are wrong to say that the process shows that every infinite binary sequence can be identified with a real number...
But zylo didn't say that.

Quote:
Originally Posted by v8archie View Post
In particular, all those ending with an infinite string of 1s are missing.
It's mostly the other way round - they are mostly present, whereas .1000000... is missing.
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April 8th, 2016, 07:53 PM   #7
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Quote:
Originally Posted by skipjack View Post
But zylo didn't say that.
No, he said is.
Quote:
The process in OP also shows that every countably infinite binary sequence is a real number.
Quote:
Originally Posted by skipjack View Post
It's mostly the other way round - they are mostly present, whereas .1000000... is missing.
My mistake. My point still stands, but in reverse. 0.1000... and 0.01000... etc. are all missing.
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April 8th, 2016, 08:10 PM   #8
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But zylo didn't say "is a unique real number".
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April 8th, 2016, 08:19 PM   #9
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No, but he's not generating every sequence even though he is using every real number. So the process doesn't show that every sequence is a real number.
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