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October 3rd, 2018, 01:33 AM   #81
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Quote:
Originally Posted by zylo View Post
The natural numbers are defined by induction.

If 222...2 is a natural number for n digits, it is a natural number for n+1 digits -> it is a natural number for all digits. ie,

222...…. is a natural number

Corollary:

ANY sequence of digits (binary, decimal, ) is a natural number.
I asked you earlier if you know the difference between a polynomial and a power series. You didn't respond. I'm curious as to whether you know the difference. Consider that every polynomial over the complex numbers has as many complex roots as its degree. That's the Fundamental Theorem of Algebra. In other words the polynomial $\displaystyle \sum_{n=0}^k c_n z^n$ where $k$ is a positive integer, must necessarily have $n$ complex roots (up to multiplicity). On the other hand an "infinite polynomial," or power series, may have no roots at all. For example $\displaystyle e^z = \sum_{n = 0}^\infty \frac{1}{n!} z^n$ has no zeros.

Another example along the same lines is that a finite sum of rational numbers, no matter how long, must be rational. But an infinite sum of rationals may be irrational.

The distinction between arbitrarily long finite strings and infinite strings is significant in many different mathematical contexts.

Do you find these examples to be of interest?

Last edited by Maschke; October 3rd, 2018 at 01:42 AM.
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October 3rd, 2018, 04:41 AM   #82
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List of binary and decimal representation of ALL The Natural and Real [0,1) Numbers:

0-----0
1-----1
2-----01
3-----11
4-----001
........
........

Comments:
1) The list is endless and contains ALL imaginable strings of binary and decimal digits.
2) Adding a radix point before the binaries and after the natural numbers gives all real numbers in [0,1) in binary and decimal form.
5000 becomes 5000., or more commonly, .0005

Like all great ideas, it is beautifully simple, transparent, foundational, and can be understood by almost everyone, making it part of the world's cultural heritage.
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October 3rd, 2018, 04:55 AM   #83
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Quote:
Originally Posted by zylo View Post
Like all great ideas, it is beautifully simple, transparent, foundational, and can be understood by almost everyone, making it part of the world's cultural heritage.
Pity it's complete nonsense, as has been shown in the preceding comments (which you conveniently seem to have ignored).
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October 3rd, 2018, 05:26 AM   #84
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Quote:
Originally Posted by zylo View Post
1) The list is endless and contains ALL imaginable strings of binary and decimal digits.
If there were any endless string in that list, there would have to be a first one, which is impossible, as the immediate preceding string would necessarily correspond to some finite integer, but each such integer has a finite successor.

Perhaps you've chosen not to imagine, for example, the endless string 111... (consisting entirely of '1's), which isn't the nth string in that list for any value of n, because you've made the nth string correspond to n-1 for n = 1, 2, 3, etc., leaving nowhere for any string that consists entirely of '1's.
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October 3rd, 2018, 05:28 AM   #85
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Originally Posted by cjem View Post
Pity it's complete nonsense, as has been shown in the preceding comments (which you conveniently seem to have ignored).
My post answers all previous questions related to the subject of natural and real numbers. Sorry you can't figure it out, but I can't compensate for, to put it politely, lack of perception.

To answer Maschke's interesting question, offhand, an infinite power series has an infinite number of roots unless it converges. That's a question for another thread.
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October 3rd, 2018, 06:00 AM   #86
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Quote:
Originally Posted by zylo View Post
My post answers all previous questions related to the subject of natural and real numbers. Sorry you can't figure it out, but I can't compensate for, to put it politely, lack of perception.
I proved that all binary strings in your construction have finite length. From this it follows that, by putting a radix point in front of each of them, you only get a subset of the rational numbers in $[0,1).

Anyway, if you actually believe what you are saying, I'd strongly recommend learning how to write proofs and communicate mathematics effectively. If you could intelligibly show that Cantor's argument is flawed, you would quickly become very rich and famous.
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October 3rd, 2018, 07:34 AM   #87
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Quote:
Originally Posted by zylo View Post
My post answers all previous questions related to the subject of natural and real numbers.
I posted two minutes before you, and you haven't responded to that post. Nor have you responded to the points that although (by virtue of their definition) there are infinitely many natural numbers, they are not bounded above and each one is finite.
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October 3rd, 2018, 08:21 AM   #88
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Quote:
Originally Posted by zylo View Post
List of binary and decimal representation of ALL The Natural and Real [0,1) Numbers:

0-----0
1-----1
2-----01
3-----11
4-----001
........
........

Comments:
1) The list is endless and contains ALL imaginable strings of binary and decimal digits.
2) Adding a radix point before the binaries and after the natural numbers gives all real numbers in [0,1) in binary and decimal form.
5000 becomes 5000., or more commonly, .0005


$0-----0.0_2 = \frac{0}{2^1}$
$1-----0.1_2 = \frac{1}{2^1}$
$2-----0.01_2 = \frac{1}{2^2}$
$3-----0.11_2 = \frac{3}{2^2}$
$4-----0.001_2 = \frac{1}{2^3}$
.
.
.


The list contains only dyadic rationals of the form $\frac{a}{2^b}$ where $a$ is an integer and $b$ is a natural number. As a result, your list is a mere subset of the rational numbers. It also contains no irrational numbers.


You now need to prove that each real number in [0,1) can be expressed in the form $\frac{a}{2^b}$ as a dyadic rational. Alternatively, you can get that sailboat.

Last edited by AplanisTophet; October 3rd, 2018 at 08:25 AM.
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October 3rd, 2018, 08:38 AM   #89
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Skipjack and CJEM. Every entry in the list is finite, but the list has no end. Just can’t get that? Every calculation for pi gives another term in the sequence, but the sequence has no end, assuming the computer runs for ever.

Maschke. The difference between a polynomial and a series is that the polynomial is a sum and the series is the limit of a sum.
e^z =0 has an infinite number of solutions.

Last edited by skipjack; October 3rd, 2018 at 11:32 AM.
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October 3rd, 2018, 08:48 AM   #90
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Quote:
Originally Posted by zylo View Post
e^z =0 has an infinite number of solutions.
such as?
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