October 3rd, 2018, 01:33 AM  #81  
Senior Member Joined: Aug 2012 Posts: 2,082 Thanks: 595  Quote:
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.  
October 3rd, 2018, 04:41 AM  #82 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
List of binary and decimal representation of ALL The Natural and Real [0,1) Numbers: 00 11 201 311 4001 ........ ........ 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. 
October 3rd, 2018, 04:55 AM  #83 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Pity it's complete nonsense, as has been shown in the preceding comments (which you conveniently seem to have ignored).

October 3rd, 2018, 05:26 AM  #84  
Global Moderator Joined: Dec 2006 Posts: 19,882 Thanks: 1835  Quote:
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 n1 for n = 1, 2, 3, etc., leaving nowhere for any string that consists entirely of '1's.  
October 3rd, 2018, 05:28 AM  #85  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115  Quote:
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.  
October 3rd, 2018, 06:00 AM  #86  
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 282 Thanks: 85 Math Focus: Algebraic Number Theory, Arithmetic Geometry  Quote:
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.  
October 3rd, 2018, 07:34 AM  #87 
Global Moderator Joined: Dec 2006 Posts: 19,882 Thanks: 1835  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.

October 3rd, 2018, 08:21 AM  #88  
Senior Member Joined: Jun 2014 From: USA Posts: 413 Thanks: 26  Quote:
$00.0_2 = \frac{0}{2^1}$ $10.1_2 = \frac{1}{2^1}$ $20.01_2 = \frac{1}{2^2}$ $30.11_2 = \frac{3}{2^2}$ $40.001_2 = \frac{1}{2^3}$ . . . 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.  
October 3rd, 2018, 08:38 AM  #89 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 
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. 
October 3rd, 2018, 08:48 AM  #90 
Senior Member Joined: Oct 2009 Posts: 611 Thanks: 187  

Tags 
binaryexpressed, cardinality, continuum hypothesis, diagonal argument, numbers, real, set 
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