January 23rd, 2019, 11:25 AM  #71  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  Quote:
But ok, I'll explain it. Suppose you claim you can measure 1/pi inches. Can you please suggest the nature of a the physical apparatus that could make such a perfectly exact measurement? You'd win the Nobel prize in physics for devising such an apparatus. All physical measurement is approximate. Are you under a different impression? Aplanis, this shows an appalling lack of understanding of the nature of physical science. Last edited by Maschke; January 23rd, 2019 at 11:35 AM.  
January 23rd, 2019, 11:32 AM  #72 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,978 Thanks: 994 
Mr. sintan, is your middle name cos? If so, then .33333333...... of your name is missing..... 
January 23rd, 2019, 11:59 AM  #73  
Senior Member Joined: Jun 2014 From: USA Posts: 479 Thanks: 36  Quote:
Answer my question on how we come up with uncountably many symbols then. That was your claim originally, not mine.  
January 23rd, 2019, 12:11 PM  #74  
Senior Member Joined: Aug 2012 Posts: 2,157 Thanks: 631  Quote:
It was yours. I defy you to quote me saying anything remotely like that. You've completely taken leave of your senses here.  
January 23rd, 2019, 12:17 PM  #75  
Senior Member Joined: Jun 2014 From: USA Posts: 479 Thanks: 36  Quote:
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But, you still have to explain this now given your amusing reversal: Quote:
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Last edited by AplanisTophet; January 23rd, 2019 at 12:26 PM.  
January 23rd, 2019, 12:36 PM  #76 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 13,978 Thanks: 994 
Geezzz....methinks time to close this thread 
January 23rd, 2019, 01:00 PM  #77  
Newbie Joined: Jan 2019 From: UK Posts: 18 Thanks: 0  Quote:
I follow most of what you've explained but I am a real novice and some of the terms and symbols you probably take for granted are new to me. To help develop my understanding I will certainly look to pick up a calculus text and would be keen to delve deeper into logic and paradoxes as I find these areas particularly fascinating. If I could please pick your collective brains one more time to recommend some good reads in these areas I would be grateful. This thread has been hugely educational for me so thanks to all who have contributed. Yours, The Troll  
January 23rd, 2019, 01:39 PM  #78  
Senior Member Joined: Oct 2009 Posts: 733 Thanks: 247  Quote:
Knowing about decimal expansions is very intimately connected with knowing precisely what a real number is. Thus it seems inevitable that you will have to learn in the future what a real number is exactly and how mathematicians construct the real numbers. You will have to learn how mathematicians deal with infinity. The following books seem relevant to you: 1) How to prove it  Velleman Details about what proofs are in mathematics, how to prove things yourselves, set theory, infinity, etc. Not very interesting topics compared to what follows, but necessary to be able to follow the rest. 2) The real numbers and real analysis  Bloch One of the most detailed books on what numbers are. Constructs, N, Z, Q, R from scratch. Decimal expansions are defined and it is proven every number has one. In order to truly understand the issues you raised in this thread, this book is a must. If you're interested in learning this just to satisfy your curiosity, please send me a private message, I might be able to help you further significantly.  
January 23rd, 2019, 02:19 PM  #79 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,600 Thanks: 2588 Math Focus: Mainly analysis and algebra 
As a further addendum to the above, something which might interest sintan is that we now have all the necessary pieces to prove that $$0.\overline9 = 1$$ a fact which causes much controversy over the internet. \begin{align}0.\overline9 = 0.999\ldots &= 0 + \frac9{10} + \frac9{100} + \frac9{1000} + \cdots \\ &= \frac9{10}\left(1 + \frac1{10} + \left(\frac1{10}\right)^2 + \left(\frac1{10}\right)^3 + \cdots \right) \\ &= \frac9{10} \left(\frac1{1  \frac1{10}}\right) \\ &= \frac9{10} \left(\frac1{\frac9{10}}\right) \\ &= \frac9{10} \cdot \frac{10}9 \\ &= 1\end{align} What is happening here is that there are two sequences, each represented by decimals, which both have the same limit. Indeed, every terminating decimal has two representations in this manner, one terminating, the other having the final digit decremented by 1 (more or less) and with an infinite tail of 9s appended. e.g. $$0.25 = 0.24999\ldots$$ 
January 23rd, 2019, 04:09 PM  #80 
Global Moderator Joined: Dec 2006 Posts: 20,298 Thanks: 1971 
Every terminating decimal except 0.
