June 26th, 2018, 09:54 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,628 Thanks: 119  An Anomaly of Decimal Representation
An Anomaly of Decimal Representation of Real Numbers 1/3=.333333333......... therefore 3(1/3)=1=.999999............... The resolution is that 1/3 $\displaystyle \neq$ .33333....... Proof: If 1/3 = .333333..... then the open interval (0,1/3) has a largest member, which violates a fundamental axiom of the real numbers. The situation is analogous to: .00000......0001, with an endless number of 0's. If this were equal to 0, the open interval (0,1/3) would have a smallest member. 
June 26th, 2018, 11:03 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 895  Quote:
There is no "resolution" needed. Both of those statements are true. But the statement making up your "proof" is incorrect. "If 1/3= 0.3333..." (and it is) it does not follow that the open interval (0, 1/3) would have a largest member. Are you thinking that 0.3333... would be in that interval? "If 1/3= 0.3333..." then obviously 0.3333... could NOT be in that interval. On the other hand ".00000.....0001 with an endless number of 0's" simply is meaningless. You can't put a "1" at the end of an endless number of 0's!  
June 26th, 2018, 11:19 AM  #3  
Senior Member Joined: Aug 2012 Posts: 2,134 Thanks: 621  Quote:
First, I hope you agree that there is no end to the sequence of positive integers 1, 2, 3, 4, 5, 6, ... Now in a decimal representation, to the right of the decimal point there is one digit for each positive integer. So there can be no "last" decimal digit, for exactly the same reason there's no last positive integer.  
June 26th, 2018, 01:09 PM  #4 
Senior Member Joined: May 2016 From: USA Posts: 1,250 Thanks: 516 
$\dfrac{1}{3} = \displaystyle \sum_{j=1}^{\infty} \dfrac{3}{10^j} \implies$ $3 * \dfrac{1}{3} = 3 * \displaystyle \sum_{j=1}^{\infty} \dfrac{3}{10^j} \implies$ $1 = \displaystyle \sum_{j=1}^{\infty} \dfrac{9}{10^j} \implies$ $1 + \displaystyle \sum_{j=1}^{\infty} \dfrac{0}{10^j} = \displaystyle \sum_{j=1}^{\infty} \dfrac{9}{10^j}.$ There is NOTHING to resolve. Line 4 follows from line 1 by simple logic. This of course does not represent a proof that 1/3 does equal 0.333.... But there is no contradiction 
June 26th, 2018, 02:00 PM  #5 
Senior Member Joined: Aug 2012 Posts: 2,134 Thanks: 621  Actually line 3 does not follow from line 2 by simple logic. Rather, termbyterm multiplication of a convergent infinite series by a constant is a theorem that must be proved. It does not follow from the axioms for real numbers without first, a careful definition of the real numbers, and second, a careful definition of the limit of a convergent infinite series.

June 26th, 2018, 10:38 PM  #6  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,557 Thanks: 2558 Math Focus: Mainly analysis and algebra  Quote:
You are treating the decimal representation of a number as a mathematical object. It isn't. It's a representation of a mathematical object. Ceci n'est pas un pipe  "this is not a pipe" it's a picture of a pipe. It's a representation of a pipe. You cannot use it as a pipe. A similar message here: the representation of the view is not the view. Although (and now I'm straying off the real point) in this case it's a little more complicated, because the painting in the picture is not a painting  it's a representation of a painting. Something that is brought into greater relief in The Two Mysteries: So the pipe is not a pipe, it's a picture of a pipe. And the picture of a pipe is not a picture of a pipe. Except that the pipe is not a pipe, but a representation of a pipe. But the representation of a pipe (the picture within the picture) is a representation of a pipe and also a representation of a representation of a pipe.  
June 27th, 2018, 12:00 AM  #7 
Senior Member Joined: Oct 2009 Posts: 687 Thanks: 223 
So we have a function $$\{0,1,...,9\}^\mathbb{N}\rightarrow [0,1]$$ which sends $(\alpha_n)_n$ to $\sum_{n=1}^{+\infty} \frac{\alpha_n}{10^n}$. This map is surjective, but not injective, so what is the big deal with this?? 
June 27th, 2018, 09:07 AM  #8 
Senior Member Joined: Sep 2016 From: USA Posts: 534 Thanks: 306 Math Focus: Dynamical systems, analytic function theory, numerics 
For the sake of continuity, please rename this thread to "Zylo doesn't understand real numbers #219"

June 27th, 2018, 10:07 AM  #9 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,628 Thanks: 119 
1/3 is a limit point of the open set (0,1/3). As such, no member of (0,1/3), including .3333333........, can equal 1/3.

June 27th, 2018, 10:27 AM  #10  
Senior Member Joined: Oct 2009 Posts: 687 Thanks: 223  Quote:
It is right that no member of (0,1/3) can equal 1/3 (although this doesn't follow from 1/3 being a limit point). It is not right that 0.3333... is an element of (0,1/3).  

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