My Math Forum An Anomaly of Decimal Representation

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June 27th, 2018, 10:59 AM   #11
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Quote:
 Originally Posted by v8archie You are treating the decimal representation of a number as a mathematical object. It isn't. It's a representation of a mathematical object.
Was this for me? You quoted me, I assume this is for me. You are wrong of course. A decimal representation is a perfectly sensible and standard mathematical object, namely a function $f : \mathbb N \to \{0, 1, \dots, 9\}$. We map decimal expressions to real numbers (in the unit interval to avoid dealing with the numerals to the left of the decimal point) in the usual manner as a convergent infinite series.

I fail to see the relevance of your pictures, especially as you are incorrect on the math.

Quote:
 Originally Posted by v8archie (and now I'm straying off the real point)
You got that right.

Quote:
 Originally Posted by v8archie Except that the pipe is not a pipe, but a representation of a pipe.
What are you going on about? A decimal representation is a perfectly valid mathematical object in its own right, and maps naturally to its corresponding real number. Whatever signal you are trying to send is lost in the noise.

Last edited by Maschke; June 27th, 2018 at 11:02 AM.

 June 27th, 2018, 11:04 AM #12 Senior Member     Joined: Sep 2015 From: USA Posts: 2,299 Thanks: 1218 Zylo posts some doggerel and you all end of at each other's throats... again, while he sits back and munches on popcorn. I'd have really thought such intelligent people would stop taking his bait by now. Thanks from Benit13
June 27th, 2018, 11:20 AM   #13
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 Originally Posted by romsek Zylo posts some doggerel and you all end of at each other's throats... again, while he sits back and munches on popcorn. I'd have really thought such intelligent people would stop taking his bait by now.
As usual, many people respond to Zylo's posts by revealing their own ignorance of the subject matter. One quotes the 3 x 1/3 proof and claims it depends on "logic" rather than on the theorem on term-by-term multiplication. Another claims decimal representations aren't mathematical objects. Zylo does the community a service by revealing the Dunning-Krugerism of some of the regulars. And this is no sporadic event. It's as reliable as the tides.

Last edited by skipjack; June 28th, 2018 at 06:26 AM.

 June 27th, 2018, 01:36 PM #14 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 Suppose n$\displaystyle _{1}$/10+n$\displaystyle _{2}$/100=1/3. Then 10n$\displaystyle _{1}$+n$\displaystyle _{2}$=100/3 where the left side is an integer and the right side is not. No matter how many terms you take, the left side is always an integer and the right side is 10$\displaystyle ^{n}$/3 which is not. Conclusion: Decimal representation defined as a sequence of digits is unique but not complete.. The problem stems from the radix of the number system. 1/3 is a number in the representation based on a radix of 3: (,1). Radix based number systems are simply a means of attaching meaningful numbers to things, with any degree of accuracy. They do not define the real numbers, you use axioms or dedekind cuts. I appreciate the opportunity to work things out in my mind using MMF. I find the comments quite invigorating.
 June 27th, 2018, 07:04 PM #15 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 .33333... to n places, represents 3/10 +3/100 +3/1000 + .... + 3/10$\displaystyle ^{n}$, which is (=), by HS algebra, to (1/3)(1-.1$\displaystyle ^{n}$), which is never 1/3 no matter what n is. Limit, limit? Who said anything about a limit. Where in .333...=3/10+3/100+3/1000 +... does it say anything about a limit? Where in the axioms* for a real number system does it say anything about a limit? *A real number is a complete ordered field. Complete: Every non-empty set of real numbers bounded above has a least upper bound. The lub of (0,1/3) is 1/3 which does not have a decimal representation. But suppose we complete our decimal representations by defining limits of infinite decimals as part of the system of decimal representations, ie, there is .3333....... and lim ,333333.... which are not the same but both belong to our system of decimal representation. That resolves all questions and should make everybody happy. Not quite, close but no cigar. To summarize: .33333..... represents two real numbers: The literal sum and the limit. And of course the generalization. So in my list of real numbers I would have .33333...., and lim.3333...., So far so good. But as real numbers they still have to satisfy the field axioms, including closure: .333... + .333... = .666... lim.333..... + lim.333.... = 1/3+1/3=2/3=lim.6666..... .33333....+lim.333..= .3333...+ 1/3 =? Note for all n, .66666 < .33333....+lim.333..
June 27th, 2018, 08:17 PM   #16
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Quote:
 Originally Posted by zylo Suppose n$\displaystyle _{1}$/10+n$\displaystyle _{2}$/100 . . .
What do you mean by n$_1$ and n$_2$?

June 27th, 2018, 09:34 PM   #17
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Quote:
 Originally Posted by zylo 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.
$\frac15$ is also a limit point of the open set $(0,\frac13)$ and $\frac15 \in (0,\frac13)$. So your inference, is false.

June 27th, 2018, 09:55 PM   #18
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No. I was addressing Zylo's post within the context of the thread using your post as a starting point. Zylo wasn't considering the decimals as functions from the naturals to digits, he was considering them as numbers. In this context, they aren't. They are, I guess, functions from the set of ordinals $\times$ the digits $\to \mathbb R$, but that only serves to obfuscate.

June 27th, 2018, 10:20 PM   #19
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 Originally Posted by v8archie No. I was addressing Zylo's post within the context of the thread using your post as a starting point. Zylo wasn't considering the decimals as functions from the naturals to digits, he was considering them as numbers. In this context, they aren't. They are, I guess, functions from the set of ordinals $\times$ the digits $\to \mathbb R$, but that only serves to obfuscate.
I got confused by your quoting me. I don't believe it's an obfuscation to recognize a decimal expansion as a function from the naturals to the digits. I regard that as essential to understanding why, for example, there is no such real number as .000....00001. People always have an intuition that there is such a thing. And the reason there isn't is EXACTLY BECAUSE a decimal expansion is a function on the natural numbers. There is no last decimal digit for exactly same reason there's no last natural number.

What you call an obfuscation I call the essence of the entire matter. I wonder if you can come around to my point of view on this. The mathematical nature of decimal expressions and how they are interpreted as real numbers is the heart of the matter.

Quote:
 Originally Posted by v8archie functions from the set of ordinals × the digits →ℝ
The way I'd put it is that a decimal expansion (again restricting attention to the right of the decimal point for simplicity) is a function $d : \mathbb N \to \{\text{digits}\}$. As a standard convention we notate $d(n)$ as $d_n$. Then we map each decimal expression $d$ to the real number $\sum \frac{d_i}{10^i}$ as usual.

Logically prior to that we construct the reals and prove the least upper bound property, so that we may be certain that the above infinite sum converges to a real number.

Last edited by Maschke; June 27th, 2018 at 10:28 PM.

 June 28th, 2018, 03:45 AM #20 Banned Camp   Joined: Mar 2015 From: New Jersey Posts: 1,720 Thanks: 124 $\displaystyle \lim_{n\rightarrow\infty}$ .33333333....n4387 = 1/3, and you can put anything you want in back of n instead of 4387.

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