My Math Forum An Anomaly of Decimal Representation

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July 9th, 2018, 11:00 AM   #71
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Quote:
 Originally Posted by zylo Induction as a means to specify and count all "infinite" place decimals (binaries, radixes). Points on a line.
Induction (recursion actually) will end you up with all finite decimals, but of arbitrary length. This way you'll never end up with infinite decimals.

July 9th, 2018, 11:10 AM   #72
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Quote:
 Originally Posted by zylo $\displaystyle \lim_{n\rightarrow \infty} .33333 = .333333....... = 3\bar{n}$ $\displaystyle \lim_{n\rightarrow \infty} .33333 = 1/3$ The first limit is a representation limit (point). The second lim is a sum limit which gives the distance of the point from the origin. I can count ALL n-place binaries (or any radix), ie, for n through "infinity."
Induction is how you specify all members of these limits. You can't write them all out.

Skipjack. If you don't like 1,2,3,4,.... as a primary counting mechanism, perhaps you prefer sticks and stones

\
\\
\\\
\\\\
.
.
or
0
00
000
0000
.
.

July 9th, 2018, 11:22 AM   #73
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Quote:
 Originally Posted by zylo 1,2,3,4,.... are the count.
That doesn't work because

(a) 1, 2, 3, 4, ... (continued without end) doesn't contain the answer for the entire list;

(b) your method doesn't satisfy the criterion "without a number system" ("counting mechanism" seems to be just an alternative phrase for "number system");

and

(c) using a symbol such as "\" or "0" seems to be equivalent to "scratches on a rock".

 July 9th, 2018, 11:32 AM #74 Senior Member   Joined: Mar 2015 From: New Jersey Posts: 1,603 Thanks: 115 a) Induction specifies all elements of a list. If I specify every element of an infinite sequence (calculus), have I specified every element of the sequence? b) 1,2,3,4,... IS the number system. You can't count something with itself. Note that the INFINITE harmonic series is specified by 1/1+1/2+1/3+...... Do you have a problem with that? Last edited by zylo; July 9th, 2018 at 11:46 AM.
July 9th, 2018, 12:15 PM   #75
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Quote:
 Originally Posted by zylo a) Induction specifies all elements of a list. If I specify every element of an infinite sequence (calculus), have I specified every element of the sequence? b) 1,2,3,4,... IS the number system. You can't count something with itself. Note that the INFINITE harmonic series is specified by 1/1+1/2+1/3+...... Do you have a problem with that?
OK, this is annoying, please learn the difference between recursion and induction. Cause right now you're just confusing me.

 July 9th, 2018, 12:26 PM #76 Global Moderator   Joined: Dec 2006 Posts: 19,888 Thanks: 1836 Your response, zylo, to (a) relates to the elements only. Specifying the elements isn't the same as counting the entire endless list. Your response to (b) is inapplicable, as I didn't ask you to count something with itself. I asked how you would count 1, 2, 3, 4, ... (continued without end) without a number system. As you didn't respond to (c), I assume you now accept that your suggestions that used repetitions of "\" and "0" violate the criterion that you don't use sticks, stones and the like, because you've already withdrawn that type of response.
July 9th, 2018, 06:30 PM   #77
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Quote:
 Originally Posted by zylo Note that the INFINITE harmonic series is specified by 1/1+1/2+1/3+...... Do you have a problem with that?
Note that the infinite harmonic series has a infinite number of terms but does not have a term $\frac1\infty$.

It's not convergent either.

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