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Author Topic: A strange number, the mathematics of.  (Read 5419 times)

Offline Atomic-S

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A strange number, the mathematics of.
« on: 13/02/2006 05:38:28 »
Let A_0 = 0 and A_1 = 1 , and for all positive integers n
let A_(2n + 2) = 1 - A_n and
let A_(2n + 3) = A_n

Then let us form the sum of all terms of the form (2^(-n))*(A_n), from n = 0 to n = infinity.

This forms a rather unusual number.

Question:  Does anyone know anything about it? Is there any literature on it? Any references?  Any brilliant insights as to what set it belongs to or what it may be the root of? Any equations?


 

sharkeyandgeorge

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Re: A strange number, the mathematics of.
« Reply #1 on: 13/02/2006 20:00:40 »
what?

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Offline neilep

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Re: A strange number, the mathematics of.
« Reply #2 on: 14/02/2006 22:26:44 »
quote:
Originally posted by Atomic-S


This forms a rather unusual number.




Can you tell what the starnge number is please ?

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Offline Atomic-S

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Re: A strange number, the mathematics of.
« Reply #3 on: 15/02/2006 04:18:28 »
Excuse me; I have committed a mathematical error in my first post. The correct formulas are (also changing to more computer-freindly notation):

Let A(0) = 0 and A(1) = 1 , and for all positive integers n
let A(2n) = A(n) and
let A(2n + 1) = 1 - A(n)

Then let us form the sum of all terms of the form (2^(-n))*A(n), from n = 0 to n = infinity.

A(0) through A(255) are (reading horizontally line by line as usual):
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

To 36 decimal places, the number = 0.824908067279466195032000541687011719

 

Offline ukmicky

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Re: A strange number, the mathematics of.
« Reply #4 on: 15/02/2006 04:31:32 »
HEY . I've been studying it for five minutes now and i still can't see any image appearing ,all i can see is a bunch of 0 and 1s, I'm afraid your illusion don't work:D


Michael
 

sharkeyandgeorge

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Re: A strange number, the mathematics of.
« Reply #5 on: 15/02/2006 21:13:14 »
sorry atomic s what are we supposed to see or understand mayby if you explained yourself more fully?

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Offline johntsang

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Re: A strange number, the mathematics of.
« Reply #6 on: 18/02/2006 04:51:07 »
0011110011000011110000110011110011000011001111000011110011000011
1100001100111100001111001100001100111100110000111100001100111100
1100001100111100001111001100001100111100110000111100001100111100
0011110011000011110000110011110011000011001111000011110011000011


That is to flip every two. there's not enough data to make sure it' repeating.

The Sum is the result of the Summation according to the above pattern in binary form ?
 

Offline Atomic-S

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Re: A strange number, the mathematics of.
« Reply #7 on: 20/02/2006 02:39:05 »
Concerning the series, consider this table:

n (decimal);     n (binary);      parity P(n) of (binary) n

{I hope you can make sense out of this table; I have had problems getting it to format.)

0;               0;               0
1;               1;               1
2;               10;              1
3;               11;              0
4;               100;             1
5;               101;             0
6;               110;             0
7;               111;             1
8;               1000;            1
9;               1001;            0
10;              1010;            0
11;              1011;            1
12;              1100;            0
13;              1101;            1
14;              1110;            1
15;              1111;            0
16;              10000;           1
17;              10001;           0
18;              10010;           0
19;              10011;           1
20;              10100;           0

etc.

You will note that the parity column matches the sequence of A(n) given earlier: i.e., A(n) is simply the parity of n when expressed as a binary integer. And the number is simply the sum from n = 0 to infinity of all terms of the form P(n)/2^n  .

Note therefore that the sequence necessarily does not have a period of length 2^{any integer}, because the way parity is formed continually defeats any tendency to repeat with such a period. An important question is whether the sequence is aperiodic also with respect to all other possible intervals of repetition.  That is because the sequence is identical to a binary representation of the number formed by the sum. If the sequence is fully aperiodic (i.e., is periodic with respect to no possible interval length), then by the rules of long division the resulting number is necessarily irrational also.

That is of interest,  because most of the irrational numbers commonly encountered, such as pi, are characterized by digits that appear according to no discernible pattern.  But here we have an [apparently] irrational number whose digits do in fact occur according to a clearly discernible pattern, at least when expressed binarily.

(To observe the pattern, go to my earlier post, and carefully examine the sequence elements in groups of 4; and also compare each row with the row above it or below it; and compare each column with the column to the left or right of it. Every row and every column repeats the same pattern, or the complement of that pattern. But it does so aperiodically.)

That raises the question: do there exist numbers which are "less irrational" that common ones? the "simplest" irrational numbers generally encountered in mathematics are integer roots of integers. These all, however, I believe, have digit that follow no discernible pattern. But because the number here does have a pattern, does that mean that it is of a simpler order of irrationality than, say, the square root of two? If so, what sort of algebraic expression might it correspond to, if any?
« Last Edit: 20/02/2006 02:48:11 by Atomic-S »
 

Offline johntsang

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Re: A strange number, the mathematics of.
« Reply #8 on: 20/02/2006 04:59:08 »
Please Generate  A(256) to A(512).
Don't have time to write a program.

Thank you
 

Offline Solvay_1927

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Re: A strange number, the mathematics of.
« Reply #9 on: 22/02/2006 01:07:23 »
Atomic (a Blondie fan, huh? :))

I’m not sure of the significance of what you’re doing.

An irrational number (i.e. one which cannot be expressed as a ratio of two integers) has an "aperiodic" decimal expansion – i.e. if written as x.xxxxxx... then the digits after the decimal point go on forever and never repeat in any sort of pattern. (So pi, e, sqrt(2), etc, when written as decimals, have an infinitely long sequence of numbers after the decimal point that appear "random" and never settle into any sort of periodic pattern.)

But that doesn’t mean that irrational numbers can’t be generated from an (infinite) sequence with a "discernible pattern".  For example:
Pi/4 = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 – 1/15 + …
(Pi^2)/6 = 1/1^2 + 1/2^2 + 1/3^2 + 1/4^2 + …
e = 1/1! + 1/2! + 1/3! + 1/4! + …

I’ve no idea whether the sum you calculated above
(0.824908067279466195032000541687011719)
is irrational or not, but I don’t recognise it as being "special" in any way (it doesn’t appear to be a square or square root or exponential or log of any integer or of any other "special" number such as pi or e).

In any case, it’s worth noting that the sequence “(2^(-n))” is just:
1, 1/2, 1/4, 1/8, 1/16, …
and the sum of all these terms = 2.

By multiplying each term by "A(n)" you’re just choosing to include only half of these terms in your sum.  If you chose every odd term (i.e. if A(n) = 1 0 1 0 1 0 ...) then the sum would be 1+1/4+1/16+... = 1.3333..., and if you chose every even term (i.e. if A(n) was 0 1 0 1 0 1 ...) then the sum would be 0.6666...

For the particular sequence A(n) you’re using, you get a sum of 0.8249... which is between 0 and 2.  But any other random sequence A(n) (which was half 0s and half 1s) would also give you a sum between 0 and 2.  And I can’t see why 0.8249... is any more "special" than any of these other sums.


Am I helping here, or am I missing the point?

Paul.
« Last Edit: 22/02/2006 01:16:36 by Solvay_1927 »
 

Offline johntsang

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Re: A strange number, the mathematics of.
« Reply #10 on: 22/02/2006 01:38:07 »
That's what the sequence trying to do, don't quite want to do the math, but judging fro the pattern, if it repeat, it should be at the binary form of  (2 ^ 256) cycle, that's why you should show me the next 256 numbers .

(2 ^256) is somewhat 77 decimal digits.

if it repeat, even in binary form, it would be a rational number.... so just check the next 256 patterns first before dig into the math, please.
 

Offline Atomic-S

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Re: A strange number, the mathematics of.
« Reply #11 on: 22/02/2006 03:16:42 »
A(0) through A(1023) are:
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1
 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0

 

Offline Atomic-S

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Re: A strange number, the mathematics of.
« Reply #12 on: 22/02/2006 03:26:06 »
quote:
An irrational number (i.e. one which cannot be expressed as a ratio of two integers) has an "aperiodic" decimal expansion – i.e. if written as x.xxxxxx... then the digits after the decimal point go on forever and never repeat in any sort of pattern. (So pi, e, sqrt(2), etc, when written as decimals, have an infinitely long sequence of numbers after the decimal point that appear "random" and never settle into any sort of periodic pattern.)


What about sum, n = 0 to infinity, of n/10^{n(n+1)/2} ?  Those digits look highly aperiodic to me, but are far from random.
 

Offline johntsang

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Re: A strange number, the mathematics of.
« Reply #13 on: 22/02/2006 04:57:45 »
That's fun, will dig into it further.

Thanks
 

Offline Solvay_1927

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Re: A strange number, the mathematics of.
« Reply #14 on: 22/02/2006 22:25:52 »
quote:
Originally posted by Atomic-S
What about sum, n = 0 to infinity, of n/10^{n(n+1)/2} ?  Those digits look highly aperiodic to me, but are far from random.



Correct, it would seem that this sum (=0.102003000400005000006...) is a number whose decimal expansion can never repeat periodically, and so it's an irrational number, but it isn't "random".

The "looking random" bit isn't a requirement for an irrational number. When I said that pi, e, sqrt(2), etc. appear "random", it was by way of an example - to keep things simple for other readers of this post.  I wasn't being mathematically precise.

(If I wanted to be mathematically precise, I could criticise your use of the phrase "sum to infinity".  But I imagine most people on TNS will find already find this thread dry & heavy-going without us getting into mathematical semantics.:()
 

Offline johntsang

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Re: A strange number, the mathematics of.
« Reply #15 on: 23/02/2006 12:24:41 »
I concur, that's more or less like what the above posting said .
Can't see what it related too.
 

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Re: A strange number, the mathematics of.
« Reply #15 on: 23/02/2006 12:24:41 »

 

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