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Old 2004-11-26, 13:41   #12
shaxper
 
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Quote:
Could you kindly explain the result you arrived at in the last line?
Whoop. There was a typo. It should be:

let

random number = q * r
where q = an irrational number, r = a rational number

Create q and r via any statistical technique to choose two primes, p1, p2

r = p2
q = pi [* p2 is incorrect at this point]
where pi = 3.14... out (p1) places

real random number = q * r = pi * p2

But as I said in the follow-up post, both q & r should be irrationalized (i.e., multiplied by a rational version of an irrational constant).

Quote:
However, I'd suspect that there must be a way to generate truly random numbers based on the Heisenberg Uncertainty Principle.
I think Luigi is right. You could create them, but once you look at them, the randomness would disappear.

Quote:
Get a radioactive nucleus with a half-life of x seconds. Wait for x seconds. If it decays, write 1. If it doesn't decay, write 0. Repeat.
That's assuming the probability of decay is exactly 50%. What if it's 50% and 49.999...%. Then it's possible there's a pattern (order) for the reason it's not exactly 50-50.

I can't think of how to prove it, but it seems to me that nothing in the universe is 0%, 50%, or 100% predictable, or in other words, nothing is completely random or completely order. That said, I haven't figured out where constants, such as pi, e, or the speed of light, lie on this continuum.

Maybe in the middle? Their exact values are random, but approximate values are order (predictable/known)?

I dunno. I've been doing a lot of thinking about pi lately (which spawned this random number creating scheme) and would like to chat about it with somebody. If you're interested, please send me a private message.

cheers,
jad
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Old 2004-11-26, 18:00   #13
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Well, what does it mean for something to be random? When we say that something is random, we mean that it is:

1) Undetermined
2) There is more than one possible outcome

If we use these two properties, then I think we must consider constants like pi as 100% order. Clearly, the decimal expansion of pi is fully determined, since there are algorithms that can approximate it to arbitrary accuracy. Furthermore, no matter how many times you calculate a set number of digits, you will always have the same answer. I think the reason pi appears to be random is that we are looking at it in an "unnatural" way. If you think about it, why should a decimal expansion always be the correct way to look at a number? In the case of pi, I think it is not. I'll leave you with a famous and beautiful equation discovered by Euler:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = (pi^2)/6
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Old 2004-11-26, 21:58   #14
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Quote:
Originally Posted by jinydu
Actually, using quantum uncertainty to generate random numbers should be quite simple.

Get a radioactive nucleus with a half-life of x seconds. Wait for x seconds. If it decays, write 1. If it doesn't decay, write 0. Repeat.
http://www.fourmilab.ch/hotbits/
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Old 2004-11-26, 22:55   #15
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There are a couple of interesting arguments presented here. First, I had not seen nor had I previously considered using radioactive decay as a means of generating random numbers but since the very nature of the Heisenberg Uncertainty Principle (HUP) ensures true randomness, it was an obvious conclusion that random numbers could be derived.

To argue that a HUP driven process is not truly random is to argue that the physical processes of our universe are not driven in a truly random fashion. A preponderance of evidence discredits this view.

The numbers generated by a computer from an algorithm inherently cannot be truly random. This has been well documented since at least the days of the Commodor 64 when some programmers recommended setting the SID chip to a high frequency and then sampling to create a random number. In effect, the randomness was significantly better than the internal random number generator.

Fusion
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Old 2004-11-27, 13:01   #16
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Quote:
Originally Posted by Fusion_power
To argue that a HUP driven process is not truly random is to argue that the physical processes of our universe are not driven in a truly random fashion. A preponderance of evidence discredits this view.
Well, think about the problem of dark matter and the related anti-matter question: it seems that at least some subjects regarding our universe are not random, unless you accept the existence of infinite "quantum-universes", where things happen in different ways according to each universe's observer.

On the other hand, irrationaality and trascendency of numbers may as well be seen as a special case of HUP. And there is more in this untechnical stream of thoughts: the demonstration of Riemann Hypotheis may be compared to the photograph of our perceived four-dimentional universe: once we have a snapshot of it, we are able to fix each and every single event in the past, present and future.

But, hey, Superstrings and M-Theory made us think that there are better ways to describe our reality than the four-dimentional concept, so there may be a better way to analyze RH as well.

Quote:
Originally Posted by Fusion_power
The numbers generated by a computer from an algorithm inherently cannot be truly random. This has been well documented since at least the days of the Commodor 64 when some programmers recommended setting the SID chip to a high frequency and then sampling to create a random number. In effect, the randomness was significantly better than the internal random number generator.
I think randomness "precision" strictly depends on machine-word precision. Computer generated random numbers are called pseud-random also because they are produced by a deterministic algorithm, that has an obviously opposite seimantic .

Luigi
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Old 2004-11-27, 17:50   #17
mfgoode
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Cool To create a real random number

[QUOTE=jinydu]Well, what does it mean for something to be random? When we say that something is random, we mean that it is:

1) Undetermined
2) There is more than one possible outcome

I'll leave you with a famous and beautiful equation discovered by Euler:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = (pi^2)/6 [unquote]

Yes the eqn is marvellous. The circle (pi) is in conjunction with the square.
But you have just scratched the surface. Euler derived many more on the same lines.
I recommend the book "Journey through Genius" by William Dunham who devotes a whole chapter on such eqns and gives the derivations of each.

There is one eqn Euler could not work out and 200 yrs. of math research has not thrown any light on. Read all about it in the book. There are conjectures made but no proof.
Here it is:

1 + 1/(2^3) + 1/(3^3) +1/(4^3) = 1 + 1/8 +1/27 + 1/64 +..........

Mally
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Old 2004-11-27, 18:05   #18
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Quote:
Originally Posted by mfgoode
Quote:
Originally Posted by jinydu
Well, what does it mean for something to be random? When we say that something is random, we mean that it is:

1) Undetermined
2) There is more than one possible outcome

I'll leave you with a famous and beautiful equation discovered by Euler:

1 + 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + ... = (pi^2)/6

Yes the eqn is marvellous. The circle (pi) is in conjunction with the square.
But you have just scratched the surface. Euler derived many more on the same lines.
I recommend the book "Journey through Genius" by William Dunham who devotes a whole chapter on such eqns and gives the derivations of each.

There is one eqn Euler could not work out and 200 yrs. of math research has not thrown any light on. Read all about it in the book. There are conjectures made but no proof.
Here it is:

1 + 1/(2^3) + 1/(3^3) +1/(4^3) = 1 + 1/8 +1/27 + 1/64 +..........

Mally
Thank you for the recommendation, but I have already read the book. It has shown me many beautiful theorems that unfortunately, aren't taught in school.

I have also heard about zeta(3). As far as I know, the only real progress on that problem (aside from brute force computer approximations) occured in 1979, when Apery proved that zeta(3) is irrational. In his honor, zeta(3) is now also known as Apery's constant.

Determining zeta(3) is such a simple and elegant problem that I couldn't resist making my own attempt at it, despite the fact that it stumped Euler himself. I posted some of my thoughts at http://www.sosmath.com/CBB/viewtopic...11387&start=30
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Old 2004-11-27, 18:13   #19
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Quote:
Originally Posted by jinydu
Thank you for the recommendation, but I have already read the book. It has shown me many beautiful theorems that unfortunately, aren't taught in school.

I have also heard about zeta(3). As far as I know, the only real progress on that problem (aside from brute force computer approximations) occured in 1979, when Apery proved that zeta(3) is irrational. In his honor, zeta(3) is now also known as Apery's constant.

Determining zeta(3) is such a simple and elegant problem that I couldn't resist making my own attempt at it, despite the fact that it stumped Euler himself. I posted some of my thoughts at http://www.sosmath.com/CBB/viewtopic...11387&start=30
:surprised
Good work Jindyu. Keep up the good work

Mally
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Old 2004-11-28, 08:13   #20
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Quote:
Originally Posted by jinydu
Well, what does it mean for something to be random?
TAOCP Chapter 3 discusses this question, of course. :)

BTW, those interested in TAOCP might enjoy Knuth's page about TAOCP at http://www-cs-faculty.stanford.edu/~knuth/taocp.html (he's actually working on Volume 4!), or even Knuth's home page at http://www-cs-faculty.stanford.edu/~knuth/index.html.

Last fiddled with by cheesehead on 2004-11-28 at 08:20
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Old 2004-11-29, 02:08   #21
shaxper
 
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Quote:
Originally Posted by jinydu
Well, what does it mean for something to be random? When we say that something is random, we mean that it is:

1) Undetermined
2) There is more than one possible outcome

If we use these two properties, then I think we must consider constants like pi as 100% order. Clearly, the decimal expansion of pi is fully determined, since there are algorithms that can approximate it to arbitrary accuracy. Furthermore, no matter how many times you calculate a set number of digits, you will always have the same answer. I think the reason pi appears to be random is that we are looking at it in an "unnatural" way.
I like your definition, but I come to the opposite conclusion: pi is not 100% order but 75%, or 37% or 99.999999%. The actual percentage is arbitrary, because pi has been determined to x decimals and x/infinity is undefined.

But even with the infinite series you offer by Euler, the exact (100%) number is undetermined since you can't write out (or conceive) of the entire expansion.

This isn't an arbitrary distinction. As I said above, I don't think anything in this universe is 0% or 100% order (including 50% was an error) unless you shrink the available outcomes to exclusively 0 or 1. Otherwise, every event has some percentage of order, of predictability (i.e., not 0%) and some percentage of chaos, of unpredictability (i.e., not 100%).
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Old 2004-11-29, 03:14   #22
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Quote:
Originally Posted by shaxper
I like your definition, but I come to the opposite conclusion: pi is not 100% order but 75%, or 37% or 99.999999%. The actual percentage is arbitrary, because pi has been determined to x decimals and x/infinity is undefined.

But even with the infinite series you offer by Euler, the exact (100%) number is undetermined since you can't write out (or conceive) of the entire expansion.
Just because you can't write out the whole decimal expansion doesn't mean its undetermined. Here's the classic example:

1/3 = 0.33333333333333333333333333333333333333333333...

Morever, I would argue that the same is true of irrational numbers that do follow a pattern, for instance:

0.1234567891011121314151617181920212223242526272829303132333435...

Thus, I would argue that pi is determined. It follows precise patterns and its value will always be the same; it doesn't matter what country you're in, what direction your computer is facing, what month it is, what color your mousepad is, whether or not the sun will rise tomorrow, etc. The value of pi will always be the same.
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