Questions on Pi

[Ziggurat]

If the series is truly infinite, then any and all such drawings are not only possible, but inevitable. An infinite series of digits must by necessity contain all possible combinations.

Nonsense. There's nothing necessary about that at all. Imagine, for example, we take pi, remove all the 9's, and call that a new number. This number is irrational, and its digits are infinite, but having no 9's, it rather clearly cannot contain all possible combinations.

Or, if you don't trust that this number will remain rational, expand pi in base 9 and then use the same digits as a base-10 number. Still irrational, still infinite, but no 9's. Hence not all possible combinations. Proof by counterexample.

 

[Molinaro]

I know there's a bit a couple of hundred digits in where it goes "999999".

There are a few web sites out there that let's you search PI for a particular string of digits.

http://www.angio.net/pi/piquery

That site let's you search the 1st 200 million digits. Your "999999" begins 762 digits after the decimal point.

My birthdate, 01301969 begins 813,012 digits after the decimal

And an interesting table from that site, regarding the % chance of finding a string of digits of a given length in the 1st 100,000,000 digits of pi:

Number Length       Chance of Finding 
1-5                           100% 
6                             Nearly 100% 
7                             99.995% 
8                             63% 
9                             9.5% 
10                            0.995%% 
11                            0.09995%

 

[TrueSceptic]

There are a few web sites out there that let's you search PI for a particular string of digits.

http://www.angio.net/pi/piquery

That site let's you search the 1st 200 million digits. Your "999999" begins 762 digits after the decimal point.

My birthdate, 01301969 begins 813,012 digits after the decimal

And an interesting table from that site, regarding the % chance of finding a string of digits of a given length in the 1st 100,000,000 digits of pi:

Number Length       Chance of Finding 
1-5                           100% 
6                             Nearly 100% 
7                             99.995% 
8                             63% 
9                             9.5% 
10                            0.995%% 
11                            0.09995%

Thanks. Very interesting.

 

[drkitten]

Nonsense. There's nothing necessary about that at all. Imagine, for example, we take pi, remove all the 9's, and call that a new number. This number is irrational, and its digits are infinite, but having no 9's, it rather clearly cannot contain all possible combinations.

Or, if you don't trust that this number will remain rational, expand pi in base 9 and then use the same digits as a base-10 number. Still irrational, still infinite, but no 9's. Hence not all possible combinations. Proof by counterexample.

I believe the first number proved transcendental was the number 0.110001000...., where the value of the n'th digit was 1 if and only if n were k! for some k.
(i.e the 1st, 2nd, 6th, 24th, 120th, &c digits were 1 and everything else was a 0).

Obviously, there are lots of patterns that appear nowhere in this number; anything like 8675309, but also simple patterns like 111.

 

[blutoski]

Maybe. But my point is that if you do geometry near a black hole then that "geometrical Pi" is not equal to the "number theory Pi". So I speculate that math may not be the same everywhere in the universe.

Math would be the same, but you may need to use different math systems, depending on situation. pi in a euclidean system is a constant. pi in a non-euclidean system is still constant, but the geometric ratios will vary with the quantity of distortion. The way you calculate this is to factor the distortion against the constant pi. That's what tensors are for.

But of course on earth both Pi are equal.

Arguably, no: the earth is in real spacetime, so its mass distorts the ratio as you describe above for black holes, but it's a very small distortion. In euclidean spacetime, tensors are effectively multipliers of one.

 

[timf1234]

Mathematical Pi is indeed an irrational number but the real pi that is the ratio of diameter and circumference of a real circle, a circle that can really exist in the natural very real world, the real universe we live in is not an irrational number.

This is my own conclusion that I arrived to 20 years ago and never faltered about it.

 

[drkitten]

Mathematical Pi is indeed an irrational number but the real pi that is the ratio of diameter and circumference of a real circle, a circle that can really exist in the natural very real world, the real universe we live in is not an irrational number.

This is my own conclusion that I arrived to 20 years ago and never faltered about it.

It may indeed be your own conclusion, but it's not a rational one.

 

[timf1234]

Drkitten,

Bring your best possible real circle, however large and I will prove to you the ratio of its circumference and diameter is a rational number.

Proof:
1. Space quantization
2. Universe is finite

 

[drkitten]

Proof:
1. Space quantization
2. Universe is finite

Neither of which are proven, and in fact, are believed to be false by most physicists.

So, basically, what you've said, is that you've made up some **** in the teeth of the best physical theories we've got, and then drawn conclusions based on your own beliefs.

I told you that your conclusion was irrational....

 

[timf1234]

Neither of which are proven, and in fact, are believed to be false by most physicists.

So, basically, what you've said, is that you've made up some **** in the teeth of the best physical theories we've got, and then drawn conclusions based on your own beliefs.

I told you that your conclusion was irrational....

Drkitten worte:

...believed to be false by most physicists.....

I disagree.

Most physicists agree that the universe is finite. It has finite mass, finite space, finite time, and what have you, everything is finite.

Most physicist agree that quantum mechanics is correct to discribe that space is quantized. Most accurate value as of today is pretty close to 10 exp -35 meter.

Another, simpler way to look at it this way:
Although when looking at universe macroscopically it gives us an illusion of analog univers but when look deeper, microscopically then we realize that the universe is digital. Everything is digital. Therefore, length of a circumference and diaeter of a circle can be expressed in integer. Therefore, it is a rational number.

 

[sol invictus]

Drkitten,

Bring your best possible real circle, however large and I will prove to you the ratio of its circumference and diameter is a rational number.

Proof:
1. Space quantization
2. Universe is finite

Nonsense. Even if space is quantized, that ratio can be irrational (in so far as it makes any sense at all). And I have no idea why you brought up the size of the unverse.

Most physicists agree that the universe is finite.

Wrong. Ask them and see, or just read a paper on cosmology.

It has finite mass, finite space, finite time, and what have you, everything is finite.

Wrong. See above.

Most physicist agree that quantum mechanics is correct to discribe that space is quantized.

Wrong. QM does not quantize space. The Schrodinger equation is a differential equation, not a difference equation.

Most accurate value as of today is pretty close to 10 exp -35 meter.

You're talking about the Planck length, which means you have no idea what you're talking about.

Another, simpler way to look at it this way:
Although when looking at universe macroscopically it gives us an illusion of analog univers but when look deeper, microscopically then we realize that the universe is digital.

Wrong.

Everything is digital. Therefore, length of a circumference and diaeter of a circle can be expressed in integer. Therefore, it is a rational number.

That doesn't follow. How do you know space is quantized in rational units? In QM, the energy levels of a quartic oscillator (or anything other than a harmonic oscillator) are not rational multiples of each other. And what do you mean by a circle, anyway, if space is quantized?

 

[timf1234]

Summary:
No real thing in the real universe that we live in is analog. It is all digital when looked deeper. All measurements are integer multiple of some small magnitude. From amount of water, to time, to length, to mass, to charge, to volume, everything is digital.

Bring a circle, triangle, or anything as you chose to define them for your advantage and I will show you it can be expressed as a ratio of two integers. The only constrain I am imposing is that things has to exist in the real world as constrained by the laws of nature.

I already have acknowledged that a mathematical pi is an irrational number.
Mathematics is blind to the limits/constraint imposed by physics/laws of nature.

In principle we can do all of our computation, design, building of worldly thing using summation -don't need no integration (calculus).

Welcome to the real digital world.

 

[sol invictus]

Summary:

nonsense
wrong
wrong
wrong
don't know what you're talking about
wrong
doesn't follow

 

[Molinaro]

Summary:
No real thing in the real universe that we live in is analog. It is all digital when looked deeper. All measurements are integer multiple of some small magnitude. From amount of water, to time, to length, to mass, to charge, to volume, everything is digital.

Bring a circle, triangle, or anything as you chose to define them for your advantage and I will show you it can be expressed as a ratio of two integers. The only constrain I am imposing is that things has to exist in the real world as constrained by the laws of nature.

I already have acknowledged that a mathematical pi is an irrational number.
Mathematics is blind to the limits/constraint imposed by physics/laws of nature.

In principle we can do all of our computation, design, building of worldly thing using summation -don't need no integration (calculus).

Welcome to the real digital world.

That's pretty easy to counter.

If the circle in question is made up of irregular little pieces preventing the ratio from being the true infinite pi, then it's not a circle.

So, in summary, your objection seems to be that you can't get the real pi as long as the thing you are calling a circle isn't realy a circle.

And to that I say, so freaking what captain obvious.

 

[Perpetual Student]

Mathematical Pi is indeed an irrational number but the real pi that is the ratio of diameter and circumference of a real circle, a circle that can really exist in the natural very real world, the real universe we live in is not an irrational number.

This is my own conclusion that I arrived to 20 years ago and never faltered about it.

If space were indeed quantized (say, into interlocking Planck length triangles), then a "real circle" (as you define it) could only approximate pi. Pi remains a mathematical object representing the ratio of two lengths that are, by definition, continuous. There is no such thing as "the real pi" since the ratio of diameter to circumference would obviously change depending on the size of the "real circle." Tiny circles would not have the same ratio as big circles. This can be easily demonstrated by making the Planck length some macroscopic length and doing the math for two circles. Of course, as mentioned above, we would have the additional problem that, if space were quantized, our circles would not be very circular.

 

[T'ai Chi]

Here's my pi page: http://www.statisticool.com/pi.htm

Calculating pi based on throwing darts randomly.

 

[Beerina]

Here's my pi page: http://www.statisticool.com/pi.htm

Calculating pi based on throwing darts randomly.

Interesting. You should easily be able to do runs into the trillions today.

In fact, you do enough runs, you should start to see un-eliminatable differences between your value and true pi, things possibly related to:

• Inaccuracy of floating point representations
• 1-off issues at boundary conditions
• Granularity of the number of bits in the random numbers generated
• Issues in the random generator itself
• Things nobody has thought of, yet.

It might also be interesting to use 2 rand()'s for the x and the y, or 1 rand(), extracting an x and a y using different rules, such as:

• Every other 10's digit is an x, the others, y
• Every other bit is an x or a y
• Every other pair of bits, etc.
• Every other digit, base 6, etc.
• Any others you can think of.

Indeed, there may be* massive reality-check tests that random number generator specialists use to verify the random numbers and their sequences. There are probably* very long lists of "true" random number sequences, too, that you can download that pass all known tests, and can be used to compare with these pseudo-random number generation algorithms.




* Well, that's what I'd do, anyway.

 

[shadron]

Here's my pi page: http://www.statisticool.com/pi.htm

Calculating pi based on throwing darts randomly.

It's called a Monte Carlo method. Another such involves simulating dropping line segments of length n on a surface marked with two lines that are parallel and n distant from each other, and counting the number of which land on one of the lines as opposed to those that land completely between the lines. The ratio of the two approach some harmonic of pi as the number of segments gets large.