Are the Digits of Pi Random?

Post Reply New Topic

Japanese researchers just calculated Pi to 2.5 trillion places. Their work sheds more light on a deep question in higher maths, particularly number theory.

Are the digits of Pi random?

Maybe I'm one of the few here who can spend hours pondering such questions. But this question extends to all the "natural" transcendental numbers, such as e, the square root of 2, the Golden Mean, et al.

Most mathematicians believe that the digits in these numbers are random, where random means that each digit from "1" through "10" will occur 1/10 of the time (equal probability), and even better, the numbers are absolutely random, meaning that the randomness will occur regardless of what base you represent the number in (limited to base 2 or higher).

Thoughts anyone?

They most probably are random, and if they're not, there are lots of possible relationships that can occur between digits.

^^ I believe I find this very interesting Aoi, thank you very much for posting this happy topic. I am sorry if this is very silly of me, however I believe I am not quite certain if this is random or not random (as defined in this post). ^^ I believe I may see arguments concerning both answers.

Firstly, Pi is a number and therefore is fixed. ^^ However, this may also be stated for all events for which probabilities are provided.

^^ Aoi, may I please ask, in the definition, I believe the probabilites for each digit may be different from 1/10 yet still be random.

For instance, consider a weighted die. When you throw the die, the outcome is randomly decided, yet probability for each number of not equal.

^^ Oh dear, I am very sorry as I have blessed myself with confusion.

Allow me to clarify:

Here, I'm simply asking if each of the 10 digits '1' through '10' occurs equally as often in the transcendental number Pi. In other words, does Pi have as many 1s as it does 2s, 3s, ..., 0s? The question can be asked of single digits, or two digits (i.e.: '01' through '99') or as many digits as you like, since the number never ends, you can choose digits of any length.

So the idea of a die that Tomasu mentions works well. Is the die for Pi completely fair or is it weighted in some at present unknown way?

Pi is already so mysterious that proving that its digits are uniformly distributed (or not) would be a major accomplishment.

Assuming that pi really is transcendental, the digits have to be uniformly distributed or you will fall into a pattern which you will not break out of.

Which, I guess, is possible. But I doubt it.

The question in my mind is.. what happens when pi is refined past the planck length?

http://en.wikipedia.org/wiki/Planck_length

Then we can call it exact pi and start using those big fancy computers for something else. Like playing Doom.

I think I remember reading that the first 16 digits of pi is sufficient for any physical calculations we would ever care to do.

Using the Kolomogoroff defintion of randomness, the digit sequence of pi is not random at all. According to Kolomogoroff and digit sequence is random if and only if the rule necessary to produce the sequence was linearly proportional to the length of the sequence itself. By this definition, pi is not random at all. In general any sequence that can be generated to an arbitrary number of elements whose rule can be finitely stated is not random, even if the distribution of single digits, pairs of single digits etc. were equally frequent. The sequence of digits of pi might be normal, but it is not random.

ruveyn

Forty digits will do it for sure. That gets well below Planck Length. There is no circular measurement that is physically doable that will yield resolution beyond thirty five to forty digits of pi.

ruveyn

if it wasn't random, wouldn't that imply a pattern? -asked 'he who's not good at math'...

I know 120-130 digits of pi. So I basically already know exact pi, for all practical purposes.

What is the application of calculating more digits of pi? Just to say we can?

Thats what I am wondering.

I guess mathematicians are looking for patterns and stuff, but if thats all well below Planck..

pi is related to other irrational numbers, so "changing" pi would require changing other constants. In other words, cosmically speaking, it could vary only a tiny amount before requiring a completely different set of physical laws. Since it is bounded (at least to some degree), it can't be random. Also, since it is possible to compute the nth digit of pi (without first computing the prior n-1 digits), it presumably must not be random.

I think that's the real question... Given ruveyn's and my explantions, couldn't the "pattern" be the entire length of pi itself? Is there an upper limit to what pattern must be present to denote randomness?
Does it ever end? Are there patterns within it (i.e. does it ever start over, or just start spitting out 1's (see Carl Sagan's Contact) )? In a very real sense, we don't understand pi and other irrational numbers, or else this thread wouldn't have been posted. Granted, computing trillions of digits probably doesn't help much (did anyone actually look at the 2.5 trillion digits to check on above possibilities?)

Last edited by roguetech on 22 Aug 2009, 2:16 am, edited 1 time in total.

If you call an arm a leg, how many legs do you have?

Answer: You have two. Calling an arm a leg does not make an arm a leg.

ruveyn

Pi never ends. It's not just irrational, it's transcendental. A transcendental number is non-algebraic, meaning you cannot calculate it using the standard operations in arithmetic or algebra (roots and squares). Pi's transcendental nature was proven in 1882 by Ferdinand Lindemann. This doesn't stop people from trying to show pi does end, but it doesn't.

As for the point of calculating pi to 2.5 trillion decimal places, doing so has value for testing computers and algorithms. And yes, the teams doing this work do check and analyze their results. A quick Google search will result in lots of pages summarizing those results, including a single digit repeating many times, or whatever other pattern you want. The number is infinitely long, and whether or not the occurrence of the digits are uniformly distributed, somewhere in pi is any series of numbers you want.

Is it practical enough to get a date with?

I thought No. But I'm a maths noob. Considering my simple brain, I am probably wrong. Therefore, you should go with Yes.