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# 11.0010010000111111011010101000100010000101101000110000100011010011

For fun, I decided to try running a test on the constant we call Pi in binary form [note headline].  Pi is the ratio of a circle’s circumference to its diameter.  It is many more things as well.  It is a unique number in mathematics.  As Linus said to Charlie Brown after meeting the kid named “Five,” “How about the name 3.14159?”  Charlie Brown says, “I think there are a lot of kids who would be named 3.14159.” (From memory, I could have botched it.)

I found on the web the first 2^15th power (32,768) binary digits for the “fractional” part of Pi. In decimal terms, it means Pi to a little more than 10,000 decimal places.

Pi is an irrational number.  That means it can’t be expressed as a fraction of two integers.  As such, in binary form, since the series does not terminate, the pattern of ones and zeroes should be random.  As such, we can do a “runs test” to see whether the number of runs is abnormal.  Too few runs: zeroes and ones alternate too frequently.  Too many runs: zeroes and ones do not alternate enough.

My expectation was that neither abnormality would occur.  But I had to follow the data to the conclusion.  As it the first 32,768 digits of Pi, it had too many runs, such that the probability of it being random was 1.26%.

I don’t know what to do with this, but my next experiment will be on the number e, 2.71828…

I’m good with math, but not great with it.  Advice is welcome…

### 4 Responses to 11.0010010000111111011010101000100010000101101000110000100011010011

1. izimbra says:

A guy name David Bailey does research on this topic and has a lot of papers you can look at:

This Mathworld article on “Normal Numberss” references one of his papers from 1988:
http://mathworld.wolfram.com/NormalNumber.html

If this is the test you are applying – http://en.wikipedia.org/wiki/Wald%E2%80%93Wolfowitz_runs_test – you could post the numbers you obtained to see whether the problem is with you sequence of applying the test.

2. mwilbert says:

I’m afraid I am not able to figure out why you got the result you did, but I ran a Wald-Wolfowitz test in R (using the “wawotest” function) and 32770 bits of pi from http://www.befria.nu/elias/pi/binpi.html

I got a p value of .294, which is lower than I would have expected, but less disturbing.

3. Steve Milos says:

Hi David,

As his superior says to Lawrence (Peter O’ Toole) in Lawrence of Arabia: “it is recognized that you have a funny sense of fun.”

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