On the Randomness of Pi's Digits

Much has been written about the randomness of the \(\pi\)'s digits but usually in the context of the decimal number system. Looking at \(\pi\) in terms of the binary number system however, introduces a simplification that can be exploited to investigate its randomness.

The first ten decimal digits of \(\pi\) are 3.141592654 which is represented in binary as:

11.00100100001111110110101010001000100

SageMathCell enables us to examine the first one million decimal digits of \(\pi\), convert these to binary and then graph the cumulative balance between the number of 0's and 1's. The first two 1's are ignored and only the 0's and 1's following the decimal point are considered. Figure 1 shows the result:

Figure 1

Figure 2 shows the code that I used to generate this graph:

Figure 2: permalink

How does this graph compare to that of a sequence of randomly generated 0's and 1's? It's easy enough to undertake this comparison. To this end, I looked at a string of three million randomly generated 0's and 1's because this is approximately the number of binary digits resulting from the conversion of one million decimal digits. The result is shown in Figure 3. Of course, successive repetitions of this algorithm will produce different graphs whereas the graph for \(\pi\) will be always the same. However, it's the general form of the two graphs that I want to compare.

Figure 3: permalink

Figure 4 shows the code that was used to produce this result.

Figure 4: permalink

Not surprisingly, the form of the two graphs is very similar. The point of the exercise is that the conversion of \(\pi\) to binary digits provides an easy visual comparison of its digits with randomly generated 0's and 1's.

ADDENDUM: March 17th 2021

There are of course many interesting facts concerning \(\pi\). Here are some of them:

• The number pi is literally infinitely long. But the number 123456 doesn’t appear anywhere in the first million digits of \( \pi \). It is a bit shocking because if a million digits of \( \pi \) don’t have the sequence 124356, it definitely is the most unique number. Source.
  
  
• Did you know there is actually an entire language based entirely on \( \pi\)? Called Pilish, the numbers of letters in successive words match the digits in \(\pi\). Mike Keith, a devoted Pilish-lover, even wrote an entire book in Pilish called "Not a Wake." The rules of the language and its variants are described here.
  
  
• Rivers bend to \( \pi\): The way a river meanders is described by its sinuosity; the length of its winding path divided by the distance from the source to the ocean as measured in a straight line. Strange as it may be, the average river has a sinuosity of around 3.14, according to the journal Science. Source.

If you hold a mirror to a circle, it looks like a circle. If you hold a mirror up to 3.14, it spells PIE! See Figure 2.

Figure 2: source

Someone has developed a search engine that can be used to find digit sequences in \( \pi \) in the first \(2 \times 10^0 \) digits of \pi. Figure 3 shows the output if 123456789 is input.

Figure 3

Pressing the next link bring brings up what is shown in Figure 4:

Figure 4

      

The site can also be used to find positions of e, \(\sqrt{2} \) and \(\phi\) where SageMath for example struggles with one million decimal places.

Here is a GeoGebra link that will allows the user to use a slider to display decimal places of \(\pi\).