Is \(\pi\) normal? An open question...
An irrational number is said to be normal in a certain base if any finite pattern occurs with an expected frequency, whatever the expansion of digits on that basis.
This means it should be as easy (or difficult, all depending on the number of decimal digits that we consider for \(\pi\)) to find the sequence \(00000000\), as the sequence \(87654321\) or even \(12121212\), or any other with the same length.
In particular, in base \(10\) and for an expansion with \(n\) digits, any number \(\left\{ 0,1,2,3,4,5,6,7,8,9\right\} \) should occur "approximately" \(\frac{n}{10}\) times. Any pair of digits \(\left\{ 00,01,...,10,11,...,99\right\} \) should occur "approximately" \(\frac{n}{100}\) times, etc
It is unknown whether \(\pi\) is normal or not.
However, the digits in \(\pi\) are very uniformly distributed in the decimal expansions currently available, as can be seen by direct inspection.
The following table presents the results of the distribution of \(\left\{ 0,1,2,3,4,5,6,7,8,9\right\} \) in the first digits of \(\pi-3\).
\(1\times10^{5}\) | \(1\times10^{6}\) | \(6\times10^{9}\) | \(5\times10^{10}\) | \(2\times10^{11}\) | |
\(0\) | \(9\, 999\) | \(99\, 959\) | \(599\, 963\, 005\) | \(5\, 000\, 012\, 647\) | \(20\, 000\, 030\, 841\) |
\(1\) | \(10\, 137\) | \(99\, 758\) | \(600\, 033\, 260\) | \(4\, 999\, 986\, 263\) | \(19\, 999\, 914\, 711\) |
\(2\) | \(9\, 908\) | \(100\, 026\) | \(599\, 999\, 169\) | \(5\, 000\, 020\, 237\) | \(20\, 000\, 136\, 978\) |
\(3\) | \(10\, 025\) | \(100\, 229\) | \(600\, 000\, 243\) | \(4\, 999\, 914\, 405\) | \(20\, 000 \,069\, 393\) |
\(4\) | \(9\, 971\) | \(100\, 230\) | \(599\, 957\, 439\) | \(5\, 000\, 023\, 598\) | \(19\, 999\, 921\, 691\) |
\(5\) | \(10\, 026\) | \(100\, 359\) | \(600\, 017\, 176\) | \(4\, 999\, 991\, 499\) | \(19\, 999\, 917\, 053\) |
\(6\) | \(10\, 029\) | \(99\, 548\) | \(600\, 016\, 588\) | \(4\, 999\, 928\, 368\) | \(19 \,999\, 881\, 515\) |
\(7\) | \(10\, 025\) | \(99\, 800\) | \(600\, 009\, 044\) | \(5\, 000\, 014\, 860\) | \(19\, 999\, 967\, 594\) |
\(8\) | \(9\, 978\) | \(99\, 985\) | \(599\, 987\, 038\) | \(5 \,000 \,117 \,637\) | \(20 \,000 \,291 \,044\) |
\(9\) | \(9 \,902\) | \(100 \,106\) | \(600 \,017 \,038\) | \(4 \,999 \,990 \,486\) | \(19 \,999 \,869 \,180\) |
With \(2 \,147 \,483 \,000\) significant digits for the the value of \(\pi\), a number with \(8\) digits, for example a date \(DDMMAAAA\), should occur approximately \[C(n)=\frac{21.5\times10^{8}}{10^{8}}\approx22\mbox{ times.}\]
For a phone number, it should be \[C(n)=\frac{2.15\times10^{9}}{10^{9}}\approx2,\]
For a sequence with \(10\) digits, \[C(n)=\frac{0.215\times10^{10}}{10^{10}}\approx0.2,\]that is, it would be convenient to have an expansion \(5\) times larger to have a reasonable probability of finding any number of \(10\) digits.
The table below appears to confirm our suspicions
Sequence | Position of the first occurrence in \(\pi-3\) |
\(0123456789\) | \(17 \,387 \,594 \,880\) |
\(9876543210\) | \(21 \,981 \,157 \,633\) |
To test this behavior interactively, follow to
this page.
The normality of the representation of \(\pi\) in base 27 is addressed in this page.