Is Pi Normal?

Open Question

It is not known whether $\pi$ (pi) is normal.


Historical Note

The question as to the normality or not of $\pi$ (pi) arose when Augustus De Morgan investigated the $707$-digit approximation to $\pi$ that had been published by William Shanks in $1853$.

De Morgan noticed that the digit $7$ occurred less frequently than the other digits: only $44$ times against the expected $61$.

He determined that the probability of this low frequency was $45$ to $1$ against.

However, as discovered by D.F. Ferguson in around $1945$, Shanks had made a mistake in the $528$th digit. The correct calculation did not show that bias against $7$.


Since that time, all tests for normality that have been done on all known digits of $\pi$ have shown no suggestion that $\pi$ is not normal.

Work continues.


Sources

  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $0 \cdotp 12345 67891 01112 13141 51617 18192 02122 \ldots$
  • 1986: David Wells: Curious and Interesting Numbers ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41972 \ldots$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $0 \cdotp 12345 \, 67891 \, 01112 \, 13141 \, 51617 \, 18192 \, 02122 \ldots$
  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $3 \cdotp 14159 \, 26535 \, 89793 \, 23846 \, 26433 \, 83279 \, 50288 \, 41971 \ldots$
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