This Python project explores normal numbers (in particular, pi, e and sqrt(2)) in different bases (base 10, hexadecimal).

1. Normality of Pi
Radhika Prasad (181109)
Vinay Gusain (181168)
(MSc Physics 1st year)
18.04.2019

2. Normal Numbers
• “Each sequence of digits is equally probable”
• Formal definition: For a real number to be
normal, every sequence of k digits has to appear
with a limiting frequency of b-k (for any base b)
• Hypothesis: π, e, 2 etc. are normal; no existing
proof!
𝑬𝒎𝒊𝒍𝒆 𝑩𝒐𝒓𝒆𝒍 (1909)

3. Frequency of
Occurrence of
Sequence
Normalised standard deviation:
2.69*10-4
Normalised standard deviation:
9.49*10-4

4. Normalised standard deviation:
0.00301
Normalised standard deviation:
0.00988

5. Hexadecimal
π e
104
digits
scanned
1 million
digits
scanned
Single Digit
Mean: 0.0625
σ: 0.0024
σ: 0.0028
σ: 0.00019
σ: 0.00028

6. Frequency
Distribution

8. Before normalisation of mean to 1:-
Mean: 0.0001
Standard deviation: 0.0000032
After normalisation of mean to 1:-
Standard deviation: 0.032
Before normalisation of mean to 1:-
Mean: 0.001
Standard deviation: 0.0000104
After normalisation of mean to 1:-
Standard deviation: 0.0104
Gaussian Fitting For Frequency Distribution

9. Comparison of Frequency Distribution

11. Comparison of Standard Deviation

12. Standard Deviation vs Number of Digits Scanned

13. Comparison of Standard Deviation

14. Standard Deviation vs Sequence Length

15. Standard Deviation of π, e, 2

16. Conclusions
• 𝜎 a
1
𝑁
; 𝑁 is the number of digits scanned
• 𝜎 a 𝑒 𝐿
; 𝐿 is the length of the sequence
• Frequency distribution curve can be fitted to a Gaussian with standard
deviation 𝜎
• No regular pattern of 𝜎 for different numbers
• π, e, 2 are normal in base-10 and base-16

17. References
• https://introcs.cs.princeton.edu/java/data/
• https://archive.org/details/Math_Constants
• https://pi2e.ch/blog/
• https://en.wikipedia.org/wiki/Normal_number
• https://arxiv.org/ftp/arxiv/papers/1612/1612.00489.pdf

18. Thank You