# Is Pi Normal?

A Simply Normal Number has all its digits spread out as if each one is chosen by a throw of a dice.

A dice can have 10 sides

And we get something like this:

Over the infinite sequence of decimal digits:

• 110 are 0, 110 are 1, 110 are 2, etc (this makes it "simply normal")

But to be "normal" (without the "simply" in front) we also need:

• 1100 are 00, 1100 are 01, 1100 are 02, etc
• 11000 are 000, 11000 are 001, 11000 are 002, etc
• and so on, for every group of digits

Is this the case for Pi ?

To investigate let us count how many of each single digit there are, and see if they appear more or less often than expected.

We also need to do this for 2-digit pairs, 3-digit groups, etc.

Let us have a go!

## A Hundred Digits

Here are the first 100 digits of Pi:

 314159265358979323846264338327950288 4197169399375105820974944592307816 406286208998628034825342117067

With 100 digits and ten possibilities (0,1,2,3,4,5,6,7,8,9) we should get 10 of each digit, right?

Well I count 8 "0"s, does that seem right?

Expecting a perfect 10 of each digit is unrealistic of course, so maybe just 8 is fine.

Here is the count for each digit:

Digit Count
0: 8
1: 8
2: 12
3: 12
4: 10
5: 8
6: 9
7: 8
8: 12
9: 13

They are mostly around 10, but wow: "9" occurred 13 times!

Should that be expected? Let us count the counts:

• 4 digits (0,1,5 and 7) occur 8 times,
• 1 digit (6) occurs 9 times,
• 1 digit (4) occurs 10 times,
• 3 digits (2,3 and 8) occur 12 times and
• 1 digits (9) occur 13 times

As a graph:

They are kind of near 10, but also a little spread out ... is this "normal"?

Let us try two-digit pairs:

Digits Count
00: 0
01: 0
02: 1
03: 1
04: 0
05: 1
06: 2
... ...
98: 1
99: 2

Mostly 0's and 1's. We need more raw data!

## A Thousand Digits

Using the first thousand digits of Pi gets us this:

Digit Count
0: 93
1: 116
2: 103
3: 103
4: 93
5: 97
6: 94
7: 95
8: 101
9: 105

Counting the counts:

Zoomed out:

It now looks more grouped around 100, our first hint that Pi might be "normal".

Let's look at two-digit pairs:

Digits Count
00: 7
01: 10
02: 10
03: 9
04: 4
05: 14
06: 8
07: 6
... ...
93: 14
94: 11
95: 16
96: 9
97: 7
98: 8
99: 11

Good, we are getting values around 10, but let us see the count of counts:

Wow, it looks nicely clustered around 10, maybe a little skewed.

Let's go for more data!

## Ten Thousand Digits

Using the first ten thousand digits of Pi gets us this:

Digit Count
0: 968
1: 1026
2: 1021
3: 975
4: 1012
5: 1046
6: 1021
7: 970
8: 947
9: 1014

And the count of counts:

And for two-digit pairs:
Digits Count
00: 85
01: 103
02: 98
03: 103
04: 98
05: 89
06: 101
07: 93
... ...
94: 98
95: 107
96: 106
97: 88
98: 100
99: 91

And as a count of counts:

It is starting to look like a normal distribution isn't it?

## Hundred Thousand Digits

Using the first hundred thousand digits of Pi gets us this:

Digit Count
0: 9999
1: 10137
2: 9908
3: 10026
4: 9971
5: 10026
6: 10028
7: 10025
8: 9978
9: 9902

The count of counts is:

And for two-digit pairs:
Digits Count
00: 998
01: 1027
02: 962
03: 993
04: 968
05: 1007
06: 1009
07: 1017
08: 1001
... ...
94: 1033
95: 1014
96: 952
97: 934
98: 997
99: 968

And the count of counts:

The numbers are all around 1000, and, using the standard deviation calculator gets a standard deviation of only 32.3, which is a reasonably tight grouping, don't you think?

## Summary

Just looking at single digits:

Digit 100 1000 10,000 100,000
expect: 10 100 1000 10000
0: 8 93 968 9999
1: 8 116 1026 10137
2: 12 103 1021 9908
3: 12 103 975 10026
4: 10 93 1012 9971
5: 8 97 1046 10026
6: 9 94 1021 10028
7: 8 95 970 10025
8: 12 101 947 9978
9: 13 105 1014 9902

Just by eye we can see that the more digits we use the more the counts tend to group around the expected value.

This is called the "Law of Large Numbers": in the long run random events tend to average out at the expected value.

We see the same thing with two-digit pairs.

So this increases our confidence that Pi is a normal number.

## Conclusion

So it seems that the single digits, and double digit pairs, occur no more frequently than random chance.  At least within the first 100,000 digits.

But we need to check more digits and also 3-digit, 4-digit etc groups. A very big job.

## Research

People with access to powerful computers have checked trillions of digits of Pi:

(external link) Digit Statistics of the First 22.4 Trillion Decimal Digits of Pi

and have also found no reason to doubt that Pi is a Normal Number. Not a proof, just a very strong suspicion that Pi is normal.

## Why is this Important?

Firstly, isn't it curious that the digits of Pi (which form an exact sequence) nonetheless seem to be randomly distributed?

But the other mind-boggling implication is that an infinite number of randomly occuring digits will contain every finite sequence of digits, including: