system
Given the following post, respond as if you're commenting on Reddit:
human
Will we ever know if Pi is a "normal" or "non-normal" irrational number? Is it within the realm of possibility that we will ever find out?

According to what I've heard, a "normal" irrational number is one whose digits contain every conceivable combination of numbers (including, for example, the social security numbers of everyone who has ever had an SSID in alphabetical order), while a "non-normal" irrational number is one whose digits are infinite but do not contain every possible combination of numbers. 

So I guess I have two questions: 

1) Will we ever know which of these categories Pi falls under

and 

2) How is it possible to have an infinite number of non-repeating numbers without any patterns that **doesn't eventually run through every possible numeral combination?**


Edit: thank you for all the awesome answers! I now totally understand question #2 and **would like to propose another question** for anyone who sees this after this edit: 

What is an example of a normal irrational number and **how can we prove that a number is normal?** (Is a normal number defined as any irrational number that contains all the digits of its respective base an infinite number of times...or...?)
gpt
Not only do normal numbers contain every possible finite combination of digits, they contain each possibility infinitely many times.

The way that this is possible is that you do it by combination-length. There are only 10^(N) different possible combinations of digits of length N. So in a normal number, you can expect there to be one copy of any particular sequence of N digits in every N\*10^(N) digits of a number. You can try to convince yourself that

* 0.1234567891011121314151617181920212223242526...

is normal because it does this for every integer N. 

We will almost surely prove that pi is normal or not sometime in the future. Though it is hard to prove things like this for numbers like pi, because it arises purely from geometry and so figuring out numerical things about it is hard. Generally, pi, as a number, is a very typical and boring. Almost every number is irrational, transcendental, normal etc, and the reason why we thought pi was these things is because pi should be like every other number. But accessing the arithmetic content of the number pi is hard to do, and so proving that pi is a boring ol' number is hard and interesting. Same goes for e and other numbers arising from other topics. Though, there are formulas for pi that are promising, as they can allow you to compute digits (which are part of the arithmetic part of pi) efficiently and directly. Not enough to prove that pi is normal, but it suggests that this information is not impossibly hard to obtain.