Coming from a post on theydidthemath talking about Pi, all the top answers begin by explaining that **if** we consider Pi to be a “normal number”, then etc. They also mention we haven’t been able to prove whether it’s a normal number or not yet. What’s that about?
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Normal just means that every digit appears roughly the same fraction of time as all the others. For example, if pi is normal in base 10, that would mean that in the infinite decimal expansion there would be approximately the same amount of 1s as 2s, 3s, etc. And that every digit from 0 to 9 would appear about 10% of the time. It would be *not normal* if, for example, there were twice as many 3s as 7s or something like that.
We can assume based on expansions that we’ve calculated (they’ve calculated pi out of tens of millions of digits or more) that a number is normal, but it’s very hard to *prove*. Like maybe after the trillionth digit all of a sudden there aren’t any more 7s. We can assume that it won’t happen, but no one has been able to prove it yet.
A normal number is comprised of the same amount of all the digits. So like the decimal representation of ⅓ isn’t normal because it’s just comprised of 3. What they are saying is if they consider the digits of pi to have a uniform distribution of all digits 0-9, then something follows from that.
A normal number is a number with infinite digits that doesn’t repeat, and those infinite digits have a completely random distribution.
So far, all the digits of Pi we’ve seen have a completely random distribution.
So far, as far as we can tell, it is a Normal number since it has a random distribution.
But Math is much more rigorous than even science.
Nobody has been able to disprove, for example, that after the 10^100000000000000^1000000000 th digit, the number 2 stops appearing, or something like that to make the distribution of digits stop being random.
There are plenty of numbers with infinite digits that don’t repeat that aren’t randomly distributed. The first number we proved was infinite and non-repeating had only the digits 1 and 0.
We don’t know one way or the other about pi.
A normal number is one that contains all numbers in its decimal representation. The simplest normal number would be 0.123456789101112…
There is a belief that Pi is normal, though we aren’t able to prove it, which is where the “the works of Shakespeare are in pi” idea comes from.
Irrational numbers go on forever without repeating, but not all irrational numbers are normal, for example 0.13456789101113… will go on forever, but you will never find a 2
Edit: it’s not enough for all numbers (sequences of digits) to exist in the number, they need to have the same probability. My example happened to fit that but another example like 0.010203040506070809010011… would contain all numbers but not in equal probability, there are too many 0s, so it wouldn’t be normal. There are also complications with different bases but I’m not knowledgeable enough to comment on that, so my answer is only accurate in base 10
A normal number is one where if you took the frequency of each digit occurring, it would form a normal distribution (a bell curve). Basically meaning that it would appear similar to if you rolled a fair die with the same number of faces as the base of the number, you would get a similar distribution. In the case of Arabic numerals that we use, that would be base 10, so a 10 sided die. Note this isn’t saying you would get pi by rolling a 10 sided die over and over, just that the distribution of the frequency of the numbers would be similar, i.e., a normal distribution.
We can’t prove pi is normal or not, because it is infinite, with no repeating pattern. Tests have been done showing that if you calculated pi to hundreds of thousands of digits, it is normal. But pi has only been calculated to trillions of digits (with no repeatable pattern detected), so maybe if it was calculated to trillions of trillions of trillions of digits, it wouldn’t be anymore so it can’t be proven.
For any practical application, you could consider pi as a normal number. But math doesn’t rely on practical applications, it only cares about what can absolutely be proved or disproved, so mathematically, it isn’t determined if pi is normal or not.
A lot of math questions sound very simple when you ask them, but are very hard to prove.
Whether pi is a normal number depends on how many of each digit 0 to 9 you get when you expand it as a decimal. That’s a very simple concept. Except that pi isn’t defined by its decimal expansion, it’s defined as a quantity relating to circles.
Knowing how big pi is doesn’t tell you about all the decimal places, because the farther along you go the less the decimal digits affect how big it is. Calculating any huge number of decimal places doesn’t prove anything because you still don’t know what happens after that. We do know that it doesn’t end in all zeros or any other repeating sequence, but that’s just a consequence of it not being a fraction, and it doesn’t help us make any positive statements about what the digits are.
It’s not even clear how you would go about saying anything meaningful about which digits appear how often in pi. Maybe if you were extremely lucky you could discover one of those infinite sums for pi where the nth term is some coefficient times 10^n or something. I believe there is one for 16^n but even then, if the coefficients are greater than 16 it’s still not strictly the properties of each digit.
What’s really interesting about normality, is that it is known that almost all numbers are normal. Which means that if you throw a dart at random on the interval [0,1] , it would fall, with probability 1, on a normal number.
Not being normal is rare, because to have a non-normal number, you need to have some kind of pattern in its expansion in some base, and patterns do not occur when you pick a number at random.
We believe that most irrational constant that we know (pi, e, sqrt(2), cos(1), solutions of interger polynomial equation, etc …) are normal, because we see no reason to have a pattern in their expansion and computer experiments suggest that there are none. But so far, we never could prove that any of these number are really normal, even in some specific base.
The only numbers that we know to be normal in some base are constructed specifically in the purpose of having that property. For example the Champernowne constant 0,123456789101112131415161718192021222324…. is normal in base 10. But we don’t even know if it’s normal in any other base …
But what’s even more striking is that we have absolutely no explicit example of a number that would be normal (in any base), even if we know that most of them are. The only “implicit” examples that I know of specific normal numbers are [Chaitin constant](https://en.wikipedia.org/wiki/Chaitin%27s_constant) which are not computable …
Follow up question: do we have any examples of infinitely long numbers which aren’t normal?
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