After spending dozens of hours watching Norman J Wildberger videos on mathematics and their history, one of course comes out with some questions.
One that comes to mind regards his section on the irrational, or as he says, meta-numbers, in which he describes how a man attempted to assume that somehow various digits corresponded to questions in the english language (or something like that). NJW then goes on to say something that essentially says an irrational has an infinite amount of information in it. He also goes on to say that perhaps these meta-numbers are a fundamentally different type of number than the rationals, almost like galaxies are different from stars although they appear similar in the night sky to the naked eye. THEN he did a proof about finding ‘9s’ inside of a metanumber if you dig down deep into the digits. Wow.
Let’s think about all this for a minute. An irrational, non-repeating decimal number like Pi, well, it actually contains an infinite amount of information in it. Many people have discussed the implications, here:
But what information? Which infinite amount? I wondered, let’s say that you assign ASCII values to every pair of digts…. will you eventually get the works of Shakespeare if you calculate Pi out far enough?
NJW also noted that there were some people who had calculated pi to a trillion digits. I then pondered, can you not just do a search and see what sorts of english words there might be in PI, assuming you interpret the digits as ASCII pairs?
But then I went further.
If Pi has an infinite amount of information, then, perhaps, doesn’t it eventually, you know, contain the irrational number e? (2.7….). Well, I don’t know… I mean, they are both infinite…how can one type of inifinity fit inside another? Certainly 0.3333 (repeating) is not found inside of Pi, because Pi is infinite and non repeating? I mean, for example, if you have the first million digits of pi, then you have ‘0.3333’ repeating for infinity, then can you have more digits after that? I mean, they are both infinity, right? Which infinity is bigger than the other?
This leads me to kind of agree with NJW’s approach, sort of, in that ‘infinite’ stuff is kind of hard to reason about. I mean, what happens if we just “dont go there” when thinking about things like Pi or e or sqrt(2)? Can we still do things with these ‘meta numbers’?
One thing NJW did that captured my imagination was that he said, well, if he is not going to “believe” in an infinite expansion of Pi, because you cannot do basic arithmetic with it in any kind of ordinary fashion, then at least, perhaps, he could believe in a number that is ‘n digits of Pi’, which he notated “Pi subscript N”. Pi sub n. That being basically the first ‘n’ digits of Pi, where n is a finite number! Not infinity. (I could be confused here, but I think thats the idea basically)
Ok. Then. So. Let’s do my question again.
Is there some n where Pi sub n will contain ‘0.33333’ repeating? no. Because 0.3333 is repeating and we are dealing with n being finite. OK!
Is there some n where Pi sub n will contain 0.3 sub m?
Ahh, now there is a question. What kind of repeating 0.3 do we find inside of pi? Is there some relationship between the digits of pi and the digits of repeating 0.3 inside of it?
This leads me to more questions. Can we write this?
Pi sub n = 3.141………. e sub m (same digits) ………… (nth digit of pi)
In other words, lets say you have n digits of pi. Will there be m digits of e inside of it? How many? Under what circumstances? And we might as well ask this too:
e sub x = 2.7………. Pi sub y (same digits) ………… (xth digit of e)
And what does it mean?
Technically when I say “the digits of e are inside the digits of pi” I am saying that there is an e divided by some power-of-ten inside of pi itself. In other words, at some point, you have
Pi sub n = 3.141….. plus e sub m / 10^q plus ……. nth digit of pi
And what is q? Well, in fact, it’s 1+ the number of digits that came before it, roughly.
So then you have to wonder, well, what if this is true for every meta-number? Take two meta-numbers, MNA and MNB. Is the following true, that there is an n and m such that …
MNA sub n = MNAsub1 . . . . . . . MNB sub m digits. . . . . . MNA nth digit
And if so, what are the values of m and n? What kind of relationship to m and n have?
I have no answers, not even a good conjecture, but I think it’s a fun question. . . .
The only thing that makes it more weird is the predilection we seem to have for base 10 digits.
I somehow think you’d find some more interesting patterns if you start using base 60, or base 2, or whatnot, don’t you think?
In fact, in NJW’s videos, he even gives a formula that calculates the hexadecimal digits of PI instead of the decimal digits. OK! If not hex, why not, you know, the base 65535 system, which we can say uses a single Unicode character for each digit? A formula for that ought to converge pretty quick dont you think?